{"id":2616,"date":"2023-01-10T22:16:10","date_gmt":"2023-01-10T21:16:10","guid":{"rendered":"https:\/\/ff.mhrooz.xyz\/?p=2616"},"modified":"2023-02-27T17:41:02","modified_gmt":"2023-02-27T16:41:02","slug":"information_theory_ding_li_zong_jie_ji_tui_dao","status":"publish","type":"post","link":"https:\/\/blog.mhrooz.xyz\/index.php\/2023\/01\/10\/information_theory_ding_li_zong_jie_ji_tui_dao\/","title":{"rendered":"Information Theory \u5b9a\u7406\u603b\u7ed3\u53ca\u63a8\u5bfc"},"content":{"rendered":"\n<!doctype html>\n<html>\n<head>\n<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>\n\n<link href='https:\/\/fonts.loli.net\/css?family=Open+Sans:400italic,700italic,700,400&#038;subset=latin,latin-ext' rel='stylesheet' type='text\/css' \/><style type='text\/css'>html {overflow-x: initial !important;}:root { --bg-color: #ffffff; --text-color: #333333; --select-text-bg-color: #B5D6FC; --select-text-font-color: auto; --monospace: \"Lucida 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class='typora-export-content'>\n<div id='write'  class=''><div class='md-toc' mdtype='toc'><p class=\"md-toc-content\" role=\"list\"><span role=\"listitem\" class=\"md-toc-item md-toc-h2\" data-ref=\"n2\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#chapter-1-information-theory-for-discrete-variables\">Chapter 1 Information Theory for Discrete Variables<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n3\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#1-message-sets\">(1) Message Sets<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n5\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#2-measuring-choice\">(2) Measuring Choice<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n11\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#3-entropy\">(3) Entropy<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n15\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-11\">Example 1.1<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n19\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-12\">Example 1.2 <\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n29\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#4-example-distributions\">(4) Example Distributions<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n30\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#5-conditional-entropy\">(5) Conditional Entropy <\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n43\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-13\">Example 1.3<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n50\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#6-joint-entropy\">(6) Joint Entropy <\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n56\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-14-chain-rule-for-entropy\">Example 1.4 Chain Rule for Entropy<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n58\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-15\">Example 1.5<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n62\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-16\">Example 1.6<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n68\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#7-information-divergence--kullback-leibler-distance\">(7) Information Divergence \/ Kullback-Leibler Distance<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n85\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-17\">Example 1.7<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n88\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-18\">Example 1.8<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n91\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-19\">Example 1.9<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n94\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-110-chain-rule-for-information-divergence\">Example 1.10 Chain Rule for Information Divergence<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n96\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-111-&#10;p&#10;1&#10;p&#10;2&#10;-instead-of-&#10;p&#10;x&#10;1&#10;x&#10;2\">Example 1.11 <span class=\"math-in-toc\"><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.88ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2157.1 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-258-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-258-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-258-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-258-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-258-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(1078.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-258-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-258-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mn>1<\/mn><\/msub><msub><mi>P<\/mi><mn>2<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_1P_2<\/script><\/span> instead of <span class=\"math-in-toc\"><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.686ex\" height=\"2.125ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2513.3 939.1\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.579ex;\"><defs><path id=\"MJX-259-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-259-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 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61V578Z\"><\/path><path id=\"MJX-259-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-259-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-259-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-259-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(1264.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-259-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-259-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><msub><mi>X<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{X_1X_2}<\/script><\/span><\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n99\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#8-mutual-information\">(8) Mutual Information<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n113\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-112\">Example 1.12<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n118\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-113-chain-rule-for-mutual-information\">Example 1.13 Chain Rule for Mutual Information<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n121\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-114\">Example 1.14<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n125\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#9-cross-entropy\">(9) Cross Entropy<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n133\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#10-inequalities\">(10) Inequalities<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n141\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-115\">Example 1.15<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n168\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#11-convexity\">(11) Convexity<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n189\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-116-jensens-inequality\">Example 1.16 Jensen&#8217;s Inequality<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n194\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-117-definitions-of-convexity\">Example 1.17 Definitions of Convexity<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h2\" data-ref=\"n200\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#chapter-2-information-theory-for-continuous-variabls\">Chapter 2 Information Theory for Continuous Variabls<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n201\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#1-differential-entropy\">(1) Differential Entropy<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n205\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-21\">Example 2.1<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n209\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#rules-for-differential-entropy\">Rules for Differential Entropy<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n214\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#2-mixed-distributions\">(2) Mixed Distributions<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n215\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#3-example-distributions\">(3) Example Distributions<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n216\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#4-informational-divergence\">(4) Informational Divergence<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n225\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#5-cross-entropy\">(5) Cross Entropy<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n231\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#6-maximum-entropy\">(6) Maximum Entropy<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n232\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#alphabet-or-volume-constraint\">Alphabet or Volume Constraint<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n236\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#first-moment-constriant\">First Moment Constriant<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n241\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#second-moment-constraint\">Second Moment Constraint<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n245\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-22\">Example 2.2<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h2\" data-ref=\"n248\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#chapter-3-coding-of-memoryless-sources\">Chapter 3 Coding of Memoryless Sources<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n253\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#1-kraft-inequality\">(1) Kraft Inequality <\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n274\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#2-entropy-bounds-on-source-coding\">(2) Entropy Bounds on Source Coding<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n304\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#3-huffman-codes\">(3) Huffman Codes<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n322\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#4-tunstall-codes\">(4) Tunstall Codes<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n347\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#5-huffman-codes-and-tunstall-codes\">(5) Huffman Codes and Tunstall Codes<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h2\" data-ref=\"n351\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#chapter-4-coding-of-stationary-sources\">Chapter 4 Coding of Stationary Sources<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n352\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#1-discrete-stationary-sources\">(1) Discrete Stationary Sources<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n376\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#2-entropy-bounds-on-source-coding-a-dss\">(2) Entropy Bounds on Source Coding a DSS<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n390\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#3-elias-code-for-positive-integers\">(3) Elias Code for Positive Integers<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n571\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#4-elias-willems-universal-source-coding\">(4) Elias-Willems Universal Source Coding<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n575\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-41\">Example 4.1<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h2\" data-ref=\"n591\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#chapter-5-channel-coding\">Chapter 5 Channel Coding<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n597\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#1-rate-reliability-and-cost\">(1) Rate, Reliability and Cost<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n599\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#2-memoryless-channels\">(2) Memoryless Channels<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n601\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#3-cost-functions\">(3) Cost Functions <\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n607\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-51-power-constraint\">Example 5.1 Power Constraint<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n609\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#4-block-and-bit-error-probability\">(4) Block and Bit Error Probability<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n646\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#5-random-coding\">(5) Random Coding<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n651\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#6-concavity-and-converse\">(6) Concavity and Converse<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n663\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#7-discrete-alphabet-examples\">(7) Discrete Alphabet Examples<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n664\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#1-binary-symmetric-channel\">&lt;1&gt; Binary Symmetric Channel<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n666\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#2-binary-erasure-channel\">&lt;2&gt; Binary Erasure Channel<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n669\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#3-strongly-symmetric-channels\">&lt;3&gt; Strongly Symmetric Channels<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n684\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#4-symmetric-channels\">&lt;4&gt; Symmetric Channels<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n690\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-52-bpsk-over-awgn\">Example 5.2 BPSK over AWGN<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h3\" data-ref=\"n694\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#8-continuous-alphabet-examples\">(8) Continuous Alphabet Examples<\/a><\/span><span role=\"listitem\" class=\"md-toc-item md-toc-h4\" data-ref=\"n704\"><a class=\"md-toc-inner\" style=\"cursor: pointer;\" href=\"#example-53-awgn-channel-with-bpsk\">Example 5.3 AWGN channel with BPSK<\/a><\/span><\/p><\/div><h2 id='chapter-1-information-theory-for-discrete-variables'><span>Chapter 1 Information Theory for Discrete Variables<\/span><\/h2><h3 id='1-message-sets'><span>(1) Message Sets<\/span><\/h3><p><span>What is a message? Shannon suggested that a message has to do with <\/span><em><span>choice<\/span><\/em><span> and <\/span><em><span>sets<\/span><\/em><span>. In his words:<\/span>\n<span>The significant aspect is that the actual message is one selected from a set of possible messages.<\/span><\/p><h3 id='2-measuring-choice'><span>(2) Measuring Choice<\/span><\/h3><p><span>Shannon was interested in defining a quantity that will measure, in some sense, how much<\/span>\n<span>information is \u201cproduced\u201d by choosing messages. He suggested that the logarithm of the number of elements in the message set can be regarded as a measure of the information produced when one message is chosen from the set, <\/span><em><span>all choices being equally likely<\/span><\/em><span>.<\/span><\/p><p><span>For example, consider a fair coin that takes on values in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"15.072ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6661.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-146-TEX-N-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 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data-c=\"1D451\" xlink:href=\"#MJX-146-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2903,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-146-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3372,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-146-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3816.7,0)\"><use data-c=\"1D447\" xlink:href=\"#MJX-146-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4520.7,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-146-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5049.7,0)\"><use data-c=\"1D456\" xlink:href=\"#MJX-146-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5394.7,0)\"><use data-c=\"1D459\" xlink:href=\"#MJX-146-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5692.7,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-146-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6161.7,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-146-TEX-N-7D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo fence=\"false\" stretchy=\"false\">{<\/mo><mi>H<\/mi><mi>e<\/mi><mi>a<\/mi><mi>d<\/mi><mi>s<\/mi><mo>,<\/mo><mi>T<\/mi><mi>a<\/mi><mi>i<\/mi><mi>l<\/mi><mi>s<\/mi><mo fence=\"false\" stretchy=\"false\">}<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\{Heads, Tails\\}<\/script><span> and suppose we flip the coin <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.357ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 600 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-479-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-479-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n<\/script><span> times. The string of coin flips takes on one of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.279ex\" height=\"1.528ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -675.5 1007.3 675.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-148-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-148-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mn\"><use data-c=\"32\" xlink:href=\"#MJX-148-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(533,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-148-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mn>2<\/mn><mi>n<\/mi><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">2^n<\/script><span> values, all equally likely, and the information produced is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"15.733ex\" height=\"2.116ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 6954 935.4\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.546ex;\"><defs><path id=\"MJX-149-TEX-N-6C\" d=\"M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z\"><\/path><path id=\"MJX-149-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-149-TEX-N-67\" d=\"M329 409Q373 453 429 453Q459 453 472 434T485 396Q485 382 476 371T449 360Q416 360 412 390Q410 404 415 411Q415 412 416 414V415Q388 412 363 393Q355 388 355 386Q355 385 359 381T368 369T379 351T388 325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z\"><\/path><path id=\"MJX-149-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-149-TEX-N-2061\" d=\"\"><\/path><path id=\"MJX-149-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-149-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-149-TEX-N-A0\" d=\"\"><\/path><path id=\"MJX-149-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path id=\"MJX-149-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-149-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-149-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-149-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-149-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-149-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-149-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-149-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1714.6,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-149-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1881.2,0)\"><g data-mml-node=\"msup\"><g data-mml-node=\"mn\"><use data-c=\"32\" xlink:href=\"#MJX-149-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(533,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-149-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3166.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-149-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4222,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-149-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(4822,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-149-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5072,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-149-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5501,0)\"><use data-c=\"1D456\" xlink:href=\"#MJX-149-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5846,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-149-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6207,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-149-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6676,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-149-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><msup><mn>2<\/mn><mi>n<\/mi><\/msup><\/mrow><mo>=<\/mo><mi>n<\/mi><mtext>&nbsp;<\/mtext><mi>b<\/mi><mi>i<\/mi><mi>t<\/mi><mi>s<\/mi><mo>.<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\"> \\log_2{2^n}=n \\ bits.<\/script><\/p><hr \/><p><span>For non-equally likely choices, Shannon developed a measure that has the same form as the entropy in <\/span><em><span>statistical mechanics<\/span><\/em><span>. For example, consider a biased coin that takes on the value Heads with probability <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.138ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 503 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-526-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-526-TEX-I-1D45D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>p<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">p<\/script><span> and Tails with probability <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.035ex\" height=\"1.946ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 2225.4 860\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-151-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-151-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-151-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-151-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(722.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-151-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1722.4,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-151-TEX-I-1D45D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">1-p<\/script><span>. The string of coin flips takes on <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.837ex\" height=\"3.013ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -849.5 1695.9 1331.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.091ex;\"><defs><path id=\"MJX-152-TEX-SO-28\" d=\"M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 56T152 251Z\"><\/path><path id=\"MJX-152-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-152-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-152-TEX-SO-29\" d=\"M305 251Q305 -145 69 -349H56Q43 -349 39 -347T35 -338Q37 -333 60 -307T108 -239T160 -136T204 27T221 250T204 473T160 636T108 740T60 807T35 839Q35 850 50 850H56H69Q197 743 256 566Q305 425 305 251Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mrow\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"OPEN\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"28\" xlink:href=\"#MJX-152-TEX-SO-28\"><\/use><\/g><\/g><g data-mml-node=\"mfrac\" transform=\"translate(458,0)\"><g data-mml-node=\"mi\" transform=\"translate(177.8,444) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-152-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(0,-345) scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-152-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(600,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-152-TEX-I-1D45D\"><\/use><\/g><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"CLOSE\" transform=\"translate(1237.9,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"29\" xlink:href=\"#MJX-152-TEX-SO-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>n<\/mi><mrow><mi>n<\/mi><mi>p<\/mi><\/mrow><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"1.2em\" maxsize=\"1.2em\">)<\/mo><\/mrow><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n\\choose np<\/script><span> values (since the expected number of Heads is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.495ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 1103 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-153-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-153-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 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xlink:href=\"#MJX-153-TEX-I-1D45D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mi>p<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">np<\/script><span>). Shannon proposed that the information produced when flipping the coin <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.357ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 600 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-479-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-479-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n<\/script><span> times is<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n10\" cid=\"n10\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 52.16ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" 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width=\"2425.4\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(19211.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(19600.9,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-1-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(19850.9,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-1-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(20279.9,0)\"><use data-c=\"1D456\" xlink:href=\"#MJX-1-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(20624.9,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-1-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(20985.9,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-1-TEX-I-1D460\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1449.5 1 2399\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>n<\/mi><mrow><mi>n<\/mi><mi>p<\/mi><\/mrow><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">)<\/mo><\/mrow><\/mrow><\/mrow><mo>\u2248<\/mo><mi>n<\/mi><mo stretchy=\"false\">(<\/mo><mi>p<\/mi><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><mn>1<\/mn><mi>p<\/mi><\/mfrac><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><mn>1<\/mn><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;<\/mtext><mi>b<\/mi><mi>i<\/mi><mi>t<\/mi><mi>s<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='3-entropy'><span>(3) Entropy<\/span><\/h3><blockquote><p><span>Let <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.636ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3375 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-155-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-155-TEX-I-1D462\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-155-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-155-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-155-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-155-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D460\" xlink:href=\"#MJX-155-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(469,0)\"><use data-c=\"1D462\" xlink:href=\"#MJX-155-TEX-I-1D462\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1041,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-155-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1544,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-155-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2047,0)\"><use data-c=\"28\" xlink:href=\"#MJX-155-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2436,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-155-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2986,0)\"><use data-c=\"29\" xlink:href=\"#MJX-155-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><mi>f<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">supp(f)<\/script><span> be the support set of the function <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.244ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 550 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-341-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-341-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f<\/script><span>, i.e., the set of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.197ex\" height=\"1.02ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -441 529 451\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-460-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-460-TEX-I-1D44E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a<\/script><span> such that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.35ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3690.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-158-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-158-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-158-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-158-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-158-TEX-N-3E\" d=\"M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z\"><\/path><path id=\"MJX-158-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-158-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(550,0)\"><use data-c=\"28\" xlink:href=\"#MJX-158-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(939,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-158-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1468,0)\"><use data-c=\"29\" xlink:href=\"#MJX-158-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2134.8,0)\"><use data-c=\"3E\" xlink:href=\"#MJX-158-TEX-N-3E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3190.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-158-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>&gt;<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f(a) > 0<\/script><span>. We define the <\/span><em><span>entropy or uncertainty<\/span><\/em><span> of the discrete random variable <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 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transform=\"translate(17992.5,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-2-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18270.5,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-2-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(19215.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-2-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-2-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-2-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-2-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(20929.7,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-2-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(21096.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-2-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-2-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(22423.8,0)\"><use data-c=\"28\" xlink:href=\"#MJX-2-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(22812.8,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-2-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(23664.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-2-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(24053.8,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-2-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1437.1 1 2374.1\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,487.1)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><mo>\u2212<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mo>\u2212<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><h4 id='example-11'><span>Example 1.1<\/span><\/h4><p><span>The entropy <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.697ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2518 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-291-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-291-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-291-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-291-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-291-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-291-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-291-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2129,0)\"><use data-c=\"29\" xlink:href=\"#MJX-291-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H(X)<\/script><span> depends only on the probability distribution <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.392ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2383.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-253-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-253-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-253-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-253-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-253-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-253-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-253-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1327.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-253-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1716.5,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-253-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1994.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-253-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X(\\cdot)<\/script><span>\uff0c and not on the letters <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.197ex\" height=\"1.02ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -441 529 451\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-460-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-460-TEX-I-1D44E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a<\/script><span> in alphabet of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span>. We thus have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n17\" cid=\"n17\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 20.87ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 20.87ex;\"><defs><path id=\"MJX-3-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-3-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-3-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-3-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-3-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-3-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3812.3 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-3-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-3-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-3-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2129,0)\"><use data-c=\"29\" xlink:href=\"#MJX-3-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2795.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-3-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3851.6,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-3-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4739.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-3-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5128.6,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-3-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5605.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-3-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5994.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-3-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6846.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-3-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7235.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-3-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>for any invertible or bijective function <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.468ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1533 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-165-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-165-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-165-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-165-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-165-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(477,0)\"><use data-c=\"28\" xlink:href=\"#MJX-165-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(866,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-165-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1144,0)\"><use data-c=\"29\" xlink:href=\"#MJX-165-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">g(\\cdot)<\/script><span>, since <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.468ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1533 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-165-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-165-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-165-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-165-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-165-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(477,0)\"><use data-c=\"28\" xlink:href=\"#MJX-165-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(866,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-165-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1144,0)\"><use data-c=\"29\" xlink:href=\"#MJX-165-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">g(\\cdot)<\/script><span> simply relabels the letters in the alphabet. Examples include <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.425ex\" height=\"2.099ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 3282 928\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-166-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-166-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-166-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-166-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-166-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1040.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-166-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(2096.6,0)\"><g data-mml-node=\"mn\"><use data-c=\"32\" xlink:href=\"#MJX-166-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(533,363) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-166-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><mo>=<\/mo><msup><mn>2<\/mn><mi>X<\/mi><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y=2^X<\/script><span>.<\/span><\/p><h4 id='example-12'><span>Example 1.2 <\/span><\/h4><p><span>Consider the Bernoulli distribution that has an alphabet <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.531ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2444.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-167-TEX-N-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path id=\"MJX-167-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-167-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-167-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-167-TEX-N-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"7B\" xlink:href=\"#MJX-167-TEX-N-7B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(500,0)\"><use data-c=\"30\" xlink:href=\"#MJX-167-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1000,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-167-TEX-N-2C\"><\/use><\/g><g 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xlink:href=\"#MJX-168-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1716.5,0)\"><use data-c=\"31\" xlink:href=\"#MJX-168-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2216.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-168-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2883.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-168-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3939,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-168-TEX-I-1D45D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>p<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X(1)=p<\/script><span>. The entropy of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 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transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-4-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-4-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(13821.1,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-4-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13821.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-4-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(14210.1,0)\"><use data-c=\"31\" xlink:href=\"#MJX-4-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14932.3,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-4-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(15932.5,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-4-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16435.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-4-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>p<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mi>p<\/mi><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mi>p<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.628ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2045.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-170-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-170-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-170-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-170-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-170-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(864,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-170-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1267.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-170-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1656.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-170-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H_2()<\/script><span> is called the <\/span><em><span>binary entropy function<\/span><\/em><span>.<\/span><\/p><blockquote><h4 id='theorem-11'><span>Theorem 1.1<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block 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id=\"MJX-5-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-5-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-5-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-5-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-5-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 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143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z\"><\/path><path id=\"MJX-5-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-5-TEX-N-2061\" d=\"\"><\/path><path id=\"MJX-5-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"4487.2 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mn\"><use data-c=\"30\" xlink:href=\"#MJX-5-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(777.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-5-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1833.6,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-5-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2721.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3110.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3962.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4629.3,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-5-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5685.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-5-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-5-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-5-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7399.7,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-5-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7566.3,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7844.3,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8696.3,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5-TEX-N-7C\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mn>0<\/mn><mo>\u2264<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>with equality on the left iff. there is one letter <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.197ex\" height=\"1.02ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -441 529 451\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-460-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-460-TEX-I-1D44E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a<\/script><span> in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.109ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4468 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-256-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-256-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-256-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 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xlink:href=\"#MJX-256-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1716.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-256-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2245.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-256-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2912.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-256-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3968,0)\"><use data-c=\"31\" xlink:href=\"#MJX-256-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X(a)=1<\/script><span>, and with equality on the right iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.225ex\" height=\"3.193ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -864.9 5403.6 1411.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.236ex;\"><defs><path id=\"MJX-255-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-255-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-255-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-255-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-255-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 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xlink:href=\"#MJX-255-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-255-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1327.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-255-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1716.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-255-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2245.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-255-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2912.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-255-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(3968,0)\"><g data-mml-node=\"mn\" transform=\"translate(541,394) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-255-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-370) scale(0.707)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-255-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(278,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-255-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1130,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-255-TEX-N-7C\"><\/use><\/g><\/g><rect width=\"1195.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mrow><\/mfrac><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X(a)=\\frac{1}{|X|}<\/script><span> for <\/span><em><span>any<\/span><\/em><span> <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.89ex\" height=\"1.636ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2603.6 723\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-289-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-289-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-289-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-289-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(806.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-289-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1751.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-289-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>\u2208<\/mo><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a\\in X<\/script><span>, i.e., <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> in uniformly distributed.<\/span><\/p><\/blockquote><p><em><span>Proof. of right equality<\/span><\/em><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n28\" cid=\"n28\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"58.652ex\" height=\"30.066ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -6894.6 25924.2 13289.1\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -14.467ex;\"><defs><path id=\"MJX-6-TEX-N-6C\" d=\"M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 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data-mjx-texclass=\"NONE\">\u2061<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><\/mtd><mtd><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>x<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='4-example-distributions'><span>(4) Example Distributions<\/span><\/h3><h3 id='5-conditional-entropy'><span>(5) Conditional Entropy <\/span><\/h3><p><span>Consider the joint distribution <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.984ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2645 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" 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461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-177-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-177-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-177-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-177-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(852,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-177-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1867,0)\"><use data-c=\"28\" xlink:href=\"#MJX-177-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2256,0)\"><use data-c=\"29\" xlink:href=\"#MJX-177-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{XY}()<\/script><span> where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-285-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-285-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y<\/script><span> are discrete random variables. <\/span><\/p><p><span>Focusing on <\/span><em><span>one line<\/span><\/em><span> of the table of joint distribution, we get the <\/span><em><span>conditional entropy of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> given the event <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.714ex\" height=\"1.756ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2525.6 776\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-181-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-181-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-181-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-181-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1040.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-181-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2096.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-181-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><mo>=<\/mo><mi>b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y=b<\/script><\/em><span> with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.84ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6117.1 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-182-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-182-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-182-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-182-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-182-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-182-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path id=\"MJX-182-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-182-TEX-N-3E\" d=\"M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z\"><\/path><path id=\"MJX-182-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-182-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(751,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-182-TEX-I-1D45F\"><\/use><\/g><g 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data-c=\"1D44C\" xlink:href=\"#MJX-7-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8491.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-7-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9547,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-7-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9976,0)\"><use data-c=\"29\" xlink:href=\"#MJX-7-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10365,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-7-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10643,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-7-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right center left\" columnspacing=\"0em 0.278em\" rowspacing=\"3pt\"><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo>=<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><mo>\u2212<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mo>\u2212<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo>=<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Similarly,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n35\" cid=\"n35\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 30.267ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 30.267ex;\"><defs><path id=\"MJX-8-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 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464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-8-TEX-N-2061\" d=\"\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"5888.9 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mn\"><use data-c=\"30\" xlink:href=\"#MJX-8-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(777.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-8-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1833.6,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-8-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2721.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-8-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3110.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-8-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3962.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-8-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4240.6,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-8-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5281.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-8-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6337.1,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-8-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6766.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-8-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7432.9,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-8-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(8488.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-8-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-8-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-8-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-8-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10203.2,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-8-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10369.9,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-8-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10647.9,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-8-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11499.9,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-8-TEX-N-7C\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mn>0<\/mn><mo>\u2264<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo>=<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>with equality on the left iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.374ex\" height=\"2.495ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5911.1 1102.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-191-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 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39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-191-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-191-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 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347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-191-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-191-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-176.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-191-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(852,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-191-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1130,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-191-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2063.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-191-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2452.6,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-191-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2981.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-191-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3259.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-191-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3688.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-191-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4355.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-191-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(5411.1,0)\"><use data-c=\"31\" xlink:href=\"#MJX-191-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{X|Y}(a|b)=1<\/script><span> for some <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.197ex\" height=\"1.02ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -441 529 451\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-460-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-460-TEX-I-1D44E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a<\/script><span>, and with equality on the right iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"15.49ex\" height=\"3.193ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -864.9 6846.7 1411.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.236ex;\"><defs><path id=\"MJX-193-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-193-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-193-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-193-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 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transform=\"translate(2063.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-193-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2452.6,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-193-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2981.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-193-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3259.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-193-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3688.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-193-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4355.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-193-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(5411.1,0)\"><g data-mml-node=\"mn\" transform=\"translate(541,394) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-193-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-370) scale(0.707)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-193-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(278,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-193-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1130,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-193-TEX-N-7C\"><\/use><\/g><\/g><rect width=\"1195.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mrow><\/mfrac><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{X|Y}(a|b)=\\frac{1}{|X|}<\/script><span> for <\/span><em><span>any<\/span><\/em><span> <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.89ex\" height=\"1.636ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2603.6 723\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-289-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 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683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-289-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(806.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-289-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1751.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-289-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>\u2208<\/mo><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a\\in X<\/script><span>.<\/span><\/p><hr \/><p><span>We average the conditional entropy on all <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.463ex\" height=\"1.661ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2414.6 734\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-187-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path id=\"MJX-187-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-187-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-187-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(706.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-187-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1651.6,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-187-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><mo>\u2208<\/mo><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">b\\in Y<\/script><span> and get the <\/span><em><span>conditional entropy of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> given <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-285-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-285-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y<\/script><\/em><span> <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n39\" 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data-c=\"2E\" xlink:href=\"#MJX-9-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right center left\" columnspacing=\"0em 0.278em\" rowspacing=\"3pt\"><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>b<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo>=<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><mo>\u2212<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mo>\u2212<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Also, <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n41\" cid=\"n41\" mdtype=\"math_block\" 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569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-10-TEX-N-2061\" d=\"\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"5007.7 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mn\"><use data-c=\"30\" xlink:href=\"#MJX-10-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(777.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-10-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1833.6,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-10-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2721.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-10-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3110.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-10-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3962.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-10-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4240.6,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-10-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5003.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-10-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5670.3,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-10-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(6726.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-10-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-10-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-10-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-10-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8440.7,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-10-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8607.3,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-10-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8885.3,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-10-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9737.3,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-10-TEX-N-7C\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mn>0<\/mn><mo>\u2264<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>with equality on the left iff. there is one <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.197ex\" height=\"1.02ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -441 529 451\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-460-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-460-TEX-I-1D44E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a<\/script><span> such that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.374ex\" height=\"2.495ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5911.1 1102.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-191-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-191-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-191-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-191-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-191-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-191-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-191-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path id=\"MJX-191-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-191-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 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xlink:href=\"#MJX-191-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1130,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-191-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2063.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-191-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2452.6,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-191-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2981.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-191-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3259.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-191-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3688.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-191-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4355.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-191-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(5411.1,0)\"><use data-c=\"31\" xlink:href=\"#MJX-191-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{X|Y}(a|b)=1<\/script><span> for every <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.971ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 429 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-313-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-313-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">b<\/script><span>. And with equality on the right iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"15.49ex\" height=\"3.193ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -864.9 6846.7 1411.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.236ex;\"><defs><path id=\"MJX-193-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 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1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-193-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-193-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-193-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-193-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-193-TEX-I-1D44F\" d=\"M73 647Q73 657 77 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id=\"MJX-193-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-193-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-176.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-193-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(852,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-193-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1130,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-193-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2063.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-193-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2452.6,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-193-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2981.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-193-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3259.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-193-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3688.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-193-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4355.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-193-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(5411.1,0)\"><g data-mml-node=\"mn\" transform=\"translate(541,394) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-193-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-370) scale(0.707)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-193-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(278,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-193-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1130,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-193-TEX-N-7C\"><\/use><\/g><\/g><rect width=\"1195.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mrow><\/mfrac><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{X|Y}(a|b)=\\frac{1}{|X|}<\/script><span> for <\/span><em><span>any<\/span><\/em><span> <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.89ex\" height=\"1.636ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2603.6 723\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-289-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-289-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-289-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-289-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(806.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-289-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1751.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-289-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>\u2208<\/mo><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a\\in X<\/script><span> and for every <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.971ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 429 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-313-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-313-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">b<\/script><span>.<\/span><\/p><h4 id='example-13'><span>Example 1.3<\/span><\/h4><p><span>We say that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-285-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-285-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y<\/script><span> essentially determines <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> if <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.2ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5392.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-198-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-198-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-198-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-198-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-198-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-198-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-198-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-198-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-198-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-198-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-198-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2129,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-198-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2407,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-198-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3170,0)\"><use data-c=\"29\" xlink:href=\"#MJX-198-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3836.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-198-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(4892.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-198-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H(X|Y)=0<\/script><span>).<\/span><\/p><p><span>Recall <\/span><em><span>Example 1.1<\/span><\/em><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.767ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2107 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-228-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-228-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-228-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-228-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-228-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(477,0)\"><use data-c=\"28\" xlink:href=\"#MJX-228-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(866,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-228-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1718,0)\"><use data-c=\"29\" xlink:href=\"#MJX-228-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">g(X)<\/script><span> is invertible, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.25ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7624.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-200-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-200-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-200-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-200-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-200-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-200-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-200-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-200-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-200-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2129,0)\"><use data-c=\"29\" xlink:href=\"#MJX-200-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2795.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-200-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3851.6,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-200-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4739.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-200-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5128.6,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-200-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5605.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-200-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5994.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-200-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6846.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-200-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7235.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-200-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H(X)= H(g(X))<\/script><span>. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> determines <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.767ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2107 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-228-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-228-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-228-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-228-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-228-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(477,0)\"><use data-c=\"28\" xlink:href=\"#MJX-228-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(866,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-228-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" 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width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n46\" cid=\"n46\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container 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id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>For a <\/span><em><span>non-invertible<\/span><\/em><span> function <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.932ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2180 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-229-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path 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1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-229-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-229-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(550,0)\"><use data-c=\"28\" xlink:href=\"#MJX-229-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(939,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-229-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1791,0)\"><use data-c=\"29\" xlink:href=\"#MJX-229-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f(X)<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.415ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7697.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-206-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-206-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-206-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-206-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-206-TEX-N-2260\" d=\"M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z\"><\/path><path id=\"MJX-206-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-206-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-206-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-206-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2129,0)\"><use data-c=\"29\" xlink:href=\"#MJX-206-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2795.8,0)\"><use data-c=\"2260\" xlink:href=\"#MJX-206-TEX-N-2260\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3851.6,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-206-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4739.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-206-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5128.6,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-206-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5678.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-206-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6067.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-206-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6919.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-206-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7308.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-206-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2260<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H(X)\\neq H(f(X))<\/script><span> since the probability distribution of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.932ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2180 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-229-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-229-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-229-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-229-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-229-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(550,0)\"><use data-c=\"28\" xlink:href=\"#MJX-229-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(939,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-229-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1791,0)\"><use data-c=\"29\" xlink:href=\"#MJX-229-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f(X)<\/script><span> might be different. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> still determines <\/span><mjx-container class=\"MathJax\" 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data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo>\u2260<\/mo><mn>0.<\/mn><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Examples include <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.929ex\" height=\"2.26ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -749.5 3504.6 999\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.564ex;\"><defs><path id=\"MJX-213-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-213-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-213-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-213-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-213-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1040.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-213-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2096.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-213-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2374.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-213-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3226.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-213-TEX-N-7C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><mo>=<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y=|X|<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.809ex\" height=\"1.731ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 4335.6 765\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-214-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-214-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-214-TEX-I-1D450\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path id=\"MJX-214-TEX-I-1D45C\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path id=\"MJX-214-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-214-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-214-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1040.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-214-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2096.6,0)\"><use data-c=\"1D450\" xlink:href=\"#MJX-214-TEX-I-1D450\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2529.6,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-214-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3014.6,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-214-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3483.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-214-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><mo>=<\/mo><mi>c<\/mi><mi>o<\/mi><mi>s<\/mi><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y=cosX<\/script><span>.<\/span><\/p><h3 id='6-joint-entropy'><span>(6) Joint Entropy <\/span><\/h3><p><span>The joint entropy of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-285-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-285-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y<\/script><span> is defined by considering the concatenation (not multiplication!) <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.654ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1615 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-217-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-217-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-217-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(852,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-217-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">XY<\/script><span> of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-285-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-285-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y<\/script><span> as a new discrete random variable, i.e., we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" 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586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z\"><\/path><path id=\"MJX-14-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-14-TEX-N-67\" d=\"M329 409Q373 453 429 453Q459 453 472 434T485 396Q485 382 476 371T449 360Q416 360 412 390Q410 404 415 411Q415 412 416 414V415Q388 412 363 393Q355 388 355 386Q355 385 359 381T368 369T379 351T388 325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 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xlink:href=\"#MJX-14-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16082.3,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-14-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(16360.3,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-14-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17212.3,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-14-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17712.6,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-14-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18212.8,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-14-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(18490.8,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-14-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(19253.8,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-14-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(19531.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-14-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><mo stretchy=\"false\">(<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u22c5<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>with equality on the left iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> essentially determines <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-285-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-285-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y<\/script><span>, or <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-285-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-285-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y<\/script><span> essentially determines <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span>, or both. And with equality on the right iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.533ex\" height=\"3.193ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -864.9 7749.5 1411.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.236ex;\"><defs><path id=\"MJX-224-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 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data-mml-node=\"mi\" transform=\"translate(852,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-224-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1867,0)\"><use data-c=\"28\" xlink:href=\"#MJX-224-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2256,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-224-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2785,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-224-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3229.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-224-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3658.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-224-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4325.4,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-224-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(5381.2,0)\"><g data-mml-node=\"mn\" transform=\"translate(1007.4,394) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-224-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-370) scale(0.707)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-224-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(278,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-224-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1130,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-224-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1408,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-224-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1686,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-224-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2449,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-224-TEX-N-7C\"><\/use><\/g><\/g><rect width=\"2128.3\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mrow><\/mfrac><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{XY}(a,b)=\\frac{1}{|X||Y|}<\/script><span> for all <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.934ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2180.7 1000\" 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-238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-225-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-225-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(918,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-225-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1362.7,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-225-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1791.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-225-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(a,b)<\/script><span>.<\/span><\/p><h4 id='example-14-chain-rule-for-entropy'><span>Example 1.4 Chain Rule for Entropy<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n57\" cid=\"n57\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"46.286ex\" height=\"9.295ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -2304.2 20458.2 4108.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -4.082ex;\"><defs><path id=\"MJX-15-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 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732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-15-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 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stretchy=\"false\">)<\/mo><mo>,<\/mo><mtext>&nbsp;<\/mtext><msup><mi>X<\/mi><mn>0<\/mn><\/msup><mtext>&nbsp;<\/mtext><mi>i<\/mi><mi>s<\/mi><mtext>&nbsp;<\/mtext><mi>a<\/mi><mtext>&nbsp;<\/mtext><mi>c<\/mi><mi>o<\/mi><mi>n<\/mi><mi>s<\/mi><mi>t<\/mi><mi>a<\/mi><mi>n<\/mi><mi>t<\/mi><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='example-15'><span>Example 1.5<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n59\" cid=\"n59\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 24.915ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 24.915ex;\"><defs><path id=\"MJX-16-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-16-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-16-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-16-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-16-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-16-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-16-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-16-TEX-N-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-16-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"4706.3 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-16-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-16-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-16-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2129,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-16-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2892,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-16-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3170,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-16-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3893,0)\"><use data-c=\"29\" xlink:href=\"#MJX-16-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4559.8,0)\"><use data-c=\"2265\" xlink:href=\"#MJX-16-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5615.6,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-16-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6503.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-16-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6892.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-16-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7744.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-16-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8022.6,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-16-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8745.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-16-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9134.6,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-16-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><em><span>Proof.<\/span><\/em><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n61\" cid=\"n61\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 48.347ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 48.347ex;\"><defs><path id=\"MJX-17-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-17-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-17-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-17-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-17-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-17-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-17-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 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18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"9884.8 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-17-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-17-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-17-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2129,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-17-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2892,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-17-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3170,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-17-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3893,0)\"><use data-c=\"29\" xlink:href=\"#MJX-17-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4559.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-17-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5615.6,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-17-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6503.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-17-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6892.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-17-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7744.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-17-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8022.6,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-17-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8745.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-17-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9356.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-17-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10357,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-17-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11245,0)\"><use data-c=\"28\" xlink:href=\"#MJX-17-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11634,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-17-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12397,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-17-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(12675,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-17-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13527,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-17-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14250,0)\"><use data-c=\"29\" xlink:href=\"#MJX-17-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14916.8,0)\"><use data-c=\"2265\" xlink:href=\"#MJX-17-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(15972.6,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-17-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16860.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-17-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(17249.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-17-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18101.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-17-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(18379.6,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-17-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(19102.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-17-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(19491.6,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-17-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='example-16'><span>Example 1.6<\/span><\/h4><p><span>Using conclusions about conditional entropy in <\/span><em><span>Example 1.3<\/span><\/em><span>, we study the joint distribution <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.629ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4698 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-226-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-226-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-226-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-226-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-226-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-226-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-226-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-226-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2129,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-226-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2679,0)\"><use data-c=\"28\" xlink:href=\"#MJX-226-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3068,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-226-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3920,0)\"><use data-c=\"29\" xlink:href=\"#MJX-226-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4309,0)\"><use data-c=\"29\" xlink:href=\"#MJX-226-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H(Xf(X))<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.464ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4625 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-227-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-227-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-227-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-227-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-227-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-227-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-227-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-227-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2129,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-227-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2606,0)\"><use data-c=\"28\" xlink:href=\"#MJX-227-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2995,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-227-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3847,0)\"><use data-c=\"29\" xlink:href=\"#MJX-227-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4236,0)\"><use data-c=\"29\" xlink:href=\"#MJX-227-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H(Xg(X))<\/script><span>.<\/span><\/p><p><span>For a <\/span><em><span>invertible<\/span><\/em><span> function <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.767ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2107 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-228-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 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stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>For a <\/span><em><span>non-invertible<\/span><\/em><span> function <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.932ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2180 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-229-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 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stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo>\u2260<\/mo><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='7-information-divergence--kullback-leibler-distance'><span>(7) Information Divergence \/ Kullback-Leibler Distance<\/span><\/h3><blockquote><p><span>The informational divergence 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data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-286-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1327.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-286-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1716.5,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-286-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1994.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-286-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X (\\cdot)<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.25ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2320.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-292-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-292-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 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data-mml-node=\"mo\" transform=\"translate(1653.5,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-292-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1931.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-292-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_Y (\\cdot)<\/script><span> is defined as<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n71\" cid=\"n71\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" 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transform=\"translate(251.5,-710)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-20-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-20-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1264.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-20-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1653.5,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-20-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2505.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-20-TEX-N-29\"><\/use><\/g><\/g><rect width=\"3157.5\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(7427.4,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-20-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -3152.1 1 5804.1\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><mrow><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msub><mi>log<\/mi><mi>x<\/mi><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><mrow><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>We define <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" 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data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-238-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4864.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-238-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5531.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-238-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6587.1,0)\"><use data-c=\"221E\" xlink:href=\"#MJX-238-TEX-N-221E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi mathvariant=\"normal\">\u221e<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D(P_X\\parallel P_Y ) = \\infin<\/script><span> if <\/span><mjx-container 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type=\"math\/tex\">P_Y (a) = 0<\/script><span> for some <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.357ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5904 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-254-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path 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data-mml-node=\"mo\" transform=\"translate(5515,0)\"><use data-c=\"29\" xlink:href=\"#MJX-254-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a\\in supp(P_X)<\/script><span>. <\/span><\/p><p><span>Observe that the definition is not symmetric in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.003ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1327.5 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-538-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-538-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-538-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-538-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.861ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1264.5 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-236-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-236-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-236-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-236-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_Y<\/script><span> , i.e., we have <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"26.789ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 11840.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-237-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-237-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-237-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-237-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-237-TEX-N-2225\" d=\"M133 736Q138 750 153 750Q164 750 170 739Q172 735 172 250T170 -239Q164 -250 152 -250Q144 -250 138 -244L137 -243Q133 -241 133 -179T132 250Q132 731 133 736ZM329 739Q334 750 346 750Q353 750 361 744L362 743Q366 741 366 679T367 250T367 -178T362 -243L361 -244Q355 -250 347 -250Q335 -250 329 -239Q327 -235 327 250T329 739Z\"><\/path><path id=\"MJX-237-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-237-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-237-TEX-N-2260\" d=\"M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-237-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(828,0)\"><use data-c=\"28\" xlink:href=\"#MJX-237-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1217,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-237-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-237-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2822.2,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-237-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3600,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-237-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-237-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4864.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-237-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5531.3,0)\"><use data-c=\"2260\" xlink:href=\"#MJX-237-TEX-N-2260\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6587.1,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-237-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7415.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-237-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(7804.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-237-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-237-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9346.4,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-237-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(10124.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-237-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-237-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(11451.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-237-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2260<\/mo><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D(P_X\\parallel P_Y ) \\neq D(P_Y\\parallel P_X )<\/script><span> in general.<\/span><\/p><\/blockquote><p><span>To avoid the case <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.165ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7587.1 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-238-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-238-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-238-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-238-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-238-TEX-N-2225\" d=\"M133 736Q138 750 153 750Q164 750 170 739Q172 735 172 250T170 -239Q164 -250 152 -250Q144 -250 138 -244L137 -243Q133 -241 133 -179T132 250Q132 731 133 736ZM329 739Q334 750 346 750Q353 750 361 744L362 743Q366 741 366 679T367 250T367 -178T362 -243L361 -244Q355 -250 347 -250Q335 -250 329 -239Q327 -235 327 250T329 739Z\"><\/path><path id=\"MJX-238-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-238-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-238-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-238-TEX-N-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-238-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(828,0)\"><use data-c=\"28\" xlink:href=\"#MJX-238-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1217,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-238-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-238-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2822.2,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-238-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3600,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-238-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-238-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4864.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-238-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5531.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-238-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6587.1,0)\"><use data-c=\"221E\" xlink:href=\"#MJX-238-TEX-N-221E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi mathvariant=\"normal\">\u221e<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D(P_X\\parallel P_Y ) = \\infin<\/script><span> we introduce the notation <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.384ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 4147.5 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-242-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-242-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-242-TEX-N-226B\" d=\"M55 539T55 547T60 561T74 567Q81 567 207 498Q297 449 365 412Q633 265 636 261Q639 255 639 250Q639 241 626 232Q614 224 365 88Q83 -65 79 -66Q76 -67 73 -67Q65 -67 60 -61T55 -47Q55 -39 61 -33Q62 -33 95 -15T193 39T320 109L321 110H322L323 111H324L325 112L326 113H327L329 114H330L331 115H332L333 116L334 117H335L336 118H337L338 119H339L340 120L341 121H342L343 122H344L345 123H346L347 124L348 125H349L351 126H352L353 127H354L355 128L356 129H357L358 130H359L360 131H361L362 132L363 133H364L365 134H366L367 135H368L369 136H370L371 137L372 138H373L374 139H375L376 140L378 141L576 251Q63 530 62 533Q55 539 55 547ZM360 539T360 547T365 561T379 567Q386 567 512 498Q602 449 670 412Q938 265 941 261Q944 255 944 250Q944 241 931 232Q919 224 670 88Q388 -65 384 -66Q381 -67 378 -67Q370 -67 365 -61T360 -47Q360 -39 366 -33Q367 -33 400 -15T498 39T625 109L626 110H627L628 111H629L630 112L631 113H632L634 114H635L636 115H637L638 116L639 117H640L641 118H642L643 119H644L645 120L646 121H647L648 122H649L650 123H651L652 124L653 125H654L656 126H657L658 127H659L660 128L661 129H662L663 130H664L665 131H666L667 132L668 133H669L670 134H671L672 135H673L674 136H675L676 137L677 138H678L679 139H680L681 140L683 141L881 251Q368 530 367 533Q360 539 360 547Z\"><\/path><path id=\"MJX-242-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-242-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-242-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1542.3,0)\"><use data-c=\"226B\" xlink:href=\"#MJX-242-TEX-N-226B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2820.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-242-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-242-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo>\u226b<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_Y\\gg P_X<\/script><span> to mean that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"23.594ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 10428.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-240-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-240-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-240-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-240-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-240-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-240-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-240-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-240-TEX-N-21D2\" d=\"M580 514Q580 525 596 525Q601 525 604 525T609 525T613 524T615 523T617 520T619 517T622 512Q659 438 720 381T831 300T927 263Q944 258 944 250T935 239T898 228T840 204Q696 134 622 -12Q618 -21 615 -22T600 -24Q580 -24 580 -17Q580 -13 585 0Q620 69 671 123L681 133H70Q56 140 56 153Q56 168 72 173H725L735 181Q774 211 852 250Q851 251 834 259T789 283T735 319L725 327H72Q56 332 56 347Q56 360 70 367H681L671 377Q638 412 609 458T580 514Z\"><\/path><path id=\"MJX-240-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-240-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-240-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1264.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-240-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1653.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-240-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2182.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-240-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2849.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-240-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3905.1,0)\"><use data-c=\"30\" xlink:href=\"#MJX-240-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4682.9,0)\"><use data-c=\"21D2\" xlink:href=\"#MJX-240-TEX-N-21D2\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5960.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-240-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-240-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7288.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-240-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7677.1,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-240-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8206.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-240-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8872.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-240-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(9928.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-240-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><mo stretchy=\"false\">\u21d2<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_Y(a)=0 \\Rightarrow P_X(a)=0<\/script><span> for all <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.89ex\" height=\"1.636ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2603.6 723\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-289-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-289-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-289-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-289-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(806.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-289-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1751.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-289-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>\u2208<\/mo><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a\\in X<\/script><span>. In other words, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.384ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 4147.5 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-242-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-242-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-242-TEX-N-226B\" d=\"M55 539T55 547T60 561T74 567Q81 567 207 498Q297 449 365 412Q633 265 636 261Q639 255 639 250Q639 241 626 232Q614 224 365 88Q83 -65 79 -66Q76 -67 73 -67Q65 -67 60 -61T55 -47Q55 -39 61 -33Q62 -33 95 -15T193 39T320 109L321 110H322L323 111H324L325 112L326 113H327L329 114H330L331 115H332L333 116L334 117H335L336 118H337L338 119H339L340 120L341 121H342L343 122H344L345 123H346L347 124L348 125H349L351 126H352L353 127H354L355 128L356 129H357L358 130H359L360 131H361L362 132L363 133H364L365 134H366L367 135H368L369 136H370L371 137L372 138H373L374 139H375L376 140L378 141L576 251Q63 530 62 533Q55 539 55 547ZM360 539T360 547T365 561T379 567Q386 567 512 498Q602 449 670 412Q938 265 941 261Q944 255 944 250Q944 241 931 232Q919 224 670 88Q388 -65 384 -66Q381 -67 378 -67Q370 -67 365 -61T360 -47Q360 -39 366 -33Q367 -33 400 -15T498 39T625 109L626 110H627L628 111H629L630 112L631 113H632L634 114H635L636 115H637L638 116L639 117H640L641 118H642L643 119H644L645 120L646 121H647L648 122H649L650 123H651L652 124L653 125H654L656 126H657L658 127H659L660 128L661 129H662L663 130H664L665 131H666L667 132L668 133H669L670 134H671L672 135H673L674 136H675L676 137L677 138H678L679 139H680L681 140L683 141L881 251Q368 530 367 533Q360 539 360 547Z\"><\/path><path id=\"MJX-242-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-242-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-242-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1542.3,0)\"><use data-c=\"226B\" xlink:href=\"#MJX-242-TEX-N-226B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2820.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-242-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-242-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo>\u226b<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_Y\\gg P_X<\/script><span> implies that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"21.664ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 9575.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-243-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-243-TEX-I-1D462\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-243-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-243-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-243-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-243-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-243-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-243-TEX-N-2286\" d=\"M84 346Q84 468 166 546T360 635Q361 635 370 635T395 635T430 636T475 636T524 636H679Q694 628 694 616Q694 607 681 597L522 596H470H441Q366 596 338 592T266 568Q244 557 224 542T179 500T139 433T124 346V341Q124 253 185 185Q244 121 328 103Q348 98 366 98T522 96H681Q694 86 694 76Q694 64 679 56H526Q510 56 480 56T434 55Q350 55 289 71T172 141Q84 223 84 346ZM104 -131T104 -118T118 -98H679Q694 -106 694 -118T679 -138H118Q104 -131 104 -118Z\"><\/path><path id=\"MJX-243-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D460\" xlink:href=\"#MJX-243-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(469,0)\"><use data-c=\"1D462\" xlink:href=\"#MJX-243-TEX-I-1D462\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1041,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-243-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1544,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-243-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2047,0)\"><use data-c=\"28\" xlink:href=\"#MJX-243-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2436,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-243-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-243-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3763.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-243-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4430.2,0)\"><use data-c=\"2286\" xlink:href=\"#MJX-243-TEX-N-2286\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5486,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-243-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5955,0)\"><use data-c=\"1D462\" xlink:href=\"#MJX-243-TEX-I-1D462\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6527,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-243-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7030,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-243-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7533,0)\"><use data-c=\"28\" xlink:href=\"#MJX-243-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(7922,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-243-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-243-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9186.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-243-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2286<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">supp(P_X)\\subseteq supp(P_Y)<\/script><span>, or <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.165ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7587.1 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-244-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-244-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-244-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 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transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-244-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2822.2,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-244-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3600,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-244-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-244-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4864.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-244-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5531.3,0)\"><use data-c=\"3C\" xlink:href=\"#MJX-244-TEX-N-3C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6587.1,0)\"><use data-c=\"221E\" xlink:href=\"#MJX-244-TEX-N-221E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>&lt;<\/mo><mi mathvariant=\"normal\">\u221e<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D(P_X\\parallel P_Y ) < \\infin<\/script><span> for finite sets.<\/span><\/p><p><span>Given a third discrete random variable <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.636ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 723 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-279-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44D\" xlink:href=\"#MJX-279-TEX-I-1D44D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Z<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Z<\/script><span>, we define the <\/span><em><span>conditional information divergence<\/span><\/em><span> between <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.994ex\" height=\"2.495ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3091.3 1102.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-246-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-246-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-246-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-246-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-246-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-246-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-246-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-246-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-176.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-246-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(852,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-246-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1130,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-246-TEX-I-1D44D\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2035.3,0)\"><use data-c=\"28\" xlink:href=\"#MJX-246-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2424.3,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-246-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2702.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-246-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{X|Z}(\\cdot)<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.851ex\" height=\"2.495ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3028.3 1102.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-247-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-247-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 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469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-247-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-247-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-247-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 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xlink:href=\"#MJX-247-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2639.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-247-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{Y|Z}(\\cdot)<\/script><span> as<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n76\" cid=\"n76\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" 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stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>Z<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mi>Z<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>c<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>Z<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mi>Z<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><mrow><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><mrow><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mfrac><mrow><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><blockquote><h4 id='theorem-12'><span>Theorem 1.2<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n80\" cid=\"n80\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 19.654ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 19.654ex;\"><defs><path id=\"MJX-22-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-22-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-22-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-22-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-22-TEX-N-2225\" d=\"M133 736Q138 750 153 750Q164 750 170 739Q172 735 172 250T170 -239Q164 -250 152 -250Q144 -250 138 -244L137 -243Q133 -241 133 -179T132 250Q132 731 133 736ZM329 739Q334 750 346 750Q353 750 361 744L362 743Q366 741 366 679T367 250T367 -178T362 -243L361 -244Q355 -250 347 -250Q335 -250 329 -239Q327 -235 327 250T329 739Z\"><\/path><path id=\"MJX-22-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-22-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-22-TEX-N-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-22-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3543.5 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-22-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(828,0)\"><use data-c=\"28\" xlink:href=\"#MJX-22-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1217,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-22-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-22-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2822.2,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-22-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3600,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-22-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-22-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4864.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-22-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5531.3,0)\"><use data-c=\"2265\" xlink:href=\"#MJX-22-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(6587.1,0)\"><use data-c=\"30\" xlink:href=\"#MJX-22-TEX-N-30\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mn>0<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>with equality iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.795ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6539.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path 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transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-248-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-248-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1327.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-248-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1716.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-248-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2245.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-248-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2912.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-248-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3968,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-248-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-248-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5232.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-248-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5621.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-248-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6150.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-248-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X (a) = P_Y (a)<\/script><span> for all <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.357ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5904 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-254-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-254-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-254-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-254-TEX-I-1D462\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 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350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-254-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-254-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-254-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-254-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-254-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(806.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-254-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1751.6,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-254-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2220.6,0)\"><use data-c=\"1D462\" xlink:href=\"#MJX-254-TEX-I-1D462\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2792.6,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-254-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3295.6,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-254-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3798.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-254-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4187.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-254-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-254-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5515,0)\"><use data-c=\"29\" xlink:href=\"#MJX-254-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a\\in supp(P_X)<\/script><span>.<\/span><\/p><\/blockquote><p><em><span>Proof.<\/span><\/em><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n83\" cid=\"n83\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"50.738ex\" 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360Q416 360 412 390Q410 404 415 411Q415 412 416 414V415Q388 412 363 393Q355 388 355 386Q355 385 359 381T368 369T379 351T388 325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z\"><\/path><path id=\"MJX-23-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 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transform=\"translate(2505.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-23-TEX-N-29\"><\/use><\/g><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-710)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-23-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-23-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1327.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-23-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1716.5,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-23-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2568.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-23-TEX-N-29\"><\/use><\/g><\/g><rect width=\"3157.5\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(7654.2,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-23-TEX-N-5D\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,232.1)\"><g data-mml-node=\"mtd\" transform=\"translate(6031.5,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(6031.5,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-23-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1333.6,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-23-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2097.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-23-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(2375.6,0)\"><g data-mml-node=\"mrow\" transform=\"translate(251.5,710)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-23-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-23-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1264.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-23-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1653.5,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-23-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2505.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-23-TEX-N-29\"><\/use><\/g><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-710)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-23-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-23-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1327.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-23-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1716.5,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-23-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2568.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-23-TEX-N-29\"><\/use><\/g><\/g><rect width=\"3157.5\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(5995.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-23-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(6995.5,0)\"><use data-c=\"31\" xlink:href=\"#MJX-23-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7495.5,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-23-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(7940.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-23-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-23-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-23-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-23-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9654.7,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-23-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9654.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-23-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10043.7,0)\"><use data-c=\"1D452\" xlink:href=\"#MJX-23-TEX-I-1D452\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10509.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-23-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-2487.9)\"><g data-mml-node=\"mtd\" transform=\"translate(6031.5,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(6031.5,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-23-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"munder\" transform=\"translate(1333.6,0)\"><g data-mml-node=\"mo\" transform=\"translate(1169,0)\"><use data-c=\"2211\" xlink:href=\"#MJX-23-TEX-LO-2211\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(0,-1147.3) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-23-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(529,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-23-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1196,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-23-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1665,0)\"><use data-c=\"1D462\" xlink:href=\"#MJX-23-TEX-I-1D462\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2237,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-23-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2740,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-23-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3243,0)\"><use data-c=\"28\" xlink:href=\"#MJX-23-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3632,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-23-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-23-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4959.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-23-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(5282.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-23-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-23-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6609.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-23-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6998.6,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-23-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7527.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-23-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7916.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-23-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(8194.6,0)\"><g data-mml-node=\"mrow\" transform=\"translate(251.5,710)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-23-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-23-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1264.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-23-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1653.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-23-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2182.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-23-TEX-N-29\"><\/use><\/g><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-710)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-23-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-23-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1327.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-23-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1716.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-23-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2245.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-23-TEX-N-29\"><\/use><\/g><\/g><rect width=\"2834.5\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(11491.3,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-23-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(12491.5,0)\"><use data-c=\"31\" xlink:href=\"#MJX-23-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12991.5,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-23-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(13436.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-23-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-23-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-23-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-23-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(15150.7,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-23-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15150.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-23-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(15539.7,0)\"><use data-c=\"1D452\" xlink:href=\"#MJX-23-TEX-I-1D452\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16005.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-23-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-4962.1)\"><g data-mml-node=\"mtd\" transform=\"translate(6031.5,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(6031.5,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-23-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1333.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-23-TEX-N-30\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>ln<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mi>x<\/mi><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>e<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mo>\u2212<\/mo><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><mrow><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mfrac><mrow><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>e<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">[<\/mo><mfrac><mrow><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>e<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mn>0<\/mn><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='example-17'><span>Example 1.7<\/span><\/h4><p><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.25ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2320.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-250-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-250-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-250-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-250-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-250-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-250-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-250-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1264.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-250-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1653.5,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-250-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1931.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-250-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_Y(\\cdot)<\/script><span> is uniform on X, for <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.357ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5904 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-254-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-254-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-254-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-254-TEX-I-1D462\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-254-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-254-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-254-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-254-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-254-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-254-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(806.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-254-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1751.6,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-254-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2220.6,0)\"><use data-c=\"1D462\" xlink:href=\"#MJX-254-TEX-I-1D462\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2792.6,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-254-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3295.6,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-254-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3798.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-254-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4187.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-254-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-254-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5515,0)\"><use data-c=\"29\" xlink:href=\"#MJX-254-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a\\in supp(P_X)<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.083ex\" height=\"3.193ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -864.9 5340.7 1411.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.236ex;\"><defs><path id=\"MJX-252-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-252-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-252-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-252-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-252-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-252-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-252-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-252-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-252-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-252-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-252-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1264.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-252-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1653.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-252-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2182.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-252-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2849.3,0)\"><use 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data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-24-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1264.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-24-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1653.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-24-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2182.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-24-TEX-N-29\"><\/use><\/g><\/g><rect width=\"2834.5\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-455)\"><g data-mml-node=\"mtd\" transform=\"translate(5253.5,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(5253.5,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-24-TEX-N-3D\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(6587.3,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"munder\" transform=\"translate(166.7,0)\"><g data-mml-node=\"mo\" transform=\"translate(1169,0)\"><use data-c=\"2211\" xlink:href=\"#MJX-24-TEX-LO-2211\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(0,-1147.3) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-24-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(529,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-24-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1196,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-24-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1665,0)\"><use data-c=\"1D462\" xlink:href=\"#MJX-24-TEX-I-1D462\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2237,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-24-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2740,0)\"><use data-c=\"1D45D\" 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xlink:href=\"#MJX-24-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5831.7,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-24-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6360.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-24-TEX-N-29\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(6916.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-24-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-24-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-24-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-24-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8630.9,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-24-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(8797.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-24-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-24-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10125.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-24-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10514.1,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-24-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11043.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-24-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11654.3,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-24-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"munder\" transform=\"translate(12654.5,0)\"><g data-mml-node=\"mo\" transform=\"translate(1169,0)\"><use data-c=\"2211\" xlink:href=\"#MJX-24-TEX-LO-2211\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(0,-1147.3) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-24-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(529,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-24-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1196,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-24-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1665,0)\"><use data-c=\"1D462\" xlink:href=\"#MJX-24-TEX-I-1D462\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2237,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-24-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2740,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-24-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3243,0)\"><use data-c=\"28\" xlink:href=\"#MJX-24-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3632,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-24-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-24-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4959.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-24-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(16603.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-24-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-24-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(17930.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-24-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(18319.6,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-24-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18848.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-24-TEX-N-29\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(19404.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-24-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-24-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-24-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-24-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(21118.8,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-24-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(21285.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-24-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-24-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(22550,0)\"><use data-c=\"28\" xlink:href=\"#MJX-24-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(22939,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-24-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(23468,0)\"><use data-c=\"29\" xlink:href=\"#MJX-24-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-2929.1)\"><g data-mml-node=\"mtd\" transform=\"translate(5253.5,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(5253.5,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-24-TEX-N-3D\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(6587.3,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(222.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-24-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1222.4,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-24-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2110.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-24-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2499.4,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-24-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3351.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-24-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3962.7,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-24-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4962.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-24-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-24-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-24-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-24-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6677.4,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-24-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6844.1,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-24-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7122.1,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-24-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7974.1,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-24-TEX-N-7C\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right center left\" columnspacing=\"0em 0.278em\" rowspacing=\"3pt\"><mtr><mtd><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><mrow><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2212<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='example-18'><span>Example 1.8<\/span><\/h4><p><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.392ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2383.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-253-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-253-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 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stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='example-19'><span>Example 1.9<\/span><\/h4><p><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.109ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4468 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-256-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-256-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-256-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-256-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-256-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-256-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-256-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-256-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-256-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1327.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-256-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1716.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-256-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2245.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-256-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2912.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-256-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3968,0)\"><use data-c=\"31\" xlink:href=\"#MJX-256-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X(a)=1<\/script><span> for some <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.89ex\" height=\"1.636ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2603.6 723\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-289-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-289-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-289-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-289-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(806.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-289-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1751.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-289-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>\u2208<\/mo><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a\\in X<\/script><span>, <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n93\" cid=\"n93\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 30.734ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 30.734ex;\"><defs><path id=\"MJX-26-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-26-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-26-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 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data-mml-node=\"msub\" transform=\"translate(7531.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-26-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-26-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-26-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-26-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9246.3,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-26-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(9413,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-26-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-26-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10677.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-26-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11066.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-26-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11595.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-26-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='example-110-chain-rule-for-information-divergence'><span>Example 1.10 Chain Rule for Information Divergence<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n95\" cid=\"n95\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"116.936ex\" height=\"60.035ex\" role=\"img\" focusable=\"false\" viewBox=\"0 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data-mjx-texclass=\"ORD\"><msup><mi>x<\/mi><mi>i<\/mi><\/msup><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msup><mi>X<\/mi><mi>i<\/mi><\/msup><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msup><mi>X<\/mi><mi>i<\/mi><\/msup><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mi>i<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><mrow><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>x<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>Y<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>x<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>\u22c5<\/mo><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><msub><mi>x<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><msub><mi>x<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow><\/msub><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><msub><mi>x<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><\/msub><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow><\/msub><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mi>i<\/mi><\/msup><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow><\/msub><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mi>i<\/mi><\/msup><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><msub><mi>x<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow><\/msub><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><msub><mi>x<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>x<\/mi><mi>i<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><msup><mi>x<\/mi><mi>i<\/mi><\/msup><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msup><mi>X<\/mi><mi>i<\/mi><\/msup><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msup><mi>X<\/mi><mi>i<\/mi><\/msup><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mi>i<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mfrac><mrow><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>x<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>Y<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>x<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/msub><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>Y<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/msub><\/mfrac><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>Y<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='example-111-\\np\\n1\\np\\n2\\n-instead-of-\\np\\nx\\n1\\nx\\n2'><span>Example 1.11 <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.88ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2157.1 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-258-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-258-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-258-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-258-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-258-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(1078.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-258-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-258-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mn>1<\/mn><\/msub><msub><mi>P<\/mi><mn>2<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_1P_2<\/script><span> instead of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.686ex\" height=\"2.125ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2513.3 939.1\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.579ex;\"><defs><path id=\"MJX-259-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-259-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-259-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-259-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-259-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-259-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-259-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(1264.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-259-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-259-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><msub><mi>X<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{X_1X_2}<\/script><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n97\" cid=\"n97\" mdtype=\"math_block\" data-math-tag-before=\"0\" 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stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>Q<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>+<\/mo><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>b<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><mrow><msub><mi>P<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>Q<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mn>1<\/mn><\/msub><mo>\u2225<\/mo><msub><mi>Q<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mn>2<\/mn><\/msub><mo>\u2225<\/mo><msub><mi>Q<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p>&nbsp;<\/p><h3 id='8-mutual-information'><span>(8) Mutual Information<\/span><\/h3><blockquote><p><span>The mutual information <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.56ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3341.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-331-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 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xlink:href=\"#MJX-331-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I(X;Y)<\/script><span> between two random variables <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 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transform=\"translate(1745,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-29-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2189.7,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-29-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2952.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-29-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3619.4,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-29-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4675.2,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-29-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5503.2,0)\"><use data-c=\"28\" xlink:href=\"#MJX-29-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5892.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-29-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-29-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(852,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-29-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8037,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-29-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(8814.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-29-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-29-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(10142.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-29-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-29-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(11406.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-29-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-517.9)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mo\"><use data-c=\"3D\" xlink:href=\"#MJX-29-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"munder\" transform=\"translate(1055.8,0)\"><g data-mml-node=\"mo\" transform=\"translate(1884.7,0)\"><use data-c=\"2211\" xlink:href=\"#MJX-29-TEX-LO-2211\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(0,-1147.3) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-29-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-29-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(918,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-29-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1196,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-29-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1625,0)\"><use data-c=\"29\" xlink:href=\"#MJX-29-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2014,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-29-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2681,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-29-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3150,0)\"><use data-c=\"1D462\" xlink:href=\"#MJX-29-TEX-I-1D462\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3722,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-29-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4225,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-29-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4728,0)\"><use data-c=\"28\" xlink:href=\"#MJX-29-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5117,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-29-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-29-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(852,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-29-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6984,0)\"><use data-c=\"29\" xlink:href=\"#MJX-29-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(6435.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-29-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-29-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(852,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-29-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8302.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-29-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8691.9,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-29-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9220.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-29-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9665.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-29-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10094.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-29-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10483.6,0)\"><use data-c=\"1D459\" xlink:href=\"#MJX-29-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10781.6,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-29-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(11266.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-29-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-29-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mfrac\" transform=\"translate(12180.1,0)\"><g data-mml-node=\"mrow\" transform=\"translate(749.2,710)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-29-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-29-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(852,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-29-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1867,0)\"><use data-c=\"28\" xlink:href=\"#MJX-29-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2256,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-29-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2785,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-29-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3229.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-29-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3658.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-29-TEX-N-29\"><\/use><\/g><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-710)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-29-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-29-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1327.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-29-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1716.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-29-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2245.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-29-TEX-N-29\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2634.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-29-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-29-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3899,0)\"><use data-c=\"28\" xlink:href=\"#MJX-29-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4288,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-29-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4717,0)\"><use data-c=\"29\" xlink:href=\"#MJX-29-TEX-N-29\"><\/use><\/g><\/g><rect width=\"5306\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -2442.1 1 4384.1\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mi>Y<\/mi><\/mrow><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mfrac><mrow><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Note that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.105ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 5792.5 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-263-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-263-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-263-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-263-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-263-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-263-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(852,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-263-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2144.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-263-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3200.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-263-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-263-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(4528,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-263-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-263-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mi>Y<\/mi><\/mrow><\/msub><mo>=<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{XY} = P_XP_Y<\/script><span> means that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-285-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-285-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y<\/script><span> are statistically independent. Thus, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.56ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3341.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-266-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-266-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-266-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-266-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-266-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-266-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-266-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(504,0)\"><use data-c=\"28\" xlink:href=\"#MJX-266-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(893,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-266-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1745,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-266-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2189.7,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-266-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2952.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-266-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I (X;Y)<\/script><span> measures the dependence of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-285-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-285-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y<\/script><span> .<\/span><\/p><\/blockquote><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n104\" cid=\"n104\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"34.725ex\" height=\"8.145ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -2050 15348.3 3600\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -3.507ex;\"><defs><path id=\"MJX-30-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-30-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-30-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-30-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 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columnalign=\"right center left\" columnspacing=\"0em 0.278em\" rowspacing=\"3pt\"><mtr><mtd><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><blockquote><h4 id='theorem-13'><span>Theorem 1.3<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n107\" cid=\"n107\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"25.028ex\" height=\"11.086ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -2700 11062.2 4900\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 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transform=\"translate(3170,0)\"><use data-c=\"29\" xlink:href=\"#MJX-31-TEX-N-29\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(3559,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-31-TEX-N-2264\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(4892.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-31-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-31-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-31-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2129,0)\"><use data-c=\"29\" xlink:href=\"#MJX-31-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-650)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" 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xlink:href=\"#MJX-31-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-31-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-31-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2040,0)\"><use data-c=\"29\" xlink:href=\"#MJX-31-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-1950)\"><g data-mml-node=\"mtd\" transform=\"translate(278,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-31-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-31-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-31-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2129,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-31-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2892,0)\"><use data-c=\"29\" xlink:href=\"#MJX-31-TEX-N-29\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(3559,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-31-TEX-N-2264\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(4892.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-31-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-31-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-31-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2129,0)\"><use data-c=\"29\" xlink:href=\"#MJX-31-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2740.2,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-31-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3740.4,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-31-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4628.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-31-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5017.4,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-31-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5780.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-31-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right center left\" columnspacing=\"0em 0.278em\" rowspacing=\"3pt\"><mtr><mtd><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2265<\/mo><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>with equality iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-285-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-285-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y<\/script><span> are statistically independent.<\/span><\/p><\/blockquote><p><em><span>Proof.<\/span><\/em><\/p><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-285-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-285-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y<\/script><span> are statistically independent, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"23.727ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 10487.2 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-273-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-273-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-273-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-273-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-273-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-273-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-273-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path id=\"MJX-273-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-273-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-273-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-273-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(852,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-273-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1867,0)\"><use data-c=\"28\" xlink:href=\"#MJX-273-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2256,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-273-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2785,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-273-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3229.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-273-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3658.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-273-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4325.4,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-273-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5381.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-273-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-273-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6708.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-273-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7097.7,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-273-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7626.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-273-TEX-N-29\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(8015.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-273-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-273-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9280.2,0)\"><use data-c=\"28\" xlink:href=\"#MJX-273-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9669.2,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-273-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10098.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-273-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{XY}(a,b)=P_X(a)P_Y(b)<\/script><span>, we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n111\" cid=\"n111\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"53.116ex\" height=\"68.477ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -15383.5 23477.4 30267\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -33.673ex;\"><defs><path id=\"MJX-32-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-32-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-32-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-32-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-32-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-32-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-32-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-32-TEX-LO-2211\" d=\"M60 948Q63 950 665 950H1267L1325 815Q1384 677 1388 669H1348L1341 683Q1320 724 1285 761Q1235 809 1174 838T1033 881T882 898T699 902H574H543H251L259 891Q722 258 724 252Q725 250 724 246Q721 243 460 -56L196 -356Q196 -357 407 -357Q459 -357 548 -357T676 -358Q812 -358 896 -353T1063 -332T1204 -283T1307 -196Q1328 -170 1348 -124H1388Q1388 -125 1381 -145T1356 -210T1325 -294L1267 -449L666 -450Q64 -450 61 -448Q55 -446 55 -439Q55 -437 57 -433L590 177Q590 178 557 222T452 366T322 544L56 909L55 924Q55 945 60 948Z\"><\/path><path id=\"MJX-32-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-32-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-32-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path id=\"MJX-32-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-32-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-32-TEX-I-1D462\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-32-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 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stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><mo>\u2212<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>+<\/mo><\/mtd><mtd><mi><\/mi><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>b<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><mo>\u2212<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>The inequality means that conditioning doesn&#39;t increase entropy, or colloquially that conditioning reduces entropy. Note, however, that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.04ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5321.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-274-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-274-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-274-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-274-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-274-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-274-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-274-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path id=\"MJX-274-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-274-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-274-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-274-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2129,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-274-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2407,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-274-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3447.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-274-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4503.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-274-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4932.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-274-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo>=<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H(X|Y=b)<\/script><span> can be larger than <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.697ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2518 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-291-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-291-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-291-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-291-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-291-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-291-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-291-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2129,0)\"><use data-c=\"29\" xlink:href=\"#MJX-291-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H(X)<\/script><span>.<\/span><\/p><h4 id='example-112'><span>Example 1.12<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n114\" cid=\"n114\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"33.258ex\" height=\"5.204ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1400 14699.9 2300\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.036ex;\"><defs><path id=\"MJX-33-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-33-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-33-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 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transform=\"translate(3041,0)\"><use data-c=\"29\" xlink:href=\"#MJX-33-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3652.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-33-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4652.4,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-33-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5540.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-33-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5929.4,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-33-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6692.4,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-33-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6970.4,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-33-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7822.4,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-33-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8545.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-33-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right center left\" columnspacing=\"0em 0.278em\" rowspacing=\"3pt\"><mtr><mtd><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>We know that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"39.25ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 17348.4 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-276-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 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id=\"MJX-276-TEX-N-6E\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-276-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-276-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"30\" xlink:href=\"#MJX-276-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(777.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-276-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1833.6,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-276-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2337.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-276-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2726.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-276-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3578.6,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-276-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4023.2,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-276-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4786.2,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-276-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5064.2,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-276-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5787.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-276-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6454,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-276-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7509.8,0)\"><use data-c=\"6D\" xlink:href=\"#MJX-276-TEX-N-6D\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-276-TEX-N-69\" transform=\"translate(833,0)\"><\/use><use data-c=\"6E\" xlink:href=\"#MJX-276-TEX-N-6E\" transform=\"translate(1111,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9176.8,0)\"><use data-c=\"28\" xlink:href=\"#MJX-276-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9565.8,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-276-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10453.8,0)\"><use data-c=\"28\" xlink:href=\"#MJX-276-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10842.8,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-276-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11694.8,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-276-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11972.8,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-276-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12695.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-276-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13084.8,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-276-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13529.4,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-276-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14417.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-276-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(14806.4,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-276-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15569.4,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-276-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(15847.4,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-276-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16570.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-276-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16959.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-276-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>\u2264<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">min<\/mo><mo stretchy=\"false\">(<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">0\\le I(X;Y|Z)\\le \\min(H(X|Z),H(Y|Z))<\/script><span>. <\/span><\/p><p><span>When <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.973ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6176.2 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-277-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-277-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-277-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-277-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-277-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-277-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-277-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-277-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-277-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-277-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-277-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(504,0)\"><use data-c=\"28\" xlink:href=\"#MJX-277-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(893,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-277-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1745,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-277-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2189.7,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-277-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2952.7,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-277-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3230.7,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-277-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3953.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-277-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4620.4,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-277-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(5676.2,0)\"><use data-c=\"30\" xlink:href=\"#MJX-277-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I(X;Y|Z)=0<\/script><span>, we say that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.821ex\" height=\"1.731ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 4782.9 765\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-278-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-278-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-278-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-278-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-278-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1074.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-278-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2074.4,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-278-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3019.7,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-278-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4019.9,0)\"><use data-c=\"1D44C\" 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transform=\"translate(0,-2204.7)\"><g data-mml-node=\"mtd\" transform=\"translate(4342.7,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(4342.7,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-34-TEX-N-3D\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(5676.4,0)\"><g data-mml-node=\"mn\"><use data-c=\"30\" xlink:href=\"#MJX-34-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(500,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-34-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right center left\" columnspacing=\"0em 0.278em\" rowspacing=\"3pt\"><mtr><mtd><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>P<\/mi><mi>Z<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mi>Z<\/mi><\/mrow><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>P<\/mi><mi>Z<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mrow data-mjx-texclass=\"ORD\"><mover><mo>=<\/mo><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><\/mtd><mtd><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Z<\/mi><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>P<\/mi><mi>Z<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mn>0<\/mn><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='example-113-chain-rule-for-mutual-information'><span>Example 1.13 Chain Rule for Mutual Information<\/span><\/h4><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16.963ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 7497.7 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-283-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-283-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-283-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-283-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-283-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-283-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-283-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(936.2,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-283-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1688.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-283-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2744,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-283-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-283-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(4008.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-283-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-283-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5273.1,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-283-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5717.8,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-283-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(6162.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-283-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-283-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>X<\/mi><mi>n<\/mi><\/msup><mo>=<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><msub><mi>X<\/mi><mn>2<\/mn><\/msub><mo>.<\/mo><mo>.<\/mo><msub><mi>X<\/mi><mi>n<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X^n=X_1X_2..X_n<\/script><span> is a string of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.357ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 600 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-479-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-479-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n<\/script><span> discrete random variables, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-285-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 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display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y<\/script><span> is some discrete random variables<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n120\" cid=\"n120\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"47.676ex\" height=\"23.361ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -5412.7 21072.9 10325.4\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -11.115ex;\"><defs><path id=\"MJX-35-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 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stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>n<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>n<\/mi><\/msup><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><mo stretchy=\"false\">(<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>;<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='example-114'><span>Example 1.14<\/span><\/h4><p><span>Using the chain rule for mutual information and conclusions of <\/span><em><span>Example 1.11<\/span><\/em><span>, we have <\/span><\/p><div contenteditable=\"false\" 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stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mi>Z<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='9-cross-entropy'><span>(9) Cross Entropy<\/span><\/h3><blockquote><p><span>The cross entropy of two distributions <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.392ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2383.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-286-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-286-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-286-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-286-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-286-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-286-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-286-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1327.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-286-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1716.5,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-286-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1994.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-286-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X (\\cdot)<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.25ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2320.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-292-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-292-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-292-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-292-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-292-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-292-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-292-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1264.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-292-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1653.5,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-292-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1931.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-292-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_Y (\\cdot)<\/script><span> with the same domain is defined as<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n128\" cid=\"n128\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 45.682ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"8.765ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -3.817ex; min-width: 45.682ex;\"><defs><path id=\"MJX-37-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-37-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-37-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-37-TEX-N-2225\" d=\"M133 736Q138 750 153 750Q164 750 170 739Q172 735 172 250T170 -239Q164 -250 152 -250Q144 -250 138 -244L137 -243Q133 -241 133 -179T132 250Q132 731 133 736ZM329 739Q334 750 346 750Q353 750 361 744L362 743Q366 741 366 679T367 250T367 -178T362 -243L361 -244Q355 -250 347 -250Q335 -250 329 -239Q327 -235 327 250T329 739Z\"><\/path><path id=\"MJX-37-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 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transform=\"translate(7818.2,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-37-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8096.2,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-37-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -2187.1 1 3874.1\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mi>X<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/munder><mo>\u2212<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mo>\u2212<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='theorem-14'><span>Theorem 1.4<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n130\" cid=\"n130\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 39.554ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 39.554ex;\"><defs><path id=\"MJX-38-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 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transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-38-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(852,0)\"><use data-c=\"28\" xlink:href=\"#MJX-38-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1241,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-38-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-38-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2846.2,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-38-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3624,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-38-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-38-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4888.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-38-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5555.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-38-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6611.1,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-38-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7499.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-38-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7888.1,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-38-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8740.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-38-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9351.3,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-38-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10351.5,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-38-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11179.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-38-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(11568.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-38-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-38-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(13173.8,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-38-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(13951.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-38-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-38-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(15216.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-38-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15605.1,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-38-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>X<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>with equality iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.795ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6539.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-288-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-288-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-288-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-288-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-288-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-288-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-288-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-288-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-288-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1327.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-288-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1716.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-288-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2245.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-288-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2912.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-288-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3968,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-288-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-288-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5232.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-288-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5621.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-288-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6150.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-288-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X(a)=P_Y(a)<\/script><span> for any <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.89ex\" height=\"1.636ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2603.6 723\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-289-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-289-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-289-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-289-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(806.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-289-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1751.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-289-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>\u2208<\/mo><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a\\in X<\/script><span>.<\/span><\/p><\/blockquote><p><span>Theorem 1.4 gives a useful tool to <\/span><strong><span>upper bound entropy<\/span><\/strong><span>: every choice of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.25ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2320.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-292-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-292-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-292-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-292-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-292-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-292-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-292-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1264.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-292-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1653.5,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-292-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1931.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-292-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_Y (\\cdot)<\/script><span> gives an upper bound on <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.697ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2518 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-291-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-291-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-291-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-291-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-291-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-291-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-291-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2129,0)\"><use data-c=\"29\" xlink:href=\"#MJX-291-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H(X)<\/script><span>. To simplify the computation, one usually chooses <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.25ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2320.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-292-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-292-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-292-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-292-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-292-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-292-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-292-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1264.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-292-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1653.5,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-292-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1931.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-292-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_Y (\\cdot)<\/script><span> to have an exponential form, such as a Gaussian distribution for continuous random variables.<\/span><\/p><h3 id='10-inequalities'><span>(10) Inequalities<\/span><\/h3><blockquote><h4 id='theorem-15-log-sum-inequality'><span>Theorem 1.5 Log-sum Inequality<\/span><\/h4><p><span>Let <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.925ex\" height=\"2.563ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -789.6 5712.9 1132.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.777ex;\"><defs><path id=\"MJX-293-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-293-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-293-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-293-TEX-SO-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path id=\"MJX-293-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-293-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-293-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-293-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-293-TEX-I-1D44E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1347.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-293-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"munderover\" transform=\"translate(2403.6,0)\"><g data-mml-node=\"mo\"><use data-c=\"2211\" xlink:href=\"#MJX-293-TEX-SO-2211\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1089,477.1) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-293-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(1089,-285.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-293-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(345,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-293-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1123,0)\"><use data-c=\"31\" xlink:href=\"#MJX-293-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(4856.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-293-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(562,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-293-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>S<\/mi><mi>a<\/mi><\/msub><mo>=<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><msub><mi>a<\/mi><mi>i<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">S_a=\\sum_{i=1}^na_i<\/script><span>, <\/span><mjx-container 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xlink:href=\"#MJX-296-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(4977,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-296-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-296-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>b<\/mi><mi>i<\/mi><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msub><mi>S<\/mi><mi>b<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_Y(i)=b_i\/S_b<\/script><span>, we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n137\" cid=\"n137\" 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xlink:href=\"#MJX-39-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(16934.9,0)\"><g data-mml-node=\"msub\" transform=\"translate(220,676)\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-39-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-39-TEX-I-1D44E\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(255.4,-686)\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-39-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-39-TEX-I-1D44F\"><\/use><\/g><\/g><rect width=\"1270.1\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(18722.7,0)\"><use data-c=\"2265\" xlink:href=\"#MJX-39-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(19778.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-39-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-39-TEX-I-1D44E\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(21015.2,0)\"><use data-c=\"6C\" xlink:href=\"#MJX-39-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-39-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-39-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(22293.2,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-39-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(22459.9,0)\"><g data-mml-node=\"msub\" transform=\"translate(220,676)\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-39-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-39-TEX-I-1D44E\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(255.4,-686)\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-39-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-39-TEX-I-1D44F\"><\/use><\/g><\/g><rect width=\"1270.1\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1654.2 1 2808.5\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,91.7)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><msub><mi>a<\/mi><mi>i<\/mi><\/msub><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><msub><mi>a<\/mi><mi>i<\/mi><\/msub><msub><mi>b<\/mi><mi>i<\/mi><\/msub><\/mfrac><mo>=<\/mo><msub><mi>S<\/mi><mi>a<\/mi><\/msub><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>S<\/mi><mi>a<\/mi><\/msub><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><msub><mi>S<\/mi><mi>a<\/mi><\/msub><msub><mi>S<\/mi><mi>b<\/mi><\/msub><\/mfrac><mo>\u2265<\/mo><msub><mi>S<\/mi><mi>a<\/mi><\/msub><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><msub><mi>S<\/mi><mi>a<\/mi><\/msub><msub><mi>S<\/mi><mi>b<\/mi><\/msub><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>with equality iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.608ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6014.9 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-297-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 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data-c=\"1D446\" xlink:href=\"#MJX-297-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-297-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>a<\/mi><mi>i<\/mi><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msub><mi>b<\/mi><mi>i<\/mi><\/msub><mo>=<\/mo><msub><mi>S<\/mi><mi>a<\/mi><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msub><mi>S<\/mi><mi>b<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a_i\/b_i=S_a\/S_b<\/script><span> for all <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.781ex\" height=\"1.52ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -661 345 672\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-456-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-456-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>i<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">i<\/script><span>.<\/span><\/p><\/blockquote><p><em><span>Proof.<\/span><\/em><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n140\" cid=\"n140\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"49.596ex\" height=\"19.099ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -4470.8 21921.2 8441.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -8.984ex;\"><defs><path id=\"MJX-40-TEX-LO-2211\" d=\"M60 948Q63 950 665 950H1267L1325 815Q1384 677 1388 669H1348L1341 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display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right center left\" columnspacing=\"0em 0.278em\" rowspacing=\"3pt\"><mtr><mtd><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><msub><mi>a<\/mi><mi>i<\/mi><\/msub><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><msub><mi>a<\/mi><mi>i<\/mi><\/msub><msub><mi>b<\/mi><mi>i<\/mi><\/msub><\/mfrac><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><msub><mi>a<\/mi><mi>i<\/mi><\/msub><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><mrow><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>\u22c5<\/mo><mfrac><msub><mi>S<\/mi><mi>a<\/mi><\/msub><msub><mi>S<\/mi><mi>b<\/mi><\/msub><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><msub><mi>S<\/mi><mi>a<\/mi><\/msub><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><mrow><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>+<\/mo><msub><mi>S<\/mi><mi>a<\/mi><\/msub><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><msub><mi>S<\/mi><mi>a<\/mi><\/msub><msub><mi>S<\/mi><mi>b<\/mi><\/msub><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><msub><mi>S<\/mi><mi>a<\/mi><\/msub><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>S<\/mi><mi>a<\/mi><\/msub><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><msub><mi>S<\/mi><mi>a<\/mi><\/msub><msub><mi>S<\/mi><mi>b<\/mi><\/msub><\/mfrac><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='example-115'><span>Example 1.15<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n142\" cid=\"n142\" 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data-c=\"6F\" xlink:href=\"#MJX-41-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-41-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15897.1,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-41-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(16063.8,0)\"><g data-mml-node=\"mrow\" transform=\"translate(220,676)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-41-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(562,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-41-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1187.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-41-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2188,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-41-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(562,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-41-TEX-N-32\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mrow\" transform=\"translate(320,-686)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-41-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(462,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-41-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1087.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-41-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2088,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-41-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(462,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-41-TEX-N-32\"><\/use><\/g><\/g><\/g><rect width=\"3353.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1297.5 1 2095\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,38.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><msub><mi>a<\/mi><mn>1<\/mn><\/msub><msub><mi>b<\/mi><mn>1<\/mn><\/msub><\/mfrac><mo>+<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><msub><mi>a<\/mi><mn>2<\/mn><\/msub><msub><mi>b<\/mi><mn>2<\/mn><\/msub><\/mfrac><mo>\u2265<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mfrac><mrow><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><\/mrow><mrow><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><\/mrow><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>with equality iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"33.92ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 14992.4 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-299-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 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transform=\"translate(11649.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-299-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(462,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-299-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(12737.7,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-299-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(13737.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-299-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(462,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-299-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(14603.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-299-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>a<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>a<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a_1\/b_1=a_2\/b_2=(a_1+a_2)(b_1+b_2)<\/script><span>.<\/span><\/p><blockquote><h4 id='theorem-16-data-processing-inequalities'><span>Theorem 1.6 Data Processing Inequalities<\/span><\/h4><p><span>Any processing decreases the mutual information:<\/span><\/p><p><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.821ex\" height=\"1.731ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 4782.9 765\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-300-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-300-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-300-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-300-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-300-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1074.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-300-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2074.4,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-300-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3059.7,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-300-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4059.9,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-300-TEX-I-1D44D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><mo>\u2212<\/mo><mi>Y<\/mi><mo>\u2212<\/mo><mi>Z<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X-Y-Z<\/script><span> forms a Markov chain, we have <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"18.047ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7976.9 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-301-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-301-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-301-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-301-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-301-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-301-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-301-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-301-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-301-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(504,0)\"><use data-c=\"28\" xlink:href=\"#MJX-301-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(893,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-301-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1745,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-301-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2189.7,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-301-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2912.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-301-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3579.4,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-301-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4635.2,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-301-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5139.2,0)\"><use data-c=\"28\" xlink:href=\"#MJX-301-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5528.2,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-301-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6380.2,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-301-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6824.9,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-301-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7587.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-301-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I(X;Z)\\le I(X;Y)<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.755ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7847.9 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-302-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-302-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-302-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 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60Z\"><\/path><path id=\"MJX-302-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-302-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-302-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 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637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-302-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(504,0)\"><use data-c=\"28\" xlink:href=\"#MJX-302-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(893,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-302-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1745,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-302-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2189.7,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-302-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2912.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-302-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3579.4,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-302-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4635.2,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-302-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5139.2,0)\"><use data-c=\"28\" xlink:href=\"#MJX-302-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5528.2,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-302-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6291.2,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-302-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6735.9,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-302-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7458.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-302-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo>;<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I(X;Z)\\le I(Y;Z)<\/script><span>.<\/span><\/p><p><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.302ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1017.6 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-303-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-303-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-303-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(614,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-303-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>Y<\/mi><mn>1<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y_1<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.302ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1017.6 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-304-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-304-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-304-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(614,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-304-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>Y<\/mi><mn>2<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y_2<\/script><span> are outputs of a channel <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.058ex\" height=\"2.495ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3119.6 1102.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-305-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-305-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-305-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-305-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-305-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-305-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-305-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-305-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-176.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-305-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(763,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-305-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1041,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-305-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2063.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-305-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2452.6,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-305-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2730.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-305-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{Y|X}(\\cdot)<\/script><span> with inputs <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.861ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1264.6 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-306-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-306-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-306-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-306-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X_1<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.861ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1264.6 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-307-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-307-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-307-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-307-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>X<\/mi><mn>2<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X_2<\/script><span>, we have <\/span><mjx-container 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data-mml-node=\"msub\" transform=\"translate(9220.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-308-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-308-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-308-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10839.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-308-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>Y<\/mi><mn>1<\/mn><\/msub><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>Y<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D(P_{Y_1}|P_{Y_2}) \\le D(P_{X_1}|P_{X_2}) <\/script><span>.<\/span><\/p><\/blockquote><p><em><span>Proof.<\/span><\/em><\/p><p><span>Using conclusions of <\/span><em><span>Example 1.13<\/span><\/em><span>,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n151\" cid=\"n151\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div 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transform=\"translate(15101.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-42-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(15490.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-42-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16342.6,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-42-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(16787.2,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-42-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17510.2,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-42-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(17788.2,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-42-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18551.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-42-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(19218,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-42-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(20273.8,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-42-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(20777.8,0)\"><use data-c=\"28\" xlink:href=\"#MJX-42-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(21166.8,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-42-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(22018.8,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-42-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(22463.4,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-42-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(23226.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-42-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(23615.4,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-42-TEX-N-2C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-750)\"><g data-mml-node=\"mtd\" transform=\"translate(129,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-42-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(504,0)\"><use data-c=\"28\" xlink:href=\"#MJX-42-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(893,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-42-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1745,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-42-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2189.7,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-42-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2912.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-42-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3579.4,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-42-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4635.2,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-42-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5139.2,0)\"><use data-c=\"28\" xlink:href=\"#MJX-42-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5528.2,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-42-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6380.2,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-42-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7143.2,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-42-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7587.9,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-42-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8310.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-42-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8977.7,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-42-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10033.4,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-42-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" 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transform=\"translate(15361.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-42-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16213.6,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-42-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(16658.2,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-42-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17381.2,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-42-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(17659.2,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-42-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18422.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-42-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(19089,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-42-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(20144.8,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-42-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(20648.8,0)\"><use data-c=\"28\" xlink:href=\"#MJX-42-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(21037.8,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-42-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(21800.8,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-42-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(22245.4,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-42-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(22968.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-42-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(23357.4,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-42-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1500 1 2500\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Z<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><\/mtd><\/mtr><mtr><mtd><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mi>Y<\/mi><mo>;<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo>;<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Z<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo>;<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Using the chain rule of information divergence, we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block 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42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-43-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-43-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 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data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><\/mrow><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>Y<\/mi><mn>1<\/mn><\/msub><\/mrow><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>Y<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>Y<\/mi><mn>1<\/mn><\/msub><\/mrow><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>Y<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>Y<\/mi><mn>1<\/mn><\/msub><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>which gives<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n155\" cid=\"n155\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 33.172ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.276ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.572ex; min-width: 33.172ex;\"><defs><path id=\"MJX-44-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-44-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-44-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-44-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-44-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-44-TEX-N-2225\" d=\"M133 736Q138 750 153 750Q164 750 170 739Q172 735 172 250T170 -239Q164 -250 152 -250Q144 -250 138 -244L137 -243Q133 -241 133 -179T132 250Q132 731 133 736ZM329 739Q334 750 346 750Q353 750 361 744L362 743Q366 741 366 679T367 250T367 -178T362 -243L361 -244Q355 -250 347 -250Q335 -250 329 -239Q327 -235 327 250T329 739Z\"><\/path><path id=\"MJX-44-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-44-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-44-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-44-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-44-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -753)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 753) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"6531 -753 1 1006.1\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,3)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-44-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(828,0)\"><use data-c=\"28\" xlink:href=\"#MJX-44-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1217,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-44-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-44-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(614,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-44-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2939.3,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-44-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3717.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-44-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-44-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(614,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-44-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5161.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-44-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5828.4,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-44-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6884.1,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-44-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7712.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-44-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(8101.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-44-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-44-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-44-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9998.1,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-44-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(10775.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-44-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-44-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-44-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(12395.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-44-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12784.1,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-44-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -753 1 1006.1\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,3)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>Y<\/mi><mn>1<\/mn><\/msub><\/mrow><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>Y<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><\/mrow><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><blockquote><h4 id='theorem-19-fanos-inequality'><span>Theorem 1.9 Fano&#39;s Inequality<\/span><\/h4><p><span>Suppose both <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"2.378ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1051 852 1051\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-322-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-322-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-322-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-322-TEX-N-5E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\hat{X}<\/script><span> take on values in the same alphabet, let <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16.253ex\" height=\"2.943ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1051 7183.6 1301\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-311-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-311-TEX-I-1D452\" d=\"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z\"><\/path><path id=\"MJX-311-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-311-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-311-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-311-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-311-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><path id=\"MJX-311-TEX-N-2260\" d=\"M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z\"><\/path><path id=\"MJX-311-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-311-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D452\" xlink:href=\"#MJX-311-TEX-I-1D452\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1332.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-311-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2388.1,0)\"><use data-c=\"1D443\" xlink:href=\"#MJX-311-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3139.1,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-311-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3590.1,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-311-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3868.1,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-311-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-311-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4997.8,0)\"><use data-c=\"2260\" xlink:href=\"#MJX-311-TEX-N-2260\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6053.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-311-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6905.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-311-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mo>=<\/mo><mi>P<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>\u2260<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_e=Pr[\\hat{X}\\neq X]<\/script><span>. We have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n159\" cid=\"n159\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 40.154ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.943ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.906ex; min-width: 40.154ex;\"><defs><path id=\"MJX-45-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-45-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-45-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-45-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-45-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><path id=\"MJX-45-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-45-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-45-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-45-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-45-TEX-I-1D452\" d=\"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z\"><\/path><path id=\"MJX-45-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-45-TEX-N-6C\" d=\"M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z\"><\/path><path id=\"MJX-45-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-45-TEX-N-67\" d=\"M329 409Q373 453 429 453Q459 453 472 434T485 396Q485 382 476 371T449 360Q416 360 412 390Q410 404 415 411Q415 412 416 414V415Q388 412 363 393Q355 388 355 386Q355 385 359 381T368 369T379 351T388 325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z\"><\/path><path id=\"MJX-45-TEX-N-2061\" d=\"\"><\/path><path id=\"MJX-45-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-45-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g 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xlink:href=\"#MJX-45-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(6638.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-45-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D452\" xlink:href=\"#MJX-45-TEX-I-1D452\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7692.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-45-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8303.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-45-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(9304.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-45-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D452\" xlink:href=\"#MJX-45-TEX-I-1D452\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(10525.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" 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xlink:href=\"#MJX-45-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14259,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-45-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(15259.2,0)\"><use data-c=\"31\" xlink:href=\"#MJX-45-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15759.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-45-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -900.5 1 1301\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-150.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>with equlity iff. for all <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.197ex\" height=\"1.02ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -441 529 451\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-460-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-460-TEX-I-1D44E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.971ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 429 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-313-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-313-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">b<\/script><span>, we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n161\" cid=\"n161\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container 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46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-46-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path id=\"MJX-46-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 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data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-46-TEX-N-5E\"><\/use><\/g><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2126.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-46-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2515.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-46-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3044.5,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-46-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3322.5,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-46-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3751.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-46-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4418.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-46-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(5474,0)\"><g data-mml-node=\"mo\"><use data-c=\"23A7\" xlink:href=\"#MJX-46-TEX-S4-23A7\" transform=\"translate(0,1160)\"><\/use><use data-c=\"23A9\" xlink:href=\"#MJX-46-TEX-S4-23A9\" transform=\"translate(0,-660)\"><\/use><use data-c=\"23A8\" xlink:href=\"#MJX-46-TEX-S4-23A8\" transform=\"translate(0,0)\"><\/use><svg width=\"889\" height=\"190\" y=\"1060\" x=\"0\" viewBox=\"0 33.5 889 190\"><use data-c=\"23AA\" xlink:href=\"#MJX-46-TEX-S4-23AA\" transform=\"scale(1,0.934)\"><\/use><\/svg><svg width=\"889\" height=\"190\" y=\"-750\" x=\"0\" viewBox=\"0 33.5 889 190\"><use data-c=\"23AA\" xlink:href=\"#MJX-46-TEX-S4-23AA\" transform=\"scale(1,0.934)\"><\/use><\/svg><\/g><g data-mml-node=\"mtable\" transform=\"translate(889,0)\"><g data-mml-node=\"mtr\" transform=\"translate(0,1309)\"><g data-mml-node=\"mtd\"><\/g><g data-mml-node=\"mtd\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-46-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(722.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-46-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1722.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-46-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D452\" xlink:href=\"#MJX-46-TEX-I-1D452\"><\/use><\/g><\/g><g data-mml-node=\"mtext\" transform=\"translate(2777,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-46-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(3027,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-46-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(3277,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-46-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(3527,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-46-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(3777,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-46-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(4027,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-46-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(4277,0)\"><use data-c=\"69\" xlink:href=\"#MJX-46-TEX-N-69\"><\/use><use data-c=\"66\" xlink:href=\"#MJX-46-TEX-N-66\" transform=\"translate(278,0)\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(4861,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-46-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5111,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-46-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5817.7,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-46-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6873.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-46-TEX-I-1D44E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-600)\"><g data-mml-node=\"mtd\"><\/g><g data-mml-node=\"mtd\"><g data-mml-node=\"mfrac\"><g data-mml-node=\"msub\" transform=\"translate(1258,676)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-46-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D452\" xlink:href=\"#MJX-46-TEX-I-1D452\"><\/use><\/g><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-709.5)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-46-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(278,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-46-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1130,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-46-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1630.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-46-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2630.4,0)\"><use data-c=\"31\" xlink:href=\"#MJX-46-TEX-N-31\"><\/use><\/g><\/g><rect width=\"3330.4\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mtext\" transform=\"translate(3570.4,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-46-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(3820.4,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-46-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(4070.4,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-46-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(4320.4,0)\"><use data-c=\"69\" xlink:href=\"#MJX-46-TEX-N-69\"><\/use><use data-c=\"66\" xlink:href=\"#MJX-46-TEX-N-66\" transform=\"translate(278,0)\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(4904.4,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-46-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5154.4,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-46-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5861.2,0)\"><use data-c=\"2260\" xlink:href=\"#MJX-46-TEX-N-2260\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6917,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-46-TEX-I-1D44E\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8335,0) translate(0 250)\"><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -2059 1 3618\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">{<\/mo><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><\/mtd><mtd><mn>1<\/mn><mo>\u2212<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>if<\/mtext><mtext>&nbsp;<\/mtext><mi>b<\/mi><mo>=<\/mo><mi>a<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mfrac><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>if<\/mtext><mtext>&nbsp;<\/mtext><mi>b<\/mi><mo>\u2260<\/mo><mi>a<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><p><em><span>Proof.<\/span><\/em><\/p><p><span>Let <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.18ex\" height=\"2.943ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1051 6267.6 1301\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-314-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-314-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-314-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-314-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-314-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><path id=\"MJX-314-TEX-N-2260\" d=\"M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z\"><\/path><path id=\"MJX-314-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-314-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1041.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-314-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2097.6,0)\"><g data-mml-node=\"mn\"><text data-variant=\"double-struck\" transform=\"scale(1,-1)\" font-size=\"884px\">????<\/text><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2452.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-314-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2841.1,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-314-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-314-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3970.9,0)\"><use data-c=\"2260\" xlink:href=\"#MJX-314-TEX-N-2260\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5026.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-314-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5878.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-314-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"double-struck\">1<\/mn><\/mrow><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>\u2260<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E=\\mathbb{1} (\\hat{X}\\neq X)<\/script><span>. Using the chain rule to expand <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.982ex\" height=\"2.943ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1051 4412 1301\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-315-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-315-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-315-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-315-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-315-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-315-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><path id=\"MJX-315-TEX-N-29\" d=\"M60 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data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3171,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-315-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-315-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4023,0)\"><use data-c=\"29\" xlink:href=\"#MJX-315-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H(EX|\\hat{X})<\/script><span> in two ways<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n164\" cid=\"n164\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"34ex\" height=\"7.584ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1926.1 15028 3352.1\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -3.226ex;\"><defs><path id=\"MJX-47-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 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43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-47-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 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d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\"><g data-mml-node=\"mtr\" transform=\"translate(0,875.1)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-47-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-47-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-47-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2041,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-47-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2893,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-47-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3171,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-47-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-47-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4023,0)\"><use data-c=\"29\" xlink:href=\"#MJX-47-TEX-N-29\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(4412,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-47-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1333.6,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-47-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2221.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-47-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2610.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-47-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3462.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-47-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3740.6,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-47-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-47-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4592.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-47-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5203.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-47-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6204,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-47-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7092,0)\"><use data-c=\"28\" xlink:href=\"#MJX-47-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7481,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-47-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8245,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-47-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8523,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-47-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(9375,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-47-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-47-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10227,0)\"><use data-c=\"29\" xlink:href=\"#MJX-47-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-1176.1)\"><g data-mml-node=\"mtd\" transform=\"translate(4412,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(4412,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mover\" transform=\"translate(277.8,0)\"><g data-mml-node=\"mo\" transform=\"translate(73.1,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-47-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(0,870.8) scale(0.707)\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-47-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-47-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(918,0)\"><use data-c=\"29\" xlink:href=\"#MJX-47-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1479.7,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-47-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2367.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-47-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2756.7,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-47-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3608.7,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-47-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3886.7,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-47-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-47-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4738.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-47-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mover><mo>=<\/mo><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mover><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.957ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1307 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-520-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-520-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-520-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-520-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-520-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(918,0)\"><use data-c=\"29\" xlink:href=\"#MJX-520-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(a)<\/script><span> follows because <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.855ex\" height=\"2.378ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1051 1704 1051\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-317-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-317-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-317-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(852,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-317-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-317-TEX-N-5E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X\\hat{X}<\/script><span> essentially determines <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.729ex\" height=\"1.538ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -680 764 680\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-318-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-318-TEX-I-1D438\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E<\/script><span>.<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n166\" cid=\"n166\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"75.982ex\" height=\"25.471ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -5879.1 33584 11258.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -12.17ex;\"><defs><path id=\"MJX-48-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 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xlink:href=\"#MJX-48-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10429.5,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-48-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10874.2,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-48-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11916,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-48-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(12971.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-48-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13471.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-48-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-2529.1)\"><g data-mml-node=\"mtd\" transform=\"translate(4412,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(4412,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mover\" transform=\"translate(277.8,0)\"><g data-mml-node=\"mo\" transform=\"translate(39.2,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-48-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(0,923.8) scale(0.707)\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-48-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D450\" xlink:href=\"#MJX-48-TEX-I-1D450\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(822,0)\"><use data-c=\"29\" xlink:href=\"#MJX-48-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1411.9,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-48-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2299.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-48-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2688.9,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-48-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3452.9,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-48-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3730.9,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-48-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-48-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4582.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-48-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5194.1,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-48-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(6194.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-48-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D452\" xlink:href=\"#MJX-48-TEX-I-1D452\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(7415.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-48-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-48-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-48-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-48-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9130,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-48-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9130,0)\"><use data-c=\"28\" xlink:href=\"#MJX-48-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9519,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-48-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9797,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-48-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10649,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-48-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11149.3,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-48-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(12149.5,0)\"><use data-c=\"31\" xlink:href=\"#MJX-48-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12649.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-48-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-3829.1)\"><g data-mml-node=\"mtd\" transform=\"translate(4412,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(4412,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-48-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1333.6,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-48-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2221.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-48-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2610.6,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-48-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3374.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-48-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3985.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-48-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4986,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-48-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D452\" xlink:href=\"#MJX-48-TEX-I-1D452\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(6207.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-48-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-48-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-48-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-48-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7921.7,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-48-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7921.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-48-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8310.7,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-48-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8588.7,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-48-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9440.7,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-48-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9941,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-48-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(10941.2,0)\"><use data-c=\"31\" xlink:href=\"#MJX-48-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11441.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-48-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-5129.1)\"><g data-mml-node=\"mtd\" transform=\"translate(4412,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(4412,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-48-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1333.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-48-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(864,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-48-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2601.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-48-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2990.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-48-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D452\" xlink:href=\"#MJX-48-TEX-I-1D452\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4044.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-48-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4655.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-48-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5656.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-48-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D452\" xlink:href=\"#MJX-48-TEX-I-1D452\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(6877.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-48-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-48-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-48-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1311,-241.4) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-48-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8591.8,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-48-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8591.8,0)\"><use data-c=\"28\" xlink:href=\"#MJX-48-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8980.8,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-48-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9258.8,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-48-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10110.8,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-48-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10611,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-48-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(11611.2,0)\"><use data-c=\"31\" xlink:href=\"#MJX-48-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12111.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-48-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mi>E<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">[<\/mo><mi>E<\/mi><mo>=<\/mo><mn>0<\/mn><mo stretchy=\"false\">]<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>,<\/mo><mi>E<\/mi><mo>=<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">[<\/mo><mi>E<\/mi><mo>=<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>,<\/mo><mi>E<\/mi><mo>=<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mover><mo>=<\/mo><mrow><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mover><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">[<\/mo><mi>E<\/mi><mo>=<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>,<\/mo><mi>E<\/mi><mo>=<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>,<\/mo><mi>E<\/mi><mo>=<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mover><mo>\u2264<\/mo><mrow><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mover><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.731ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1207 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-472-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-472-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path id=\"MJX-472-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-472-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-472-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(818,0)\"><use data-c=\"29\" xlink:href=\"#MJX-472-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(b)<\/script><span> follows because <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.877ex\" height=\"1.724ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -680 2597.6 762\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-320-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-320-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-320-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-320-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1041.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-320-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2097.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-320-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo>=<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E=0<\/script><span> means that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.872ex\" height=\"2.563ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1051 3037.6 1133\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-321-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-321-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-321-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-321-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1129.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-321-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2185.6,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-321-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-321-TEX-N-5E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X=\\hat{X}<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"2.378ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1051 852 1051\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-322-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-322-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-322-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-322-TEX-N-5E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\hat{X}<\/script><span> determines <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span>. And <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.74ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1211 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-324-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-324-TEX-I-1D450\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path id=\"MJX-324-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-324-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D450\" xlink:href=\"#MJX-324-TEX-I-1D450\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(822,0)\"><use data-c=\"29\" xlink:href=\"#MJX-324-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(c)<\/script><span> follows because <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.877ex\" height=\"1.724ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -680 2597.6 762\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-325-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-325-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-325-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-325-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1041.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-325-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2097.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-325-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo>=<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E=1<\/script><span> means that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.872ex\" height=\"2.864ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1051 3037.6 1266\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.486ex;\"><defs><path id=\"MJX-326-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 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data-c=\"1D44B\" xlink:href=\"#MJX-326-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(515.3,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-326-TEX-N-5E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><mo>\u2260<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X\\neq\\hat{X}<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" 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xlink:href=\"#MJX-49-TEX-I-1D444\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(824,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-49-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(21328.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-49-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(21717.2,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-49-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>\u03bb<\/mi><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>Q<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>\u03bb<\/mi><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>Q<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><blockquote><h4 id='theorem-111-convexity-of-information-divergence'><span>Theorem 1.11 Convexity of Information Divergence<\/span><\/h4><p><span>For any <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" 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xlink:href=\"#MJX-50-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15202.3,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-50-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(16202.5,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-50-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16785.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-50-TEX-N-29\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(17174.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D444\" xlink:href=\"#MJX-50-TEX-I-1D444\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(824,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-50-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(18588.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-50-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18977.1,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-50-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1500 1 2500\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mi>\u03bb<\/mi><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>Q<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>Q<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mo>\u2265<\/mo><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><mi>\u03bb<\/mi><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>Q<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><mi>\u03bb<\/mi><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>Q<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><blockquote><h4 id='theorem-112-mutual-information'><span>Theorem 1.12 Mutual Information<\/span><\/h4><p><span>We write <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.56ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3341.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-331-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-331-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-331-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-331-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-331-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-331-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-331-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(504,0)\"><use data-c=\"28\" xlink:href=\"#MJX-331-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(893,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-331-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1745,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-331-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2189.7,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-331-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2952.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-331-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I(X;Y)<\/script><span> as <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.578ex\" height=\"2.495ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5117.7 1102.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-337-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-337-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-337-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-337-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-337-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-337-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-337-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-337-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-337-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(504,0)\"><use data-c=\"28\" xlink:href=\"#MJX-337-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(893,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-337-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-337-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2220.5,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-337-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2665.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-337-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-176.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-337-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(763,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-337-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1041,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-337-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4728.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-337-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>;<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I(P_X;P_{Y|X})<\/script><span>. The function <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.578ex\" height=\"2.495ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5117.7 1102.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-337-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-337-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-337-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-337-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-337-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-337-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-337-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-337-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-337-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(504,0)\"><use data-c=\"28\" xlink:href=\"#MJX-337-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(893,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-337-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-337-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2220.5,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-337-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2665.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-337-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-176.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-337-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(763,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-337-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1041,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-337-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4728.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-337-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>;<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I(P_X;P_{Y|X})<\/script><span> is concave in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.003ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1327.5 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-538-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-538-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-538-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-538-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X<\/script><span>, i.e., the input distribution if <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.669ex\" height=\"2.343ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2063.6 1035.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-338-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-338-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 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data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>\u03bb<\/mi><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>;<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>Q<\/mi><mi>X<\/mi><\/msub><mo>;<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>\u03bb<\/mi><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>Q<\/mi><mi>X<\/mi><\/msub><mo>,<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>for any <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.616ex\" height=\"1.661ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 4250.1 734\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-340-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-340-TEX-N-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><path id=\"MJX-340-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path id=\"MJX-340-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"30\" xlink:href=\"#MJX-340-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(777.8,0)\"><use data-c=\"3C\" xlink:href=\"#MJX-340-TEX-N-3C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1833.6,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-340-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2694.3,0)\"><use data-c=\"3C\" xlink:href=\"#MJX-340-TEX-N-3C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3750.1,0)\"><use data-c=\"31\" xlink:href=\"#MJX-340-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>&lt;<\/mo><mi>\u03bb<\/mi><mo>&lt;<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">0<\\lambda <1<\/script><span>.<\/span><\/p><p><span>The function <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.578ex\" height=\"2.495ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5117.7 1102.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-337-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-337-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-337-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-337-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-337-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-337-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-337-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-337-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-337-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(504,0)\"><use data-c=\"28\" xlink:href=\"#MJX-337-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(893,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-337-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-337-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2220.5,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-337-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2665.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-337-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-176.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-337-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(763,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-337-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1041,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-337-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4728.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-337-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>;<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I(P_X;P_{Y|X})<\/script><span> is convex in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.669ex\" height=\"2.343ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2063.6 1035.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-338-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-338-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-338-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-338-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-338-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-176.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-338-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(763,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-338-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1041,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-338-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{Y|X}<\/script><span>, i.e., the channel if <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.003ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1327.5 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-538-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 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stretchy=\"false\">|<\/mo><mi>X<\/mi><\/mrow><\/msub><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>Q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>for any <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.616ex\" height=\"1.661ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 4250.1 734\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-340-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-340-TEX-N-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><path id=\"MJX-340-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path id=\"MJX-340-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"30\" xlink:href=\"#MJX-340-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(777.8,0)\"><use data-c=\"3C\" xlink:href=\"#MJX-340-TEX-N-3C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1833.6,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-340-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2694.3,0)\"><use data-c=\"3C\" xlink:href=\"#MJX-340-TEX-N-3C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3750.1,0)\"><use data-c=\"31\" xlink:href=\"#MJX-340-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>&lt;<\/mo><mi>\u03bb<\/mi><mo>&lt;<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">0<\\lambda <1<\/script><span>.<\/span><\/p><\/blockquote><blockquote><h4 id='theorem-113-linearity-and-convexity-of-cross-entropy'><span>Theorem 1.13 Linearity and Convexity of Cross Entropy<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n187\" cid=\"n187\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 68.257ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.656ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" 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xlink:href=\"#MJX-53-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(28180.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-53-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-750)\"><g data-mml-node=\"mtd\" transform=\"translate(31.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-53-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(583,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-53-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1435,0)\"><use data-c=\"28\" xlink:href=\"#MJX-53-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1824,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-53-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-53-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3429.2,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-53-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4207,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-53-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-53-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5471.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-53-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6082.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-53-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7083,0)\"><use data-c=\"28\" xlink:href=\"#MJX-53-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(7472,0)\"><use data-c=\"31\" xlink:href=\"#MJX-53-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8194.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-53-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9194.4,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-53-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9777.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-53-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10166.4,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-53-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11018.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-53-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(11407.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-53-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-53-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(13012.7,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-53-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"msub\" 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data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-53-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(19772.7,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-53-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(20550.5,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-53-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(21133.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-53-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-53-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(22620.3,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-53-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(23620.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-53-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(24009.5,0)\"><use data-c=\"31\" xlink:href=\"#MJX-53-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(24731.7,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-53-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(25731.9,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-53-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(26314.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-53-TEX-N-29\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(26703.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D444\" xlink:href=\"#MJX-53-TEX-I-1D444\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(824,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-53-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(28117.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-53-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1500 1 2500\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mi>\u03bb<\/mi><mi>X<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><mi>X<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>Q<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>X<\/mi><mo stretchy=\"false\">(<\/mo><mi>\u03bb<\/mi><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>Q<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mi>\u03bb<\/mi><mi>X<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><mi>X<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>Q<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>Y<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mi>X<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><mi>\u03bb<\/mi><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>Q<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p>&nbsp;<\/p><\/blockquote><h4 id='example-116-jensens-inequality'><span>Example 1.16 Jensen&#39;s Inequality<\/span><\/h4><p><span> Let <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.244ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 550 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-341-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-341-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f<\/script><span> be a convex function on <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.14ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 504 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-497-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-497-TEX-I-1D43C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> be a random variable taking values in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.14ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 504 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-497-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-497-TEX-I-1D43C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I<\/script><span>, we have <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"18.854ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 8333.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-345-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-345-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-345-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-345-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-345-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-345-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-345-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-345-TEX-N-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-345-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-345-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-345-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1592,0)\"><use data-c=\"28\" xlink:href=\"#MJX-345-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1981,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-345-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2833,0)\"><use data-c=\"29\" xlink:href=\"#MJX-345-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3222,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-345-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3777.8,0)\"><use data-c=\"2265\" xlink:href=\"#MJX-345-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4833.6,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-345-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5383.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-345-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5772.6,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-345-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6536.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-345-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6814.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-345-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7666.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-345-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7944.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-345-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>\u2265<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[f(X)]\\ge f(E[X])<\/script><span>.<\/span><\/p><p><em><span>Proof.<\/span><\/em><\/p><p><span>Since <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.633ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1606 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-346-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-346-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-346-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-346-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-346-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(550,0)\"><use data-c=\"28\" xlink:href=\"#MJX-346-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(939,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-346-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1217,0)\"><use data-c=\"29\" xlink:href=\"#MJX-346-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f(\\cdot)<\/script><span> is convex, there exists a real number <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.986ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 878 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-379-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45A\" xlink:href=\"#MJX-379-TEX-I-1D45A\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>m<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">m<\/script><span> such that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"25.456ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 11251.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-348-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-348-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-348-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-348-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-348-TEX-N-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-348-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-348-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-348-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-348-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-348-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(550,0)\"><use data-c=\"28\" xlink:href=\"#MJX-348-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(939,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-348-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1511,0)\"><use data-c=\"29\" xlink:href=\"#MJX-348-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2177.8,0)\"><use data-c=\"2265\" xlink:href=\"#MJX-348-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3233.6,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-348-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3783.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-348-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4172.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-348-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"30\" xlink:href=\"#MJX-348-TEX-N-30\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5181.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-348-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5792.3,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-348-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6792.6,0)\"><use data-c=\"1D45A\" xlink:href=\"#MJX-348-TEX-I-1D45A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7670.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-348-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8059.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-348-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8853.8,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-348-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(9854,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-348-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"30\" xlink:href=\"#MJX-348-TEX-N-30\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10862.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-348-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>0<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>m<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>0<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f(x)\\ge f(x_0)+m(x-x_0)<\/script><span>. We choose <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.282ex\" height=\"1.375ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 1008.6 607.6\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.375ex;\"><defs><path id=\"MJX-349-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-349-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-349-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"30\" xlink:href=\"#MJX-349-TEX-N-30\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>x<\/mi><mn>0<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x_0<\/script><span> to be <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.914ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2172 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-350-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-350-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-350-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-350-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-350-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-350-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-350-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1894,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-350-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[X]<\/script><span> and get <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"30.72ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 13578.4 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-351-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-351-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-351-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-351-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-351-TEX-N-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-351-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-351-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-351-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-351-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-351-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-351-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-351-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-351-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(550,0)\"><use data-c=\"28\" xlink:href=\"#MJX-351-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(939,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-351-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1511,0)\"><use data-c=\"29\" xlink:href=\"#MJX-351-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2177.8,0)\"><use data-c=\"2265\" xlink:href=\"#MJX-351-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3233.6,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-351-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3783.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-351-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4172.6,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-351-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4936.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-351-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5214.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-351-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6066.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-351-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6344.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-351-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6955.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-351-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7956,0)\"><use data-c=\"1D45A\" xlink:href=\"#MJX-351-TEX-I-1D45A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8834,0)\"><use data-c=\"28\" xlink:href=\"#MJX-351-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9223,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-351-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10017.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-351-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11017.4,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-351-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11781.4,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-351-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(12059.4,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-351-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12911.4,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-351-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13189.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-351-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>m<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f(x)\\ge f(E[X])+m(x-E[X])<\/script><span>. This holds for every <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.294ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 572 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-532-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-532-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x<\/script><span>, we take the expectation over <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.294ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 572 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-532-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-532-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x<\/script><span>:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n193\" cid=\"n193\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 53.145ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 53.145ex;\"><defs><path id=\"MJX-54-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 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xlink:href=\"#MJX-54-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>\u2265<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>m<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo>\u2212<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='example-117-definitions-of-convexity'><span>Example 1.17 Definitions of Convexity<\/span><\/h4><p><span>These definitions are equivalent<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n196\" cid=\"n196\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg 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data-mml-node=\"mo\" transform=\"translate(1074.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-55-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1463.9,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-55-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2035.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-55-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2702.7,0)\"><use data-c=\"2265\" xlink:href=\"#MJX-55-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3758.5,0)\"><use data-c=\"30\" xlink:href=\"#MJX-55-TEX-N-30\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right center\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><\/mtd><mtd><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>0<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>m<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>0<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi>\u03bb<\/mi><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>\u03bb<\/mi><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><msup><mi>f<\/mi><mo data-mjx-alternate=\"1\">\u2033<\/mo><\/msup><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mn>0<\/mn><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><em><span>Proof.<\/span><\/em><\/p><p><span>Using Jensen&#39;s Inequality,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n199\" cid=\"n199\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 69.4ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" 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xlink:href=\"#MJX-56-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(24704.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-56-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(25426.9,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-56-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(26427.1,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-56-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(27010.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-56-TEX-N-29\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(27399.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-56-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-56-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(28407.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-56-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(28796.7,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-56-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>\u03bb<\/mi><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>\u2265<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>\u03bb<\/mi><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h2 id='chapter-2-information-theory-for-continuous-variabls'><span>Chapter 2 Information Theory for Continuous Variabls<\/span><\/h2><h3 id='1-differential-entropy'><span>(1) Differential Entropy<\/span><\/h3><blockquote><p><span>Consider a real-valued random variable <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> with density <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.078ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2244.5 1000\" 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98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-355-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-355-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-355-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-355-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-355-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1188.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-355-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1577.5,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-355-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1855.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-355-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">p_X(\\cdot)<\/script><span>. The differential entropy of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 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stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>a<\/mi><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mo>\u2212<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><h4 id='example-21'><span>Example 2.1<\/span><\/h4><p><span>Consider the uniform distribution with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.622ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5579 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-357-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-357-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-357-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-357-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-357-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-357-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 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155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-357-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-357-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1188.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-357-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1577.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-357-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2106.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-357-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2773.2,0)\"><use data-c=\"3D\" 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xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.306ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4113.2 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-358-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-358-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 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xlink:href=\"#MJX-359-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">h(X)<\/script><span> is <\/span><em><span>negative<\/span><\/em><span> for <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.845ex\" height=\"1.71ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 2583.6 756\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-360-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 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stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>l<\/mi><mi>o<\/mi><mi>g<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>c<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><\/mtr><mtr><mtd><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>c<\/mi><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><blockquote><p><span>Conditional Entropy<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" 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xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/msub><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>a<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><h3 id='2-mixed-distributions'><span>(2) Mixed Distributions<\/span><\/h3><h3 id='3-example-distributions'><span>(3) Example Distributions<\/span><\/h3><h3 id='4-informational-divergence'><span>(4) Informational Divergence<\/span><\/h3><blockquote><p><span>The information divergence between the densities <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.689ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 1188.5 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-365-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 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type=\"math\/tex\">I(X;Y)=D(p_{XY}\\parallel p_Xp_Y) \\ge 0<\/script><\/p><\/li><li><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"15.405ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6809.1 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-364-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-364-TEX-N-28\" 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transform=\"translate(2683.2,0)\"><use data-c=\"2225\" xlink:href=\"#MJX-364-TEX-N-2225\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3461,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-364-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-364-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4586.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-364-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5253.3,0)\"><use data-c=\"2265\" xlink:href=\"#MJX-364-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(6309.1,0)\"><use data-c=\"30\" xlink:href=\"#MJX-364-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>p<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D(p_X\\parallel p_Y)\\ge 0<\/script><span> <\/span><\/p><\/li><\/ul><h3 id='5-cross-entropy'><span>(5) Cross Entropy<\/span><\/h3><blockquote><p><span>The cross-entropy of two densities <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.689ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 1188.5 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-365-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 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transform=\"translate(6333.1,0)\"><use data-c=\"210E\" xlink:href=\"#MJX-63-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6909.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-63-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7298.1,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-63-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8150.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-63-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -2250 1 4000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mi>X<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>p<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>p<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mi>D<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>p<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mi>X<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>p<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>with equality iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.166ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6261.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-367-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 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display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>\u2208<\/mo><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a\\in supp(p_X)<\/script><span>.<\/span><\/p><h3 id='6-maximum-entropy'><span>(6) Maximum Entropy<\/span><\/h3><h4 id='alphabet-or-volume-constraint'><span>Alphabet or Volume Constraint<\/span><\/h4><p><span>Uniform distribution maximizes the entropy. If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 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xlink:href=\"#MJX-64-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(576,0)\"><use data-c=\"28\" xlink:href=\"#MJX-64-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(965,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-64-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1817,0)\"><use data-c=\"29\" xlink:href=\"#MJX-64-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2483.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-64-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3539.6,0)\"><use data-c=\"6C\" xlink:href=\"#MJX-64-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-64-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-64-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4817.6,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-64-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4984.2,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-64-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5262.2,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-64-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5907.2,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-64-TEX-N-7C\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mi>log<\/mi><mo 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data-c=\"3D\" xlink:href=\"#MJX-371-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3829,0)\"><use data-c=\"31\" xlink:href=\"#MJX-371-TEX-N-31\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4329,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-371-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4829,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-371-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5107,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-371-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5752,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-371-TEX-N-7C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>S<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">p_X(a)=1\/|S|<\/script><span> for almost all <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.422ex\" height=\"1.686ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 2396.6 745\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-372-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 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xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>\u2208<\/mo><mi>S<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a\\in S<\/script><span>.<\/span><\/p><h4 id='first-moment-constriant'><span>First Moment Constriant<\/span><\/h4><p><span>Under the first-moment constraint <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.918ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4383.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-373-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-373-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-373-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 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-118Z\"><\/path><path id=\"MJX-373-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" 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stretchy=\"false\">]<\/mo><mo>\u2264<\/mo><mi>m<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[X]\\le m<\/script><span>, where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> is non-negative, independent <\/span><strong><span>exponential random variables<\/span><\/strong><span> maximize (differential) entropy under the first moment and non-negativity constraints<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n238\" cid=\"n238\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 29.951ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"4.945ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.907ex; min-width: 29.951ex;\"><defs><path id=\"MJX-65-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 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xlink:href=\"#MJX-65-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(10860.5,0)\"><use data-c=\"30\" xlink:href=\"#MJX-65-TEX-N-30\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-65-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1342.9 1 2185.8\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,0.9)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>p<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>E<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><msub><mi>m<\/mi><mi>i<\/mi><\/msub><\/mfrac><msup><mi>e<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mi>a<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msub><mi>m<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msup><mo>,<\/mo><mtext>&nbsp;<\/mtext><mi>a<\/mi><mo>\u2265<\/mo><mn>0.<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>We use cross entropy to upper bound the entropy<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n240\" cid=\"n240\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg 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rowspacing=\"3pt\"><mtr><mtd><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mi>X<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>p<\/mi><mi>E<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2212<\/mo><msub><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/msub><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><msub><mi>p<\/mi><mi>E<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mi>d<\/mi><mi>a<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2212<\/mo><msub><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/msub><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mi>i<\/mi><\/munder><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mfrac><msub><mi>a<\/mi><mi>i<\/mi><\/msub><msub><mi>m<\/mi><mi>i<\/mi><\/msub><\/mfrac><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mo>\u2212<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>m<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>a<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mi>i<\/mi><\/munder><mo stretchy=\"false\">(<\/mo><mfrac><mrow><msub><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/msub><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>a<\/mi><mi>i<\/mi><\/msub><mi>d<\/mi><mi>a<\/mi><\/mrow><msub><mi>m<\/mi><mi>i<\/mi><\/msub><\/mfrac><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mo>+<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>m<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><\/mtd><mtd><mi><\/mi><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mi>i<\/mi><\/munder><mo stretchy=\"false\">(<\/mo><mfrac><msub><mi>m<\/mi><mi>i<\/mi><\/msub><msub><mi>m<\/mi><mi>i<\/mi><\/msub><\/mfrac><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mo>+<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><msub><mi>m<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mi>i<\/mi><\/munder><mi>l<\/mi><mi>o<\/mi><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>e<\/mi><msub><mi>m<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='second-moment-constraint'><span>Second Moment Constraint<\/span><\/h4><p><span>Under the second-moment constraint <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.405ex\" height=\"2.26ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -749.5 4157 999\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.564ex;\"><defs><path id=\"MJX-375-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-375-TEX-I-1D444\" d=\"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z\"><\/path><path id=\"MJX-375-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-375-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-375-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(278,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D444\" xlink:href=\"#MJX-375-TEX-I-1D444\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(824,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-375-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1754.5,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-375-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2310.2,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-375-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3366,0)\"><use data-c=\"1D444\" xlink:href=\"#MJX-375-TEX-I-1D444\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>Q<\/mi><mi>X<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2264<\/mo><mi>Q<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">|Q_X|\\le Q<\/script><span>, where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.873ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 828 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-376-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-376-TEX-I-1D437\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D<\/script><span> is some constant. Let <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.778ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 786 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-377-TEX-I-1D43A\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43A\" xlink:href=\"#MJX-377-TEX-I-1D43A\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>G<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">G<\/script><span> be Gaussian with covariance matrix <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.34ex\" height=\"2.032ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -704 1476.5 898\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-378-TEX-I-1D444\" d=\"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z\"><\/path><path id=\"MJX-378-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D444\" xlink:href=\"#MJX-378-TEX-I-1D444\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(824,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-378-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>Q<\/mi><mi>X<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Q_X<\/script><span> and entries with same mean <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.986ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 878 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-379-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45A\" xlink:href=\"#MJX-379-TEX-I-1D45A\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>m<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">m<\/script><span>.<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n243\" cid=\"n243\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"76.478ex\" height=\"19.901ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -4648.1 33803.2 8796.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -9.385ex;\"><defs><path id=\"MJX-67-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-67-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-67-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 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stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mi>X<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo>\u2225<\/mo><msub><mi>p<\/mi><mi>G<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mo>\u2212<\/mo><msub><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/msub><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><msub><mi>p<\/mi><mi>G<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mi>d<\/mi><mi>a<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mo>\u2212<\/mo><msub><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>s<\/mi><mi>u<\/mi><mi>p<\/mi><mi>p<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/msub><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">[<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><msup><mo stretchy=\"false\">)<\/mo><mi>n<\/mi><\/msup><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>Q<\/mi><mi>X<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>\u2212<\/mo><mi>m<\/mi><msup><mo stretchy=\"false\">)<\/mo><mi>T<\/mi><\/msup><msubsup><mi>Q<\/mi><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>\u2212<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mi>e<\/mi><mo stretchy=\"false\">]<\/mo><mi>d<\/mi><mi>a<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mi>e<\/mi><msup><mo stretchy=\"false\">)<\/mo><mi>n<\/mi><\/msup><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>Q<\/mi><mi>X<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><strong><span>Gaussian random variables<\/span><\/strong><span> maximize differential entropy under the second moment contraint.<\/span><\/p><h4 id='example-22'><span>Example 2.2<\/span><\/h4><p><span>The second moment contraint for a scalar is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.033ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5318.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-380-TEX-I-1D449\" d=\"M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z\"><\/path><path id=\"MJX-380-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-380-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 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display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mi>e<\/mi><mi>D<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h2 id='chapter-3-coding-of-memoryless-sources'><span>Chapter 3 Coding of Memoryless Sources<\/span><\/h2><p><span>We encode a random variable source of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.103ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3581.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" 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400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-465-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 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xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>K<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">K-ary<\/script><span> alphabet into a sequence of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> digits.<\/span><\/p><ul><li><p><span>A <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> code is said to be uniquely decodable if <\/span><em><span>every<\/span><\/em><span> finite <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> sequence has at most one interpretation as a sequence of its codewords. A prefix-free code is automatically also a uniquely decodable code.<\/span><\/p><\/li><\/ul><h3 id='1-kraft-inequality'><span>(1) Kraft Inequality <\/span><\/h3><blockquote><h4 id='theorem-31-kraft-inequality'><span>Theorem 3.1 Kraft Inequality<\/span><\/h4><p><span>A <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> prefix-free code with codeword length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.207ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 7605.6 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-386-TEX-I-1D459\" d=\"M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path id=\"MJX-386-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-386-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 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data-mml-node=\"mo\" transform=\"translate(1992.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-386-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3048.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-386-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3548.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-386-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3993.3,0)\"><use data-c=\"32\" xlink:href=\"#MJX-386-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4493.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-386-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4938,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-386-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5382.6,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-386-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5827.3,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-386-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6272,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-386-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6716.6,0)\"><use data-c=\"1D43E\" xlink:href=\"#MJX-386-TEX-I-1D43E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>l<\/mi><mi>k<\/mi><\/msub><mo>,<\/mo><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mn>2<\/mn><mo>,<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mo>,<\/mo><mi>K<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">l_k, k=1, 2, ..., K<\/script><span>, exists iff.<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n257\" cid=\"n257\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container 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26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path id=\"MJX-69-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-69-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" 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xlink:href=\"#MJX-69-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(407.7,1150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D43E\" xlink:href=\"#MJX-69-TEX-I-1D43E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msup\" transform=\"translate(1610.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-69-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(861,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-69-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(778,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D459\" xlink:href=\"#MJX-69-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(331,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-69-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3879.5,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-69-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(4935.3,0)\"><use data-c=\"31\" xlink:href=\"#MJX-69-TEX-N-31\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-69-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1749.3 1 2998.7\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,16.4)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>K<\/mi><\/mrow><\/munderover><msup><mi>D<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><msub><mi>l<\/mi><mi>k<\/mi><\/msub><\/mrow><\/msup><mo>\u2264<\/mo><mn>1.<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Equality holds iff. the code has no unused leaves in its <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 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-160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> tree.<\/span><\/p><\/blockquote><ul><li><p><span>Kraft&#39;s inequality provides a necessary condition for the existence of a prefix-free code. A <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> uniquely decodable code with a given list of codeword lengths exists if and only if these lengths satisfy Kraft\u2019s inequality.<\/span><\/p><\/li><\/ul><p><em><span>Proof.<\/span><\/em><\/p><ul><li><p><span>Path Length Lemma<\/span><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n266\" cid=\"n266\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 20.079ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"6.935ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.902ex; min-width: 20.079ex;\"><defs><path id=\"MJX-70-TEX-LO-2211\" d=\"M60 948Q63 950 665 950H1267L1325 815Q1384 677 1388 669H1348L1341 683Q1320 724 1285 761Q1235 809 1174 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639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path id=\"MJX-70-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 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686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path id=\"MJX-70-TEX-I-1D457\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path id=\"MJX-70-TEX-I-1D43D\" d=\"M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z\"><\/path><path id=\"MJX-70-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -1782.6)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1782.6) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3637.5 -1782.6 1 3065.1\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,49.6)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"munderover\"><g data-mml-node=\"mo\"><use data-c=\"2211\" xlink:href=\"#MJX-70-TEX-LO-2211\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(86,-1107.7) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-70-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(521,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-70-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1299,0)\"><use data-c=\"31\" xlink:href=\"#MJX-70-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(407.7,1150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D43E\" xlink:href=\"#MJX-70-TEX-I-1D43E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(1610.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-70-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-70-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(2565.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D459\" xlink:href=\"#MJX-70-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(331,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-70-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3592.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-70-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"munderover\" transform=\"translate(4648,0)\"><g data-mml-node=\"mo\"><use data-c=\"2211\" xlink:href=\"#MJX-70-TEX-LO-2211\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(124.5,-1087.9) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D457\" xlink:href=\"#MJX-70-TEX-I-1D457\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(412,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-70-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1190,0)\"><use data-c=\"31\" xlink:href=\"#MJX-70-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(498.2,1150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D43D\" xlink:href=\"#MJX-70-TEX-I-1D43D\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(6258.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-70-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D457\" xlink:href=\"#MJX-70-TEX-I-1D457\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1782.6 1 3065.1\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,49.6)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>K<\/mi><\/mrow><\/munderover><msub><mi>p<\/mi><mi>k<\/mi><\/msub><msub><mi>l<\/mi><mi>k<\/mi><\/msub><mo>=<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>j<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>J<\/mi><\/mrow><\/munderover><msub><mi>P<\/mi><mi>j<\/mi><\/msub><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Proof.<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n268\" cid=\"n268\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"61.449ex\" height=\"22.162ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -5147.7 27160.3 9795.4\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -10.515ex;\"><defs><path id=\"MJX-71-TEX-LO-2211\" d=\"M60 948Q63 950 665 950H1267L1325 815Q1384 677 1388 669H1348L1341 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display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>H<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>l<\/mi><mi>e<\/mi><mi>a<\/mi><mi>f<\/mi><\/mrow><\/msub><mo>=<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>j<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>J<\/mi><\/mrow><\/munderover><msub><mi>P<\/mi><mi>j<\/mi><\/msub><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>Q<\/mi><mi>j<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='2-entropy-bounds-on-source-coding'><span>(2) Entropy Bounds on Source Coding<\/span><\/h3><blockquote><h4 id='theorem-32-coding-theorem-for-a-single-random-variable'><span>Theorem 3.2 Coding theorem for a Single Random Variable.<\/span><\/h4><p><span>The average code word length 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transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-464-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-464-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-464-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1723,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-464-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[L]<\/script><span> of an optimum <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" 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26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> prefix-free code for a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.103ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3581.4 888\" 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382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-465-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 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xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>K<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">K-ary<\/script><span> random variable <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 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type=\"math\/tex\">X<\/script><span> satisfies<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n278\" cid=\"n278\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 31.492ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.319ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.094ex; min-width: 31.492ex;\"><defs><path id=\"MJX-73-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 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xlink:href=\"#MJX-73-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(298,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-73-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(783,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-73-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-73-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1696.6,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-73-TEX-I-1D437\"><\/use><\/g><\/g><rect width=\"2724.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(10819.4,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-73-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(11819.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-73-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1425.5 1 2351\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-34.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>\u2264<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><mo>&lt;<\/mo><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mn>1<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Equality holds iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.681ex\" height=\"2.37ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 4279 1047.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-393-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 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data-mml-node=\"mo\" transform=\"translate(1232.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-393-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(2288,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-393-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(861,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-393-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(778,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D459\" xlink:href=\"#MJX-393-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(331,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-393-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>p<\/mi><mi>k<\/mi><\/msub><mo>=<\/mo><msup><mi>D<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><msub><mi>l<\/mi><mi>k<\/mi><\/msub><\/mrow><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">p_k=D^{-l_k}<\/script><span> for all <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-477-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-477-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span>.<\/span><\/p><\/blockquote><p><em><span>Proof.<\/span><\/em><\/p><p><span>For the lower bound, we use Leaf Length Lemma and Leaf-Entropy Lemma, <\/span><\/p><div 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data-mml-node=\"mi\" transform=\"translate(3091.7,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-74-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3576.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-74-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-74-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(4490.2,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-74-TEX-I-1D437\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-4037.6)\"><g data-mml-node=\"mtd\" transform=\"translate(2518,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(2518,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-74-TEX-N-3D\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(3851.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-74-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-74-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-74-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1723,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-74-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2001,0)\"><use data-c=\"1D459\" xlink:href=\"#MJX-74-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2299,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-74-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2784,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-74-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-74-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(3697.6,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-74-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(4525.6,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-74-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4775.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-74-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5204.6,0)\"><use data-c=\"1D456\" xlink:href=\"#MJX-74-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5549.6,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-74-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5910.6,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-74-TEX-I-1D460\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right center left\" columnspacing=\"0em 0.278em\" rowspacing=\"3pt\"><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><msub><mi>H<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>l<\/mi><mi>e<\/mi><mi>a<\/mi><mi>f<\/mi><\/mrow><\/msub><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>j<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>J<\/mi><\/mrow><\/munderover><msub><mi>P<\/mi><mi>j<\/mi><\/msub><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>Q<\/mi><mi>j<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><\/mtd><mtd><mi><\/mi><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>j<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>J<\/mi><\/mrow><\/munderover><msub><mi>P<\/mi><mi>j<\/mi><\/msub><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><mtext>&nbsp;<\/mtext><mi>b<\/mi><mi>i<\/mi><mi>t<\/mi><mi>s<\/mi><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>and get<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n284\" cid=\"n284\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 18.5ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.319ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.094ex; min-width: 18.5ex;\"><defs><path id=\"MJX-75-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-75-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-75-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-75-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path 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422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-75-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-75-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-75-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-75-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-75-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-75-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-75-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-75-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -1425.5)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1425.5) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3288.6 -1425.5 1 2351\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g 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data-c=\"1D454\" xlink:href=\"#MJX-75-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-75-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1696.6,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-75-TEX-I-1D437\"><\/use><\/g><\/g><rect width=\"2724.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(3242.3,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-75-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4298.1,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-75-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5062.1,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-75-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5340.1,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-75-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6021.1,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-75-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6299.1,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-75-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1425.5 1 2351\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-34.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>\u2264<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>For the upper bound, we use Kraft&#39;s Inequality,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n286\" cid=\"n286\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 25.366ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"6.784ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.827ex; min-width: 25.366ex;\"><defs><path id=\"MJX-76-TEX-LO-2211\" d=\"M60 948Q63 950 665 950H1267L1325 815Q1384 677 1388 669H1348L1341 683Q1320 724 1285 761Q1235 809 1174 838T1033 881T882 898T699 902H574H543H251L259 891Q722 258 724 252Q725 250 724 246Q721 243 460 -56L196 -356Q196 -357 407 -357Q459 -357 548 -357T676 -358Q812 -358 896 -353T1063 -332T1204 -283T1307 -196Q1328 -170 1348 -124H1388Q1388 -125 1381 -145T1356 -210T1325 -294L1267 -449L666 -450Q64 -450 61 -448Q55 -446 55 -439Q55 -437 57 -433L590 177Q590 178 557 222T452 366T322 544L56 909L55 924Q55 945 60 948Z\"><\/path><path id=\"MJX-76-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-76-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-76-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-76-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path id=\"MJX-76-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-76-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-76-TEX-I-1D459\" d=\"M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path id=\"MJX-76-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-76-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-76-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -1749.3)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1749.3) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"4805.9 -1749.3 1 2998.7\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,16.4)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"munderover\"><g data-mml-node=\"mo\"><use data-c=\"2211\" xlink:href=\"#MJX-76-TEX-LO-2211\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(86,-1107.7) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-76-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(521,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-76-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1299,0)\"><use data-c=\"31\" xlink:href=\"#MJX-76-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(407.7,1150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D43E\" xlink:href=\"#MJX-76-TEX-I-1D43E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msup\" transform=\"translate(1610.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-76-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(861,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-76-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(778,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D459\" xlink:href=\"#MJX-76-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(331,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-76-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3879.5,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-76-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(4935.3,0)\"><use data-c=\"31\" xlink:href=\"#MJX-76-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5713,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-76-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"munderover\" transform=\"translate(6768.8,0)\"><g data-mml-node=\"mo\"><use data-c=\"2211\" xlink:href=\"#MJX-76-TEX-LO-2211\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(86,-1107.7) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-76-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(521,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-76-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1299,0)\"><use data-c=\"31\" xlink:href=\"#MJX-76-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(407.7,1150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D43E\" xlink:href=\"#MJX-76-TEX-I-1D43E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(8379.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-76-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-76-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9333.9,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-76-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1749.3 1 2998.7\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,16.4)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>K<\/mi><\/mrow><\/munderover><msup><mi>D<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><msub><mi>l<\/mi><mi>k<\/mi><\/msub><\/mrow><\/msup><mo>\u2264<\/mo><mn>1<\/mn><mo>=<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>K<\/mi><\/mrow><\/munderover><msub><mi>p<\/mi><mi>k<\/mi><\/msub><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Let<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n288\" cid=\"n288\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 17.244ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"6.004ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.436ex; min-width: 17.244ex;\"><defs><path id=\"MJX-77-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-77-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-77-TEX-I-1D459\" d=\"M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path id=\"MJX-77-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-77-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-77-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-77-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-77-TEX-N-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-77-TEX-I-1D45C\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path id=\"MJX-77-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-77-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -1576.9)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1576.9) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3010.9 -1576.9 1 2653.7\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mtable\"><g data-mml-node=\"mtr\" transform=\"translate(0,673.1)\"><g data-mml-node=\"mtd\" transform=\"translate(732.4,0)\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-77-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(861,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-77-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(778,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D459\" xlink:href=\"#MJX-77-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(331,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-77-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2268.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-77-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3324.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-77-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-77-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4279,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-77-TEX-N-2C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-826.9)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D459\" xlink:href=\"#MJX-77-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(331,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-77-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1027.2,0)\"><use data-c=\"2265\" xlink:href=\"#MJX-77-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2083,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-77-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2861,0)\"><use data-c=\"1D459\" xlink:href=\"#MJX-77-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3159,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-77-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3644,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-77-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"1D437\" xlink:href=\"#MJX-77-TEX-I-1D437\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(4789.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-77-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-77-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5743.8,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-77-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1576.9 1 2653.7\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><msup><mi>D<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><msub><mi>l<\/mi><mi>k<\/mi><\/msub><\/mrow><\/msup><mo>\u2264<\/mo><msub><mi>p<\/mi><mi>k<\/mi><\/msub><mo>,<\/mo><\/mtd><\/mtr><mtr><mtd><msub><mi>l<\/mi><mi>k<\/mi><\/msub><mo>\u2265<\/mo><mo>\u2212<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mi>D<\/mi><\/msub><msub><mi>p<\/mi><mi>k<\/mi><\/msub><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>We choose the shortest codes<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n290\" cid=\"n290\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 19.253ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 19.253ex;\"><defs><path id=\"MJX-78-TEX-I-1D459\" d=\"M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path id=\"MJX-78-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 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transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3454.9 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D459\" xlink:href=\"#MJX-78-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(331,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-78-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1027.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-78-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2083,0)\"><use data-c=\"2308\" xlink:href=\"#MJX-78-TEX-N-2308\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2527,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-78-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3305,0)\"><use data-c=\"1D459\" xlink:href=\"#MJX-78-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3603,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-78-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4088,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-78-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"1D437\" xlink:href=\"#MJX-78-TEX-I-1D437\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(5233.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-78-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-78-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6187.8,0)\"><use data-c=\"2309\" xlink:href=\"#MJX-78-TEX-N-2309\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6631.8,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-78-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>l<\/mi><mi>k<\/mi><\/msub><mo>=<\/mo><mo fence=\"false\" stretchy=\"false\">\u2308<\/mo><mo>\u2212<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mi>D<\/mi><\/msub><msub><mi>p<\/mi><mi>k<\/mi><\/msub><mo fence=\"false\" stretchy=\"false\">\u2309<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Therefore, <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block 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46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-79-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3872.1 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D459\" xlink:href=\"#MJX-79-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(331,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-79-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1027.2,0)\"><use data-c=\"3C\" xlink:href=\"#MJX-79-TEX-N-3C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2083,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-79-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2861,0)\"><use data-c=\"1D459\" xlink:href=\"#MJX-79-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3159,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-79-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3644,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-79-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"1D437\" xlink:href=\"#MJX-79-TEX-I-1D437\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(4789.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-79-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-79-TEX-I-1D458\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5966.1,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-79-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(6966.3,0)\"><use data-c=\"31\" xlink:href=\"#MJX-79-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7466.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-79-TEX-N-2C\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>l<\/mi><mi>k<\/mi><\/msub><mo>&lt;<\/mo><mo>\u2212<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mi>D<\/mi><\/msub><msub><mi>p<\/mi><mi>k<\/mi><\/msub><mo>+<\/mo><mn>1<\/mn><mo>,<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>and we get<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n295\" cid=\"n295\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"31.205ex\" height=\"26.929ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -6201.3 13792.7 11902.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -12.899ex;\"><defs><path id=\"MJX-80-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-80-TEX-N-5B\" d=\"M118 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0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-464-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-464-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 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xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[L]<\/script><span> of an optimum <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 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441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> prefix-free code for a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.103ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3581.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-465-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 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transform=\"translate(2111.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-465-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2640.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-465-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3091.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-465-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>K<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">K-ary<\/script><span> DMS <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.77ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2550.1 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-398-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-398-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 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xlink:href=\"#MJX-398-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2161.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-398-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mo>.<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X(.)<\/script><span> that is parsed into <\/span><em><span>blocks of length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-469-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 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xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>B<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">B<\/script><\/em><span> satisfies<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n299\" cid=\"n299\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 34.069ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.319ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.094ex; min-width: 34.069ex;\"><defs><path id=\"MJX-81-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 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data-mml-node=\"mrow\" transform=\"translate(220,-686)\"><g data-mml-node=\"mi\"><use data-c=\"1D459\" xlink:href=\"#MJX-81-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(298,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-81-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(783,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-81-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-81-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1696.6,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-81-TEX-I-1D437\"><\/use><\/g><\/g><rect width=\"2724.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(11259.4,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-81-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(12259.7,0)\"><g data-mml-node=\"mn\" transform=\"translate(349.5,676)\"><use data-c=\"31\" xlink:href=\"#MJX-81-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(220,-686)\"><use data-c=\"1D435\" xlink:href=\"#MJX-81-TEX-I-1D435\"><\/use><\/g><rect width=\"959\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1425.5 1 2351\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-34.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>\u2264<\/mo><mfrac><mrow><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mi>B<\/mi><\/mfrac><mo>&lt;<\/mo><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mn>1<\/mn><mi>B<\/mi><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><p><em><span>Proof.<\/span><\/em><\/p><p><span>We already have the coding theorem for a single random variable <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span>, here we treat a block of length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-469-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-469-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>B<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">B<\/script><span> as a new single random variable <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.445ex\" height=\"1.914ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 1522.9 846\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-402-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 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xlink:href=\"#MJX-82-TEX-N-29\"><\/use><\/g><\/g><g data-mml-node=\"mrow\" transform=\"translate(552.2,-686)\"><g data-mml-node=\"mi\"><use data-c=\"1D459\" xlink:href=\"#MJX-82-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(298,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-82-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(783,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-82-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-82-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1696.6,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-82-TEX-I-1D437\"><\/use><\/g><\/g><rect width=\"3388.9\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(12588.1,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-82-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(13588.3,0)\"><use data-c=\"31\" xlink:href=\"#MJX-82-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1473.5 1 2447\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-82.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>\u2264<\/mo><mfrac><mrow><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mi>B<\/mi><\/mfrac><mo>&lt;<\/mo><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mn>1<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Since the source is DMS, we have <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.646ex\" height=\"2.48ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 7799.4 1096\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-403-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-403-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-403-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-403-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-403-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-403-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-403-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-403-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(1277,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-403-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(936.2,363) scale(0.707)\"><use data-c=\"1D435\" xlink:href=\"#MJX-403-TEX-I-1D435\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2799.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-403-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3466.7,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-403-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4522.4,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-403-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5281.4,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-403-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6169.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-403-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6558.4,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-403-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7410.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-403-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>B<\/mi><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H(X^B)=BH(X)<\/script><span> and finish the proof.<\/span><\/p><h3 id='3-huffman-codes'><span>(3) Huffman Codes<\/span><\/h3><blockquote><p><span>The <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> prefix-free codes that minimize <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.527ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2001 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-464-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-464-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-464-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-464-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-464-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-464-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-464-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1723,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-464-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[L]<\/script><span> for a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.103ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3581.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-465-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path id=\"MJX-465-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-465-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-465-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-465-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43E\" xlink:href=\"#MJX-465-TEX-I-1D43E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1111.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-465-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2111.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-465-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2640.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-465-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3091.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-465-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>K<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">K-ary<\/script><span> random variable are called <\/span><em><span>Huffman codes<\/span><\/em><span>. <\/span><\/p><p><span>The branches of the tree are <\/span><strong><span>codewords<\/span><\/strong><span> with minimal <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.527ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2001 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-464-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-464-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-464-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-464-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-464-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-464-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-464-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1723,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-464-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[L]<\/script><span>.<\/span><\/p><\/blockquote><ul><li><p><span>Optimal <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> prefix-free codes assign shorter code words to more probable letters.<\/span><\/p><\/li><li><p><span>All used leaves in the tree of an optimal <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> prefix-free code has a sibling.<\/span><\/p><\/li><li><p><span>The number of unused leaves in the tree of an optimal <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> prefix-free code is<\/span><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n315\" cid=\"n315\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 32.483ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 32.483ex;\"><defs><path id=\"MJX-83-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-83-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 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-64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-83-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-83-TEX-N-A0\" d=\"\"><\/path><path id=\"MJX-83-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-83-TEX-I-1D45C\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path id=\"MJX-83-TEX-I-1D451\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 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data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"6378.7 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-83-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D43E\" xlink:href=\"#MJX-83-TEX-I-1D43E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1500.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-83-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2500.4,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-83-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3328.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-83-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3717.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-83-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4106.4,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-83-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5156.7,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-83-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(6156.9,0)\"><use data-c=\"32\" xlink:href=\"#MJX-83-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6656.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-83-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(7045.9,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-83-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7295.9,0)\"><use data-c=\"1D45A\" xlink:href=\"#MJX-83-TEX-I-1D45A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8173.9,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-83-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8658.9,0)\"><use data-c=\"1D451\" xlink:href=\"#MJX-83-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(9178.9,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-83-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9428.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-83-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9817.9,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-83-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10868.1,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-83-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(11868.3,0)\"><use data-c=\"31\" xlink:href=\"#MJX-83-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12368.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-83-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mo stretchy=\"false\">(<\/mo><mi>K<\/mi><mo>\u2212<\/mo><mi>D<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>D<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;<\/mtext><mi>m<\/mi><mi>o<\/mi><mi>d<\/mi><mtext>&nbsp;<\/mtext><mo stretchy=\"false\">(<\/mo><mi>D<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Proof.<\/span><\/p><p><span>A <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> tree must have <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"19.259ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 8512.3 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-412-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-412-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-412-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-412-TEX-I-1D43D\" d=\"M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z\"><\/path><path id=\"MJX-412-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-412-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-412-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-412-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-412-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2050.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-412-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2439.4,0)\"><use data-c=\"1D43D\" xlink:href=\"#MJX-412-TEX-I-1D43D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3294.7,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-412-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(4294.9,0)\"><use data-c=\"31\" xlink:href=\"#MJX-412-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4794.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-412-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5183.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-412-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5572.9,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-412-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6623.1,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-412-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(7623.3,0)\"><use data-c=\"31\" xlink:href=\"#MJX-412-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8123.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-412-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>J<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>D<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D+(J-1)(D-1)<\/script><span> leaves, in which <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.011ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 889 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-413-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43E\" xlink:href=\"#MJX-413-TEX-I-1D43E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>K<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">K<\/script><span> leaves are codewords. <\/span><\/p><p><span>The number of unused leaves is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"28.073ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 12408.3 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-414-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-414-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-414-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-414-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-414-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-414-TEX-I-1D43D\" d=\"M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z\"><\/path><path id=\"MJX-414-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-414-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-414-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-414-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45F\" xlink:href=\"#MJX-414-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(728.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-414-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1784.6,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-414-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2834.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-414-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3835,0)\"><use data-c=\"28\" xlink:href=\"#MJX-414-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4224,0)\"><use data-c=\"1D43D\" xlink:href=\"#MJX-414-TEX-I-1D43D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5079.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-414-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(6079.4,0)\"><use data-c=\"31\" xlink:href=\"#MJX-414-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6579.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-414-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6968.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-414-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7357.4,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-414-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8407.7,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-414-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(9407.9,0)\"><use data-c=\"31\" xlink:href=\"#MJX-414-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9907.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-414-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10519.1,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-414-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11519.3,0)\"><use data-c=\"1D43E\" xlink:href=\"#MJX-414-TEX-I-1D43E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>r<\/mi><mo>=<\/mo><mi>D<\/mi><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>J<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>D<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>K<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">r=D+(J-1)(D-1)-K<\/script><span>, i.e.,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n319\" cid=\"n319\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 35.213ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 35.213ex;\"><defs><path id=\"MJX-84-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-84-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-84-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-84-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path id=\"MJX-84-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-84-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-84-TEX-I-1D43D\" d=\"M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z\"><\/path><path id=\"MJX-84-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-84-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-84-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"6982.2 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-84-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-84-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1439.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-84-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2439.4,0)\"><use data-c=\"1D43E\" xlink:href=\"#MJX-84-TEX-I-1D43E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3328.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-84-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3995.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-84-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5051,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-84-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5829,0)\"><use data-c=\"28\" xlink:href=\"#MJX-84-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6218,0)\"><use data-c=\"1D43D\" xlink:href=\"#MJX-84-TEX-I-1D43D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7073.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-84-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(8073.4,0)\"><use data-c=\"31\" xlink:href=\"#MJX-84-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8573.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-84-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8962.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-84-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9351.4,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-84-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10401.7,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-84-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(11401.9,0)\"><use data-c=\"31\" xlink:href=\"#MJX-84-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11901.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-84-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12513.1,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-84-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13513.3,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-84-TEX-I-1D45F\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mo stretchy=\"false\">(<\/mo><mi>D<\/mi><mo>\u2212<\/mo><mi>K<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>J<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>D<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>r<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Since <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.808ex\" height=\"1.731ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 4335 765\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-415-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-415-TEX-N-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><path id=\"MJX-415-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-415-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-415-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45F\" xlink:href=\"#MJX-415-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(728.8,0)\"><use data-c=\"3C\" xlink:href=\"#MJX-415-TEX-N-3C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1784.6,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-415-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2834.8,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-415-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3835,0)\"><use data-c=\"31\" xlink:href=\"#MJX-415-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>r<\/mi><mo>&lt;<\/mo><mi>D<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">r<D-1<\/script><span>, we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n321\" cid=\"n321\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 29.619ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 29.619ex;\"><defs><path id=\"MJX-85-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-85-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-85-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-85-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-85-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-85-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path id=\"MJX-85-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-85-TEX-N-A0\" d=\"\"><\/path><path id=\"MJX-85-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-85-TEX-I-1D45C\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path id=\"MJX-85-TEX-I-1D451\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-85-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-85-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"5745.7 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D45F\" xlink:href=\"#MJX-85-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(728.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-85-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1784.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-85-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2173.6,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-85-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3223.8,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-85-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4224,0)\"><use data-c=\"1D43E\" xlink:href=\"#MJX-85-TEX-I-1D43E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5113,0)\"><use data-c=\"29\" xlink:href=\"#MJX-85-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(5502,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-85-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5752,0)\"><use data-c=\"1D45A\" xlink:href=\"#MJX-85-TEX-I-1D45A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6630,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-85-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7115,0)\"><use data-c=\"1D451\" xlink:href=\"#MJX-85-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(7635,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-85-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7885,0)\"><use data-c=\"28\" xlink:href=\"#MJX-85-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8274,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-85-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9324.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-85-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(10324.4,0)\"><use data-c=\"31\" xlink:href=\"#MJX-85-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10824.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-85-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11213.4,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-85-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>r<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>D<\/mi><mo>\u2212<\/mo><mi>K<\/mi><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;<\/mtext><mi>m<\/mi><mi>o<\/mi><mi>d<\/mi><mtext>&nbsp;<\/mtext><mo stretchy=\"false\">(<\/mo><mi>D<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='4-tunstall-codes'><span>(4) Tunstall Codes<\/span><\/h3><blockquote><p><span>The proper <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> dictionary codes with fixed block length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.541ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 681 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-417-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-417-TEX-I-1D43F\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>L<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">L<\/script><span> that maximize <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.704ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2079 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-420-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-420-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-420-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-420-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-420-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-420-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-420-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1801,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-420-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>B<\/mi><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[B]<\/script><span> for a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.103ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3581.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-465-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path id=\"MJX-465-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-465-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-465-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-465-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43E\" xlink:href=\"#MJX-465-TEX-I-1D43E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1111.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-465-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2111.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-465-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2640.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-465-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3091.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-465-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>K<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">K-ary<\/script><span> random variable are called <\/span><em><span>Tunstall codes<\/span><\/em><span>.<\/span><\/p><p><span>The branches of the tree are <\/span><strong><span>source sequence<\/span><\/strong><span> with maximal <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.704ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2079 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-420-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-420-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-420-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-420-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-420-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-420-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-420-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1801,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-420-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>B<\/mi><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[B]<\/script><span>.<\/span><\/p><\/blockquote><ul><li><p><span>A <\/span><strong><span>proper dictionary<\/span><\/strong><span> satisfies the following two conditions: The dictionary (or message set) satisfies the prefix-free condition, i.e., the dictionary words are leaves of a rooted tree. No leaves are unused.<\/span><\/p><\/li><li><p><span>Choose the largest <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.432ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 633 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-421-TEX-I-1D43D\" d=\"M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43D\" xlink:href=\"#MJX-421-TEX-I-1D43D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>J<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">J<\/script><span> that satisfies <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"25.702ex\" height=\"2.48ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 11360.4 1096\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-422-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path id=\"MJX-422-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-422-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-422-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-422-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-422-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-422-TEX-I-1D43D\" d=\"M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z\"><\/path><path id=\"MJX-422-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-422-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-422-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43E\" xlink:href=\"#MJX-422-TEX-I-1D43E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1111.2,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-422-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2111.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-422-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2500.4,0)\"><use data-c=\"1D43E\" xlink:href=\"#MJX-422-TEX-I-1D43E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3611.7,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-422-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(4611.9,0)\"><use data-c=\"31\" xlink:href=\"#MJX-422-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5111.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-422-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5500.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-422-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5889.9,0)\"><use data-c=\"1D43D\" xlink:href=\"#MJX-422-TEX-I-1D43D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6745.1,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-422-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(7745.3,0)\"><use data-c=\"31\" xlink:href=\"#MJX-422-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8245.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-422-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8912.1,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-422-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(9967.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-422-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,363) scale(0.707)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-422-TEX-I-1D43F\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>K<\/mi><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>K<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>J<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><msup><mi>D<\/mi><mi>L<\/mi><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">K+(K-1)(J-1)\\le D^L<\/script><span>.<\/span><\/p><\/li><li><p><span>Similar to Huffman code, we now have <\/span><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n333\" cid=\"n333\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 19.043ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.475ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.172ex; min-width: 19.043ex;\"><defs><path id=\"MJX-86-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-86-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-86-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-86-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-86-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-86-TEX-N-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-86-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-86-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-86-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-86-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-86-TEX-I-1D459\" d=\"M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path id=\"MJX-86-TEX-I-1D45C\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path id=\"MJX-86-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-86-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-86-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -1460)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1460) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3408.6 -1460 1 2420\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mfrac\"><g data-mml-node=\"mi\" transform=\"translate(919,676)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-86-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-710)\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-86-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-86-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-86-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1801,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-86-TEX-N-5D\"><\/use><\/g><\/g><rect width=\"2279\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(2796.8,0)\"><use data-c=\"2265\" xlink:href=\"#MJX-86-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(3852.6,0)\"><g data-mml-node=\"mrow\" transform=\"translate(223.3,710)\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-86-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-86-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-86-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2129,0)\"><use data-c=\"29\" xlink:href=\"#MJX-86-TEX-N-29\"><\/use><\/g><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-686)\"><g data-mml-node=\"mi\"><use data-c=\"1D459\" xlink:href=\"#MJX-86-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(298,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-86-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(783,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-86-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-86-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1696.6,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-86-TEX-I-1D437\"><\/use><\/g><\/g><rect width=\"2724.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1460 1 2420\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mfrac><mi>L<\/mi><mrow><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>B<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/mfrac><mo>\u2265<\/mo><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Proof. <\/span><\/p><p><span>Using Leaf Entropy Lemma, we get<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n336\" 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xlink:href=\"#MJX-87-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6179.6,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-87-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(6664.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-87-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-87-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(7578.2,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-87-TEX-I-1D437\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right center left\" columnspacing=\"0em 0.278em\" rowspacing=\"3pt\"><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>U<\/mi><mi>L<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi><\/mi><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>j<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>J<\/mi><\/mrow><\/munderover><msub><mi>P<\/mi><mi>j<\/mi><\/msub><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>Q<\/mi><mi>j<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>j<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>J<\/mi><\/mrow><\/munderover><msub><mi>P<\/mi><mi>j<\/mi><\/msub><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>B<\/mi><mo stretchy=\"false\">]<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>U<\/mi><mi>L<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><\/mtd><mtd><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msup><mi>D<\/mi><mi>L<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>L<\/mi><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Combine both and finish the proof.<\/span><\/p><blockquote><h4 id='theorem-34-converse-to-the-coding-theorem-for-a-dms'><span>Theorem 3.4 Converse to the Coding Theorem for a DMS.<\/span><\/h4><p><span>A <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 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xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> prefix-free encoding of a proper dictionary for a DMS satisfies<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n341\" cid=\"n341\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container 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stretchy=\"false\">]<\/mo><\/mrow><mrow><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>B<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/mfrac><mo>\u2265<\/mo><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><p><span>This is a more general lower bound that applies to first using a variable-length-to-block encoder with a proper dictionary followed by a prefix-free block-to-variable-length encoder.<\/span><\/p><p><span>First, we apply Tunstall code to encode a source block of length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 683\" 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transform=\"translate(6382.5,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-89-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6660.5,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-89-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7548.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-89-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7937.5,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-89-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8789.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-89-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9178.5,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-89-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -849.8 1 1199.7\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-99.8)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>U<\/mi><mrow data-mjx-texclass=\"ORD\"><msup><mi>L<\/mi><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>B<\/mi><mo stretchy=\"false\">]<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Then, we apply Huffman code to encode a single <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.686ex\" height=\"2.499ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -899.7 4281.2 1104.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-426-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-426-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-426-TEX-V-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><path id=\"MJX-426-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-426-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-426-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-426-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-426-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(861,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-426-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(714,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-426-TEX-V-2032\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1811,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-426-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2811.2,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-426-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3340.2,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-426-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3791.2,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-426-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>D<\/mi><mrow data-mjx-texclass=\"ORD\"><msup><mi>L<\/mi><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><\/mrow><\/msup><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D^{L'}-ary<\/script><span> random variable into a sequence of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> letters,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n346\" cid=\"n346\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 20.137ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.658ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.263ex; min-width: 20.137ex;\"><defs><path id=\"MJX-90-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 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height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(7022.5,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-90-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1500.3 1 2500.7\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-109.3)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><mo>\u2265<\/mo><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>U<\/mi><mrow data-mjx-texclass=\"ORD\"><msup><mi>L<\/mi><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='5-huffman-codes-and-tunstall-codes'><span>(5) Huffman Codes and Tunstall Codes<\/span><\/h3><p><span>Huffman codes and Tunstall codes are both types of prefix-free codes, which are coding schemes in which no code is a prefix of any other code, allowing messages to be uniquely decoded. However, there are some key differences between these two types of codes:<\/span><\/p><p><span>Huffman codes (block to variable length) are constructed using a method called <\/span><em><span>Huffman coding<\/span><\/em><span>, which is a lossless data compression algorithm that assigns shorter codes to more probable symbols and longer codes to less probable symbols, in order to minimize the expected length of the encoded message. <\/span><\/p><p><span>Tunstall codes (variable length to block), on the other hand, are constructed using a method called the <\/span><em><span>Tunstall algorithm<\/span><\/em><span>, which is a lossless data compression algorithm that generates a prefix-free code by decomposing code into overlapping blocks and assigning a unique message to each block.<\/span><\/p><h2 id='chapter-4-coding-of-stationary-sources'><span>Chapter 4 Coding of Stationary Sources<\/span><\/h2><h3 id='1-discrete-stationary-sources'><span>(1) Discrete Stationary Sources<\/span><\/h3><p><span>A DSS output <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"15.033ex\" height=\"1.984ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 6644.8 877\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-428-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 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scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-428-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2973.8,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-428-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3418.4,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-428-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3863.1,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-428-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4307.8,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-428-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4752.4,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-428-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5197.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-428-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D435\" xlink:href=\"#MJX-428-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><msub><mi>X<\/mi><mn>2<\/mn><\/msub><mo>,<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>B<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X_1, X_2, ..., X_B<\/script><span>, we define the entropy per source letter as <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n354\" cid=\"n354\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 24.163ex; position: relative;\"><svg 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666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-91-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -1264)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1264) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"4540.1 -1264 1 2028\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-78)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-91-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(864,-150) scale(0.707)\"><use data-c=\"1D435\" xlink:href=\"#MJX-91-TEX-I-1D435\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1450.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-91-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1839.7,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-91-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2691.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-91-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3358.5,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-91-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(4414.2,0)\"><g data-mml-node=\"mn\" transform=\"translate(349.5,676)\"><use data-c=\"31\" xlink:href=\"#MJX-91-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(220,-686)\"><use data-c=\"1D435\" xlink:href=\"#MJX-91-TEX-I-1D435\"><\/use><\/g><rect width=\"959\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mi\" transform=\"translate(5613.2,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-91-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6501.2,0)\"><use data-c=\"28\" xlink:href=\"#MJX-91-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(6890.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-91-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(936.2,413) scale(0.707)\"><use data-c=\"1D435\" xlink:href=\"#MJX-91-TEX-I-1D435\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8413.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-91-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8802.1,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-91-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1264 1 2028\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-78)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>H<\/mi><mi>B<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mi>B<\/mi><\/mfrac><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>The entropy rate is defined as <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n356\" cid=\"n356\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 29.34ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"4.751ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.81ex; min-width: 29.34ex;\"><defs><path id=\"MJX-92-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-92-TEX-N-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><path id=\"MJX-92-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-92-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-92-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-92-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-92-TEX-N-6C\" d=\"M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z\"><\/path><path id=\"MJX-92-TEX-N-69\" d=\"M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z\"><\/path><path id=\"MJX-92-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-92-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-92-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-92-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-92-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -1299.9)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1299.9) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"5684.1 -1299.9 1 2099.7\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-42.1)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" 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transform=\"translate(278,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-92-TEX-N-6D\" transform=\"translate(556,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(0,-650) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-92-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(759,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-92-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1759,0)\"><use data-c=\"221E\" xlink:href=\"#MJX-92-TEX-N-221E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mfrac\" transform=\"translate(6702.2,0)\"><g data-mml-node=\"mn\" transform=\"translate(349.5,676)\"><use data-c=\"31\" xlink:href=\"#MJX-92-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(220,-686)\"><use data-c=\"1D435\" xlink:href=\"#MJX-92-TEX-I-1D435\"><\/use><\/g><rect width=\"959\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mi\" transform=\"translate(7901.2,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-92-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8789.2,0)\"><use data-c=\"28\" xlink:href=\"#MJX-92-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(9178.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-92-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(936.2,413) scale(0.707)\"><use data-c=\"1D435\" xlink:href=\"#MJX-92-TEX-I-1D435\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10701.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-92-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11090.1,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-92-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1299.9 1 2099.7\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-42.1)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>H<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u221e<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">lim<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo accent=\"false\" stretchy=\"false\">\u2192<\/mo><mi mathvariant=\"normal\">\u221e<\/mi><\/mrow><\/munder><mfrac><mn>1<\/mn><mi>B<\/mi><\/mfrac><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>It has property<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n359\" cid=\"n359\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 32.576ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"3.741ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.305ex; min-width: 32.576ex;\"><defs><path id=\"MJX-93-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 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499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-93-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-93-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-93-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-93-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" 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data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>H<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u221e<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">lim<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo accent=\"false\" stretchy=\"false\">\u2192<\/mo><mi mathvariant=\"normal\">\u221e<\/mi><\/mrow><\/munder><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>B<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><blockquote><h4 id='theorem-41-entropy-rate-for-a-dss'><span>Theorem 4.1 Entropy Rate for a DSS<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n362\" cid=\"n362\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 52.868ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.593ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.731ex; min-width: 52.868ex;\"><defs><path id=\"MJX-94-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 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xlink:href=\"#MJX-96-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18752.1,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-96-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>H<\/mi><mi>B<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;<\/mtext><mi>i<\/mi><mi>s<\/mi><mtext>&nbsp;<\/mtext><mi>n<\/mi><mi>o<\/mi><mi>n<\/mi><mo>\u2212<\/mo><mi>i<\/mi><mi>n<\/mi><mi>c<\/mi><mi>r<\/mi><mi>e<\/mi><mi>a<\/mi><mi>s<\/mi><mi>i<\/mi><mi>n<\/mi><mi>g<\/mi><mtext>&nbsp;<\/mtext><mi>a<\/mi><mi>s<\/mi><mtext>&nbsp;<\/mtext><mi>B<\/mi><mtext>&nbsp;<\/mtext><mi>i<\/mi><mi>n<\/mi><mi>c<\/mi><mi>r<\/mi><mi>e<\/mi><mi>a<\/mi><mi>s<\/mi><mi>e<\/mi><mi>s<\/mi><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><p><em><span>Proof.<\/span><\/em><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n367\" cid=\"n367\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: 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data-c=\"7C\" xlink:href=\"#MJX-98-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3906.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-98-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(936.2,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-98-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(759,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-98-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1537,0)\"><use data-c=\"33\" xlink:href=\"#MJX-98-TEX-N-33\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6333,0)\"><use data-c=\"29\" xlink:href=\"#MJX-98-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right center left\" columnspacing=\"0em 0.278em\" rowspacing=\"3pt\"><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><msub><mi>X<\/mi><mn>2<\/mn><\/msub><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mn>2<\/mn><\/msub><msub><mi>X<\/mi><mn>3<\/mn><\/msub><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><msub><mi>X<\/mi><mn>2<\/mn><\/msub><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Thus, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.164ex\" height=\"2.48ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 5818.3 1096\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-429-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-429-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-429-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-429-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-429-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-429-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-429-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-429-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-429-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-429-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1277,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-429-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D435\" xlink:href=\"#MJX-429-TEX-I-1D435\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2724.7,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-429-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3002.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-429-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(936.2,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-429-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(759,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-429-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1537,0)\"><use data-c=\"31\" xlink:href=\"#MJX-429-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5429.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-429-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>B<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H(X_B|X^{B-1})<\/script><span> is non-increasing as <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-469-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-469-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>B<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">B<\/script><span> increases.<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n371\" cid=\"n371\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 58.263ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.593ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.731ex; min-width: 58.263ex;\"><defs><path id=\"MJX-99-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-99-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-99-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-99-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 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stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>B<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2265<\/mo><\/mtd><mtd><mi>B<\/mi><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>B<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><em><span>Proof.<\/span><\/em><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n375\" cid=\"n375\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" 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data-c=\"28\" xlink:href=\"#MJX-101-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1839.7,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-101-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2691.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-101-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3080.7,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-101-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right center left\" columnspacing=\"0em 0.278em\" rowspacing=\"3pt\"><mtr><mtd><msub><mi>H<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mfrac><mn>1<\/mn><mrow><mi>B<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>B<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><\/mrow><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>B<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>B<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><\/mtd><mtd><mfrac><mrow><msub><mi>H<\/mi><mi>B<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>B<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><\/mtd><mtd><msub><mi>H<\/mi><mi>B<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='2-entropy-bounds-on-source-coding-a-dss'><span>(2) Entropy Bounds on Source Coding a DSS<\/span><\/h3><blockquote><h4 id='theorem-42-coding-theorem-for-a-dss'><span>Theorem 4.2 Coding Theorem for a DSS<\/span><\/h4><p><span>The average code word length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.527ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2001 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-464-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-464-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-464-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-464-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-464-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-464-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-464-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1723,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-464-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[L]<\/script><span> of an optimum <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 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xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> prefix-free code for a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.103ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3581.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-465-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 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transform=\"translate(2111.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-465-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2640.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-465-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3091.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-465-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>K<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">K-ary<\/script><span> <\/span><strong><span>DSS<\/span><\/strong><span> that is parsed into blocks of length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 683\" 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displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mfrac><mrow><msub><mi>H<\/mi><mi mathvariant=\"normal\">\u221e<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>\u2264<\/mo><mfrac><mrow><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mi>B<\/mi><\/mfrac><mo>&lt;<\/mo><mfrac><mrow><msub><mi>H<\/mi><mi>B<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mn>1<\/mn><mi>B<\/mi><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><p><em><span>Proof.<\/span><\/em><\/p><p><span>We already have the coding theorem for a single random variable <\/span><mjx-container class=\"MathJax\" 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xlink:href=\"#MJX-103-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(783,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-103-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-103-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1696.6,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-103-TEX-I-1D437\"><\/use><\/g><\/g><rect width=\"2724.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(10819.4,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-103-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(11819.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-103-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12319.7,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-103-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" 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stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mn>1<\/mn><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Now we replace the single random variable with a block of length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-469-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 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xlink:href=\"#MJX-104-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(298,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-104-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(783,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-104-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-104-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1696.6,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-104-TEX-I-1D437\"><\/use><\/g><\/g><rect width=\"3388.9\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(12148.1,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-104-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(13148.3,0)\"><use data-c=\"31\" xlink:href=\"#MJX-104-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13648.3,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-104-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1473.5 1 2447\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-82.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>\u2264<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><mo>&lt;<\/mo><mfrac><mrow><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mn>1<\/mn><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Using the notation <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"19.157ex\" height=\"2.755ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -864.9 8467.4 1217.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-437-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-437-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-437-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-437-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-437-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 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73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-437-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(864,-150) scale(0.707)\"><use data-c=\"1D435\" xlink:href=\"#MJX-437-TEX-I-1D435\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1450.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-437-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1839.7,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-437-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2691.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-437-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3358.5,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-437-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(4414.2,0)\"><g data-mml-node=\"mn\" transform=\"translate(255.4,394) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-437-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(220,-345) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-437-TEX-I-1D45B\"><\/use><\/g><rect width=\"624.3\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mi\" transform=\"translate(5278.5,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-437-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6166.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-437-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(6555.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-437-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(936.2,363) scale(0.707)\"><use data-c=\"1D435\" xlink:href=\"#MJX-437-TEX-I-1D435\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8078.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-437-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>H<\/mi><mi>B<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mi>n<\/mi><\/mfrac><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H_B(X)=\\frac{1}{n}H(X^B)<\/script><span>, i.e., entropy per source symbol, we get <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n387\" cid=\"n387\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 37.215ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.319ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.094ex; min-width: 37.215ex;\"><defs><path id=\"MJX-105-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-105-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-105-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-105-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-105-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-105-TEX-I-1D459\" d=\"M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path id=\"MJX-105-TEX-I-1D45C\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path id=\"MJX-105-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 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660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-105-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -1425.5)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1425.5) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"7424.5 -1425.5 1 2351\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-34.5)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mfrac\"><g data-mml-node=\"mrow\" transform=\"translate(220,710)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" 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data-c=\"1D44B\" xlink:href=\"#MJX-105-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2691.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-105-TEX-N-29\"><\/use><\/g><\/g><g data-mml-node=\"mrow\" transform=\"translate(498.1,-686)\"><g data-mml-node=\"mi\"><use data-c=\"1D459\" xlink:href=\"#MJX-105-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(298,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-105-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(783,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-105-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-105-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1696.6,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-105-TEX-I-1D437\"><\/use><\/g><\/g><rect width=\"3280.7\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(12371.7,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-105-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(13371.9,0)\"><g data-mml-node=\"mn\" transform=\"translate(349.5,676)\"><use data-c=\"31\" xlink:href=\"#MJX-105-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(220,-686)\"><use data-c=\"1D435\" xlink:href=\"#MJX-105-TEX-I-1D435\"><\/use><\/g><rect width=\"959\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(14570.9,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-105-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1425.5 1 2351\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-34.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mfrac><mrow><msub><mi>H<\/mi><mi>B<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>\u2264<\/mo><mfrac><mrow><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mi>B<\/mi><\/mfrac><mo>&lt;<\/mo><mfrac><mrow><msub><mi>H<\/mi><mi>B<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mn>1<\/mn><mi>B<\/mi><\/mfrac><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Since <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.97ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3080.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-438-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-438-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-438-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-438-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-438-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-438-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(864,-150) scale(0.707)\"><use data-c=\"1D435\" xlink:href=\"#MJX-438-TEX-I-1D435\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1450.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-438-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1839.7,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-438-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2691.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-438-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>H<\/mi><mi>B<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H_B(X)<\/script><span> is non-increasing as <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-469-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-469-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>B<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">B<\/script><span> increases, we have <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.342ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7665.4 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-440-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-440-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-440-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-440-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 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viewBox=\"0 -1425.5 1 2351\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-34.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mfrac><mrow><msub><mi>H<\/mi><mi mathvariant=\"normal\">\u221e<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>\u2264<\/mo><mfrac><mrow><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mi>B<\/mi><\/mfrac><mo>&lt;<\/mo><mfrac><mrow><msub><mi>H<\/mi><mi>B<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mi>D<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mn>1<\/mn><mi>B<\/mi><\/mfrac><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='3-elias-code-for-positive-integers'><span>(3) Elias Code for Positive Integers<\/span><\/h3><figure><table><thead><tr><th><span>x<\/span><\/th><th style='text-align:center;' ><span>Elias Code <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.189ex\" height=\"2.48ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 2735.7 1096\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-441-TEX-I-1D448\" d=\"M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z\"><\/path><path id=\"MJX-441-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-441-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-441-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-441-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D448\" xlink:href=\"#MJX-441-TEX-I-1D448\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(854.2,363) scale(0.707)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-441-TEX-I-1D43F\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1385.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-441-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1774.7,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-441-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2346.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-441-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>U<\/mi><mi>L<\/mi><\/msup><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">U^L(x)<\/script><\/th><th><span>Elias Code Lenght <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.595ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2031 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-442-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-442-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-442-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-442-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-442-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(681,0)\"><use data-c=\"28\" xlink:href=\"#MJX-442-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1070,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-442-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1642,0)\"><use data-c=\"29\" xlink:href=\"#MJX-442-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">L(x)<\/script><\/th><\/tr><\/thead><tbody><tr><td><span>1<\/span><\/td><td style='text-align:center;' ><span>1<\/span><\/td><td><span>1<\/span><\/td><\/tr><tr><td><span>2<\/span><\/td><td style='text-align:center;' ><span>010.0<\/span><\/td><td><span>4<\/span><\/td><\/tr><tr><td><span>3<\/span><\/td><td style='text-align:center;' ><span>010.1<\/span><\/td><td><span>4<\/span><\/td><\/tr><tr><td><span>4<\/span><\/td><td style='text-align:center;' ><span>011.00<\/span><\/td><td><span>5<\/span><\/td><\/tr><tr><td><span>5<\/span><\/td><td style='text-align:center;' ><span>011.01<\/span><\/td><td><span>5<\/span><\/td><\/tr><tr><td><span>6<\/span><\/td><td style='text-align:center;' ><span>011.10<\/span><\/td><td><span>5<\/span><\/td><\/tr><tr><td><span>7<\/span><\/td><td style='text-align:center;' ><span>011.11<\/span><\/td><td><span>5<\/span><\/td><\/tr><tr><td><span>8<\/span><\/td><td style='text-align:center;' ><span>00100.000<\/span><\/td><td><span>8<\/span><\/td><\/tr><tr><td><span>9<\/span><\/td><td style='text-align:center;' ><span>00100.001<\/span><\/td><td><span>8<\/span><\/td><\/tr><\/tbody><\/table><\/figure><p><em><span>Proof.<\/span><\/em><\/p><hr \/><p><span>For <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> representation of integers, we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n435\" cid=\"n435\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 25.543ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 25.543ex;\"><defs><path id=\"MJX-107-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-107-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-107-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-107-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-107-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-107-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-107-TEX-N-230A\" d=\"M174 734Q174 735 175 737T177 740T180 744T184 747T189 749T196 750Q206 748 214 735V-210H310H373Q401 -210 411 -213T422 -230T411 -247T369 -251Q362 -251 338 -251T298 -250H190Q178 -246 174 -234V734Z\"><\/path><path id=\"MJX-107-TEX-I-1D459\" d=\"M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path id=\"MJX-107-TEX-I-1D45C\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path id=\"MJX-107-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-107-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-107-TEX-N-230B\" d=\"M229 734Q229 735 230 737T232 740T235 744T239 747T244 749T251 750Q262 748 269 735V-235Q266 -240 256 -249L147 -250H77Q43 -250 32 -247T21 -230T32 -213T72 -209Q79 -209 99 -209T133 -210H229V734Z\"><\/path><path id=\"MJX-107-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"4845 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-107-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(714,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-107-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1117.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-107-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1506.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-107-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2078.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-107-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2745.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-107-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3801.1,0)\"><use data-c=\"230A\" xlink:href=\"#MJX-107-TEX-N-230A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4245.1,0)\"><use data-c=\"1D459\" xlink:href=\"#MJX-107-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4543.1,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-107-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5028.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-107-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"1D437\" xlink:href=\"#MJX-107-TEX-I-1D437\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6173.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-107-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6562.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-107-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7134.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-107-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7523.6,0)\"><use data-c=\"230B\" xlink:href=\"#MJX-107-TEX-N-230B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8189.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-107-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(9190,0)\"><use data-c=\"31\" xlink:href=\"#MJX-107-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>L<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo fence=\"false\" stretchy=\"false\">\u230a<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mi>D<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo fence=\"false\" stretchy=\"false\">\u230b<\/mo><mo>+<\/mo><mn>1<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><figure><table><thead><tr><th><span>x<\/span><\/th><th style='text-align:center;' ><span>Binary Representation  <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.888ex\" height=\"2.48ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 3044.4 1096\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-446-TEX-I-1D448\" d=\"M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z\"><\/path><path id=\"MJX-446-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-446-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-446-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-446-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-446-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D448\" xlink:href=\"#MJX-446-TEX-I-1D448\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(854.2,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-446-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(714,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-446-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1694.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-446-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2083.4,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-446-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2655.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-446-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>U<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>L<\/mi><mn>1<\/mn><\/msub><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">U^{L_1}(x)<\/script><\/th><th><span>Code Lenght <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.583ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2467.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-445-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-445-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-445-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-445-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-445-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-445-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(714,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-445-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1117.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-445-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1506.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-445-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2078.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-445-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>L<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">L_1(x)<\/script><\/th><\/tr><\/thead><tbody><tr><td><span>1<\/span><\/td><td style='text-align:center;' ><span>1<\/span><\/td><td><span>1<\/span><\/td><\/tr><tr><td><span>2<\/span><\/td><td style='text-align:center;' ><span>10<\/span><\/td><td><span>2<\/span><\/td><\/tr><tr><td><span>3<\/span><\/td><td style='text-align:center;' ><span>11<\/span><\/td><td><span>2<\/span><\/td><\/tr><tr><td><span>4<\/span><\/td><td style='text-align:center;' ><span>100<\/span><\/td><td><span>3<\/span><\/td><\/tr><tr><td><span>5<\/span><\/td><td style='text-align:center;' ><span>101<\/span><\/td><td><span>3<\/span><\/td><\/tr><tr><td><span>6<\/span><\/td><td style='text-align:center;' ><span>110<\/span><\/td><td><span>3<\/span><\/td><\/tr><tr><td><span>7<\/span><\/td><td style='text-align:center;' ><span>111<\/span><\/td><td><span>3<\/span><\/td><\/tr><tr><td><span>8<\/span><\/td><td style='text-align:center;' ><span>1000<\/span><\/td><td><span>4<\/span><\/td><\/tr><tr><td><span>9<\/span><\/td><td style='text-align:center;' ><span>1001<\/span><\/td><td><span>4<\/span><\/td><\/tr><\/tbody><\/table><\/figure><hr \/><p><span>This base-2 code is not prefix-free, so does a ternary, etc. representation. Therefore, we pre-pad each code <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.888ex\" height=\"2.48ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 3044.4 1096\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-446-TEX-I-1D448\" d=\"M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z\"><\/path><path id=\"MJX-446-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-446-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-446-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-446-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-446-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D448\" xlink:href=\"#MJX-446-TEX-I-1D448\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(854.2,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-446-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(714,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-446-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1694.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-446-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2083.4,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-446-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2655.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-446-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>U<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>L<\/mi><mn>1<\/mn><\/msub><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">U^{L_1}(x)<\/script><span> with a certain number of zeros to make it prefix-free. The decoder counts the number of zeros to determine how many subsequent <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> symbols it should consider after the first digit.<\/span><\/p><p><span>For <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> representation of integers, we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n480\" cid=\"n480\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 40.302ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 40.302ex;\"><defs><path id=\"MJX-108-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-108-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-108-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-108-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-108-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-108-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-108-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-108-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-108-TEX-N-230A\" d=\"M174 734Q174 735 175 737T177 740T180 744T184 747T189 749T196 750Q206 748 214 735V-210H310H373Q401 -210 411 -213T422 -230T411 -247T369 -251Q362 -251 338 -251T298 -250H190Q178 -246 174 -234V734Z\"><\/path><path id=\"MJX-108-TEX-I-1D459\" d=\"M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path id=\"MJX-108-TEX-I-1D45C\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path id=\"MJX-108-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 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737T232 740T235 744T239 747T244 749T251 750Q262 748 269 735V-235Q266 -240 256 -249L147 -250H77Q43 -250 32 -247T21 -230T32 -213T72 -209Q79 -209 99 -209T133 -210H229V734Z\"><\/path><path id=\"MJX-108-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"8106.8 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-108-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(714,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-108-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1117.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-108-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1506.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-108-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2078.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-108-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2745.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-108-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3801.1,0)\"><use data-c=\"32\" xlink:href=\"#MJX-108-TEX-N-32\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4301.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-108-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(714,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-108-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5418.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-108-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5807.7,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-108-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6379.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-108-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6990.9,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-108-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(7991.1,0)\"><use data-c=\"31\" xlink:href=\"#MJX-108-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8768.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-108-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(9824.7,0)\"><use data-c=\"32\" xlink:href=\"#MJX-108-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10324.7,0)\"><use data-c=\"230A\" xlink:href=\"#MJX-108-TEX-N-230A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10768.7,0)\"><use data-c=\"1D459\" xlink:href=\"#MJX-108-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11066.7,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-108-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(11551.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-108-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(510,-150) scale(0.707)\"><use data-c=\"1D437\" xlink:href=\"#MJX-108-TEX-I-1D437\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(12697.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-108-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13086.1,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-108-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13658.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-108-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14047.1,0)\"><use data-c=\"230B\" xlink:href=\"#MJX-108-TEX-N-230B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14713.4,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-108-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(15713.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-108-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>L<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>2<\/mn><msub><mi>L<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>2<\/mn><mo fence=\"false\" stretchy=\"false\">\u230a<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mi>D<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo fence=\"false\" stretchy=\"false\">\u230b<\/mo><mo>+<\/mo><mn>1<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><figure><table><thead><tr><th><span>x<\/span><\/th><th style='text-align:center;' ><span>Zero-pre-padded Binary Representation  <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.888ex\" height=\"2.48ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 3044.4 1096\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-449-TEX-I-1D448\" d=\"M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z\"><\/path><path id=\"MJX-449-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-449-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-449-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-449-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-449-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D448\" xlink:href=\"#MJX-449-TEX-I-1D448\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(854.2,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-449-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(714,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-449-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1694.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-449-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2083.4,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-449-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2655.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-449-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>U<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>L<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">U^{L_2}(x)<\/script><\/th><th><span>Code Lenght <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.583ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2467.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-450-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-450-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-450-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-450-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-450-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-450-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(714,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-450-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1117.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-450-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1506.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-450-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2078.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-450-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>L<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">L_2(x)<\/script><\/th><\/tr><\/thead><tbody><tr><td><span>1<\/span><\/td><td style='text-align:center;' ><span>1<\/span><\/td><td><span>1<\/span><\/td><\/tr><tr><td><span>2<\/span><\/td><td style='text-align:center;' ><span>0.10<\/span><\/td><td><span>3<\/span><\/td><\/tr><tr><td><span>3<\/span><\/td><td style='text-align:center;' ><span>0.11<\/span><\/td><td><span>3<\/span><\/td><\/tr><tr><td><span>4<\/span><\/td><td style='text-align:center;' ><span>00.100<\/span><\/td><td><span>5<\/span><\/td><\/tr><tr><td><span>5<\/span><\/td><td style='text-align:center;' ><span>00.101<\/span><\/td><td><span>5<\/span><\/td><\/tr><tr><td><span>6<\/span><\/td><td style='text-align:center;' ><span>00.110<\/span><\/td><td><span>5<\/span><\/td><\/tr><tr><td><span>7<\/span><\/td><td style='text-align:center;' ><span>00.111<\/span><\/td><td><span>5<\/span><\/td><\/tr><tr><td><span>8<\/span><\/td><td style='text-align:center;' ><span>000.1000<\/span><\/td><td><span>7<\/span><\/td><\/tr><tr><td><span>9<\/span><\/td><td style='text-align:center;' ><span>000.1001<\/span><\/td><td><span>7<\/span><\/td><\/tr><\/tbody><\/table><\/figure><hr \/><p><span>Elias pre-pads differently to shorten this code by using <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.834ex\" height=\"1.964ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 1694.4 868\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-451-TEX-I-1D448\" d=\"M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z\"><\/path><path id=\"MJX-451-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-451-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D448\" xlink:href=\"#MJX-451-TEX-I-1D448\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(854.2,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-451-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(714,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-451-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>U<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>L<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">U^{L_2}<\/script><span> to count the number of digits (including the beginning <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.507ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 500 666\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-530-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-530-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">1<\/script><span>). <\/span><\/p><p><span>For <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.965ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3520.4 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-453-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-453-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-453-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-453-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-453-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-453-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1050.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-453-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2050.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-453-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2579.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-453-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3030.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-453-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>D<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">D-ary<\/script><span> representation of integers, we have <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n525\" cid=\"n525\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 76.029ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 76.029ex;\"><defs><path id=\"MJX-109-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-109-TEX-N-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z\"><\/path><path id=\"MJX-109-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-109-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-109-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-109-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-109-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-109-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-109-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-109-TEX-N-230A\" d=\"M174 734Q174 735 175 737T177 740T180 744T184 747T189 749T196 750Q206 748 214 735V-210H310H373Q401 -210 411 -213T422 -230T411 -247T369 -251Q362 -251 338 -251T298 -250H190Q178 -246 174 -234V734Z\"><\/path><path id=\"MJX-109-TEX-I-1D459\" d=\"M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path id=\"MJX-109-TEX-I-1D45C\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path id=\"MJX-109-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-109-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-109-TEX-N-230B\" d=\"M229 734Q229 735 230 737T232 740T235 744T239 747T244 749T251 750Q262 748 269 735V-235Q266 -240 256 -249L147 -250H77Q43 -250 32 -247T21 -230T32 -213T72 -209Q79 -209 99 -209T133 -210H229V734Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -750)\"><g 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data-mml-node=\"mo\" transform=\"translate(27949.3,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-109-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(28949.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-109-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(29449.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-109-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(29838.6,0)\"><use data-c=\"230B\" xlink:href=\"#MJX-109-TEX-N-230B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(30504.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-109-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(31505,0)\"><use data-c=\"31\" xlink:href=\"#MJX-109-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>L<\/mi><mn>3<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>L<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>L<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>L<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo fence=\"false\" stretchy=\"false\">\u230a<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mi>D<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo fence=\"false\" stretchy=\"false\">\u230b<\/mo><mo>+<\/mo><mn>1<\/mn><mo>+<\/mo><mn>2<\/mn><mo fence=\"false\" stretchy=\"false\">\u230a<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mi>D<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mo fence=\"false\" stretchy=\"false\">\u230a<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mi>D<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo fence=\"false\" stretchy=\"false\">\u230b<\/mo><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo fence=\"false\" stretchy=\"false\">\u230b<\/mo><mo>+<\/mo><mn>1<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><figure><table><thead><tr><th><span>x<\/span><\/th><th style='text-align:center;' ><span>Elias-pre-padded Binary Representation  <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.888ex\" height=\"2.48ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 3044.4 1096\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-454-TEX-I-1D448\" d=\"M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z\"><\/path><path id=\"MJX-454-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-454-TEX-N-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z\"><\/path><path id=\"MJX-454-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-454-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-454-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D448\" xlink:href=\"#MJX-454-TEX-I-1D448\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(854.2,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-454-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(714,-150) scale(0.707)\"><use data-c=\"33\" xlink:href=\"#MJX-454-TEX-N-33\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1694.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-454-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2083.4,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-454-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2655.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-454-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>U<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>L<\/mi><mn>3<\/mn><\/msub><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">U^{L_3}(x)<\/script><\/th><th><span>Code Lenght <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.583ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2467.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-455-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-455-TEX-N-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z\"><\/path><path id=\"MJX-455-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-455-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-455-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-455-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(714,-150) scale(0.707)\"><use data-c=\"33\" xlink:href=\"#MJX-455-TEX-N-33\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1117.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-455-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1506.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-455-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2078.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-455-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>L<\/mi><mn>3<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">L_3(x)<\/script><\/th><\/tr><\/thead><tbody><tr><td><span>1<\/span><\/td><td style='text-align:center;' ><span>1<\/span><\/td><td><span>1<\/span><\/td><\/tr><tr><td><span>2<\/span><\/td><td style='text-align:center;' ><span>010.10<\/span><\/td><td><span>5<\/span><\/td><\/tr><tr><td><span>3<\/span><\/td><td style='text-align:center;' ><span>010.11<\/span><\/td><td><span>5<\/span><\/td><\/tr><tr><td><span>4<\/span><\/td><td style='text-align:center;' ><span>011.100<\/span><\/td><td><span>6<\/span><\/td><\/tr><tr><td><span>5<\/span><\/td><td style='text-align:center;' ><span>011.101<\/span><\/td><td><span>6<\/span><\/td><\/tr><tr><td><span>6<\/span><\/td><td style='text-align:center;' ><span>011.110<\/span><\/td><td><span>6<\/span><\/td><\/tr><tr><td><span>7<\/span><\/td><td style='text-align:center;' ><span>011.111<\/span><\/td><td><span>6<\/span><\/td><\/tr><tr><td><span>8<\/span><\/td><td style='text-align:center;' ><span>00100.1000<\/span><\/td><td><span>9<\/span><\/td><\/tr><tr><td><span>9<\/span><\/td><td style='text-align:center;' ><span>00100.1001<\/span><\/td><td><span>9<\/span><\/td><\/tr><\/tbody><\/table><\/figure><hr \/><p><span>Finally, we get the Elias code by omitting the first &quot;1&quot; after &quot;.&quot; with length <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n570\" cid=\"n570\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 49.908ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 49.908ex;\"><defs><path id=\"MJX-110-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 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xlink:href=\"#MJX-110-TEX-N-230B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16403.7,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-110-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(17403.9,0)\"><use data-c=\"31\" xlink:href=\"#MJX-110-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17903.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-110-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18292.9,0)\"><use data-c=\"230B\" xlink:href=\"#MJX-110-TEX-N-230B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18959.1,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-110-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(19959.3,0)\"><use data-c=\"31\" xlink:href=\"#MJX-110-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo fence=\"false\" stretchy=\"false\">\u230a<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mi>D<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo fence=\"false\" stretchy=\"false\">\u230b<\/mo><mo>+<\/mo><mn>2<\/mn><mo fence=\"false\" stretchy=\"false\">\u230a<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mi>D<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mo fence=\"false\" stretchy=\"false\">\u230a<\/mo><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mi>D<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo fence=\"false\" stretchy=\"false\">\u230b<\/mo><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo fence=\"false\" stretchy=\"false\">\u230b<\/mo><mo>+<\/mo><mn>1<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='4-elias-willems-universal-source-coding'><span>(4) Elias-Willems Universal Source Coding<\/span><\/h3><p><span>We now describe an specific algorithm for source coding when the source statistics are unknown. An important result that we state without proof is that the expected recurrence time of a stationary and ergodic source satisfies<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n573\" cid=\"n573\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 27.446ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.222ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.045ex; min-width: 27.446ex;\"><defs><path id=\"MJX-111-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 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349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-111-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-111-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-111-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 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xlink:href=\"#MJX-111-TEX-N-25B3\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(922,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-111-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2258,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-111-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2536,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-111-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-111-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3968.7,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-111-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5024.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-111-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5553.5,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-111-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6109.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-111-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(7165,0)\"><g data-mml-node=\"mn\" transform=\"translate(1433.1,676)\"><use data-c=\"31\" xlink:href=\"#MJX-111-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-710)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-111-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-111-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-111-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1619.2,0)\"><use data-c=\"28\" xlink:href=\"#MJX-111-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2008.2,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-111-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2537.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-111-TEX-N-29\"><\/use><\/g><\/g><rect width=\"3126.2\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1404 1 2308.1\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,62)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msub><mi mathvariant=\"normal\">\u25b3<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>for all times <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.781ex\" height=\"1.52ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -661 345 672\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-456-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-456-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>i<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">i<\/script><span> and all letters <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.197ex\" height=\"1.02ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -441 529 451\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-460-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-460-TEX-I-1D44E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a<\/script><span>.<\/span><\/p><h4 id='example-41'><span>Example 4.1<\/span><\/h4><p><span>Consider a DMS <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.003ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1327.5 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-538-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-538-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-538-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-538-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X<\/script><span> for which the probability of waiting for <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.932ex\" height=\"1.957ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -661 412 865\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.462ex;\"><defs><path id=\"MJX-459-TEX-I-1D457\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D457\" xlink:href=\"#MJX-459-TEX-I-1D457\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>j<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">j<\/script><span> symbols before seeing the letter <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.197ex\" height=\"1.02ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -441 529 451\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-460-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-460-TEX-I-1D44E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">a<\/script><span> is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"20.469ex\" height=\"2.452ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 9047.4 1083.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-461-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-461-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path 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xlink:href=\"#MJX-461-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(6412.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-461-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-461-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7740.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-461-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8129.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-461-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8658.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-461-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><msup><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>j<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(1-P_X(a))^{j-1}P_X(a)<\/script><span>. The expected waiting time is therefore<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n577\" cid=\"n577\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 53.539ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"6.549ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.709ex; min-width: 53.539ex;\"><defs><path id=\"MJX-112-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 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data-mml-node=\"mi\"><use data-c=\"1D457\" xlink:href=\"#MJX-112-TEX-I-1D457\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(412,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-112-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1190,0)\"><use data-c=\"31\" xlink:href=\"#MJX-112-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(15021.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-112-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-112-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(16349.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-112-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(16738.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-112-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17267.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-112-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17934.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-112-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(18989.9,0)\"><g data-mml-node=\"mn\" transform=\"translate(1287.2,676)\"><use data-c=\"31\" xlink:href=\"#MJX-112-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-710)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-112-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-112-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1327.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-112-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1716.5,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-112-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2245.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-112-TEX-N-29\"><\/use><\/g><\/g><rect width=\"2834.5\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1697.4 1 2894.7\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,134.8)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msub><mi mathvariant=\"normal\">\u25b3<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>j<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi mathvariant=\"normal\">\u221e<\/mi><\/munderover><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><msup><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>j<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>We apply the binary Elias code to <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.751ex\" height=\"1.977ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 1216 873.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-462-TEX-N-25B3\" d=\"M75 0L72 2Q69 3 67 5T62 11T59 20Q59 24 62 30Q65 37 245 370T428 707Q428 708 430 710T436 714T444 716Q451 716 455 712Q459 710 644 368L828 27V20Q828 7 814 0H75ZM610 347L444 653Q443 653 278 347T113 40H775Q775 42 610 347Z\"><\/path><path id=\"MJX-462-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"25B3\" xlink:href=\"#MJX-462-TEX-N-25B3\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(922,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-462-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi mathvariant=\"normal\">\u25b3<\/mi><mi>i<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\triangle_i<\/script><span> and compute<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block 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stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>The second inequality follows from the concavity of logarithm and Jensen&#39;s inequality.<\/span><\/p><p><span>Taking expectations we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n582\" cid=\"n582\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 43.083ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 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xlink:href=\"#MJX-114-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17164.5,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-114-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msub><mi>L<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">]<\/mo><mo>\u2264<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>2<\/mn><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Consider a block of length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-469-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-469-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>B<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">B<\/script><span> instead of a single random variable, we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block 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transform=\"translate(15569.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-115-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16181,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-115-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(17181.2,0)\"><use data-c=\"31\" xlink:href=\"#MJX-115-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17681.2,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-115-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -823 1 1146\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-73)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msub><mi>L<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">]<\/mo><mo>\u2264<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>2<\/mn><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>X<\/mi><mi>B<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><blockquote><h4 id='theorem-43-universal-source-coding-theorem-for-a-dss'><span>Theorem 4.3 Universal Source Coding Theorem for a DSS<\/span><\/h4><p><span>The average code word length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.527ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2001 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-464-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 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data-c=\"1D438\" xlink:href=\"#MJX-464-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-464-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-464-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1723,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-464-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[L]<\/script><span> of a binary code for a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.103ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3581.4 888\" 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382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-465-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43E\" xlink:href=\"#MJX-465-TEX-I-1D43E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1111.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-465-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2111.4,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-465-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2640.4,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-465-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3091.4,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-465-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>K<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>r<\/mi><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">K-ary<\/script><span> DSS that is parsed into blocks of length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-469-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 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xlink:href=\"#MJX-116-TEX-I-1D435\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(13571.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-116-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13960.9,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-116-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14812.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-116-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15424.2,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-116-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(16424.4,0)\"><use data-c=\"31\" xlink:href=\"#MJX-116-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16924.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-116-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17535.6,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-116-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(18535.8,0)\"><g data-mml-node=\"mn\" transform=\"translate(349.5,676)\"><use data-c=\"31\" xlink:href=\"#MJX-116-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(220,-686)\"><use data-c=\"1D435\" xlink:href=\"#MJX-116-TEX-I-1D435\"><\/use><\/g><rect width=\"959\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1323 1 2146\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-137)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mfrac><mrow><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>L<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mi>B<\/mi><\/mfrac><mo>&lt;<\/mo><msub><mi>H<\/mi><mi>B<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mfrac><mn>2<\/mn><mi>B<\/mi><\/mfrac><mi>l<\/mi><mi>o<\/mi><msub><mi>g<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>B<\/mi><msub><mi>H<\/mi><mi>B<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mfrac><mn>1<\/mn><mi>B<\/mi><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><p><span>Moreover, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.376ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3260 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path 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39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-467-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-467-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-467-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1723,0)\"><use 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xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>B<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">B<\/script><span> sufficiently large.<\/span><\/p><h2 id='chapter-5-channel-coding'><span>Chapter 5 Channel Coding<\/span><\/h2><blockquote><h4 id='data-processing'><span>Data Processing<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n594\" cid=\"n594\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"44.843ex\" height=\"28.121ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -6464.7 19820.5 12429.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 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data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mi>X<\/mi><mi>n<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mover><mo>\u2264<\/mo><mrow><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mover><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><mo stretchy=\"false\">(<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>;<\/mo><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.957ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1307 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-520-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-520-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-520-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-520-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-520-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(918,0)\"><use data-c=\"29\" xlink:href=\"#MJX-520-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(a)<\/script><span> follows from <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.4ex\" height=\"2.943ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1051 5038.9 1301\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-471-TEX-I-1D44A\" d=\"M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 628T831 637Q817 637 817 647Q817 650 819 660Q823 676 825 679T839 682Q842 682 856 682T895 682T949 681Q1015 681 1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 154 304T131 622Q129 631 121 633T82 637H58Q51 644 51 648Q52 671 64 683H76Q118 680 176 680Q301 680 313 683H323Q329 677 329 674T327 656Q322 641 318 637H297Q236 634 232 620Q262 160 266 136L501 550L499 587Q496 629 489 632Q483 636 447 637Q428 637 422 639T416 648Q416 650 418 660Q419 664 420 669T421 676T424 680T428 682T436 683Z\"><\/path><path id=\"MJX-471-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><path id=\"MJX-471-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-471-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-471-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-471-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-471-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-471-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44A\" xlink:href=\"#MJX-471-TEX-I-1D44A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(550,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-471-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1325.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-471-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2381.6,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-471-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2931.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-471-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3320.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-471-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(855.1,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-471-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4649.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-471-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mover><mi>W<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>=<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>Y<\/mi><mi>n<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\hat{W}=f(Y^n)<\/script><span> and data processing doesn&#39;t increase mutual information. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.731ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1207 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-472-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-472-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path id=\"MJX-472-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-472-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-472-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(818,0)\"><use data-c=\"29\" xlink:href=\"#MJX-472-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(b)<\/script><span> follows from that the channel is memoryless, i.e., <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.054ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 908 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-473-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-473-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-473-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(614,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-473-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y_i<\/script><span> only depends on <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.613ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1155 840.8\" 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xlink:href=\"#MJX-118-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6317.6,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-118-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7080.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-118-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-3414.3)\"><g data-mml-node=\"mtd\" transform=\"translate(3822.7,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(3822.7,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-118-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1333.6,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-118-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1933.6,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-118-TEX-I-1D436\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo>;<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>W<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>;<\/mo><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mi>n<\/mi><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><\/mrow><\/munder><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>;<\/mo><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mi>n<\/mi><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><mrow data-mjx-texclass=\"ORD\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><\/mrow><\/munder><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>n<\/mi><mi>C<\/mi><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><h3 id='1-rate-reliability-and-cost'><span>(1) Rate, Reliability and Cost<\/span><\/h3><p><span>By reducing rate, one can increase reliability. We thus expect a <\/span><em><span>rate-reliability<\/span><\/em><span> tradeoff. If on introduces a cost <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.459ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 645 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-512-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-512-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>S<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">S<\/script><span> associated with transmission, then there is a <\/span><em><span>capacity-cost<\/span><\/em><span> tradeoff.<\/span><\/p><h3 id='2-memoryless-channels'><span>(2) Memoryless Channels<\/span><\/h3><p><span>An <\/span><strong><span>encoder<\/span><\/strong><span> maps a source message <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.371ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1048 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-476-TEX-I-1D44A\" d=\"M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 628T831 637Q817 637 817 647Q817 650 819 660Q823 676 825 679T839 682Q842 682 856 682T895 682T949 681Q1015 681 1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 154 304T131 622Q129 631 121 633T82 637H58Q51 644 51 648Q52 671 64 683H76Q118 680 176 680Q301 680 313 683H323Q329 677 329 674T327 656Q322 641 318 637H297Q236 634 232 620Q262 160 266 136L501 550L499 587Q496 629 489 632Q483 636 447 637Q428 637 422 639T416 648Q416 650 418 660Q419 664 420 669T421 676T424 680T428 682T436 683Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44A\" xlink:href=\"#MJX-476-TEX-I-1D44A\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>W<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">W<\/script><span> of length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-477-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-477-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span> into a string <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.729ex\" height=\"1.781ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 4742 787\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-478-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-478-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-478-TEX-I-1D44A\" d=\"M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 628T831 637Q817 637 817 647Q817 650 819 660Q823 676 825 679T839 682Q842 682 856 682T895 682T949 681Q1015 681 1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 154 304T131 622Q129 631 121 633T82 637H58Q51 644 51 648Q52 671 64 683H76Q118 680 176 680Q301 680 313 683H323Q329 677 329 674T327 656Q322 641 318 637H297Q236 634 232 620Q262 160 266 136L501 550L499 587Q496 629 489 632Q483 636 447 637Q428 637 422 639T416 648Q416 650 418 660Q419 664 420 669T421 676T424 680T428 682T436 683Z\"><\/path><path id=\"MJX-478-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-478-TEX-I-1D43A\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-478-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1129.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-478-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2185.6,0)\"><use data-c=\"1D44A\" xlink:href=\"#MJX-478-TEX-I-1D44A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3455.8,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-478-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3956,0)\"><use data-c=\"1D43A\" xlink:href=\"#MJX-478-TEX-I-1D43A\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><mo>=<\/mo><mi>W<\/mi><mo>\u22c5<\/mo><mi>G<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X=W\\cdot G<\/script><span> of length <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.357ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 600 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-479-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-479-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n<\/script><span>. We view <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.191ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1410.5 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-480-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-480-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-480-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(936.2,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-480-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>X<\/mi><mi>n<\/mi><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X^n<\/script><span> as a string <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.236ex\" height=\"1.964ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 1872.3 868\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-481-TEX-I-1D449\" d=\"M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z\"><\/path><path id=\"MJX-481-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-481-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D449\" xlink:href=\"#MJX-481-TEX-I-1D449\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(861.3,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-481-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(600,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-481-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>V<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mi>R<\/mi><\/mrow><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">V^{nR}<\/script><span> of independent bits. The rate <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.048ex;\"><defs><path id=\"MJX-482-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-482-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>R<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">R<\/script><span> measures how many bits are needed to represent a symbol <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-483-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-483-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span>.<\/span><\/p><h3 id='3-cost-functions'><span>(3) Cost Functions <\/span><\/h3><p><span>Suppose that transmitting <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.442ex\" height=\"1.553ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -675.5 1079.3 686.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-484-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-484-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-484-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-484-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>x<\/mi><mi>n<\/mi><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x^n<\/script><span> and receiving <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.256ex\" height=\"1.992ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -675.5 997.3 880.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-485-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-485-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-485-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(523,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-485-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>y<\/mi><mi>n<\/mi><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">y^n<\/script><span> incurs with a cost of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.673ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4275.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-486-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-486-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-486-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-486-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-486-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-486-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path 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transform=\"translate(605,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-486-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2444.5,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-486-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(2889.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-486-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(523,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-486-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3886.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-486-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>s<\/mi><mi>n<\/mi><\/msup><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mi>n<\/mi><\/msup><mo>,<\/mo><msup><mi>y<\/mi><mi>n<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">s^n(x^n,y^n)<\/script><span> units. Naturally, we want to bound the average cost, otherwise we can send information with infinite energy and ignore all noises<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n603\" cid=\"n603\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 20.756ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 20.756ex;\"><defs><path id=\"MJX-119-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-119-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-119-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-119-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-119-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-119-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-119-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 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d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-119-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-119-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-119-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 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xlink:href=\"#MJX-119-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-119-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(1042,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D460\" xlink:href=\"#MJX-119-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(502,413) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-119-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2018.3,0)\"><use data-c=\"28\" xlink:href=\"#MJX-119-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(2407.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-119-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,413) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-119-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3486.5,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-119-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3931.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-119-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(523,413) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-119-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4928.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-119-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5317.5,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-119-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5873.2,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-119-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6929,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-119-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msup><mi>s<\/mi><mi>n<\/mi><\/msup><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mi>n<\/mi><\/msup><mo>,<\/mo><msup><mi>y<\/mi><mi>n<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>\u2264<\/mo><mi>S<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>We consider <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.598ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2032.3 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-487-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-487-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-487-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-487-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-487-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D460\" xlink:href=\"#MJX-487-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(502,363) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-487-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(976.3,0)\"><use data-c=\"28\" xlink:href=\"#MJX-487-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1365.3,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-487-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1643.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-487-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>s<\/mi><mi>n<\/mi><\/msup><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">s^n(\\cdot)<\/script><span> as a per-letter cost function <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.45ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1525 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-488-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-488-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-488-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-488-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D460\" xlink:href=\"#MJX-488-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(469,0)\"><use data-c=\"28\" xlink:href=\"#MJX-488-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(858,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-488-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1136,0)\"><use data-c=\"29\" xlink:href=\"#MJX-488-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">s(\\cdot)<\/script><span>:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n605\" cid=\"n605\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div 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data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msup><mi>s<\/mi><mi>n<\/mi><\/msup><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mi>n<\/mi><\/msup><mo>,<\/mo><msup><mi>y<\/mi><mi>n<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mi>n<\/mi><\/mfrac><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo>,<\/mo><msub><mi>y<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>The largest rate <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.719ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 760 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-518-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-518-TEX-I-1D436\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>C<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">C<\/script><span> as a function of the cost <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.459ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 645 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-512-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-512-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>S<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">S<\/script><span> is called the <\/span><em><span>capacity cost function<\/span><\/em><span> denoted by <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.939ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2183 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-521-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-521-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-521-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-521-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-521-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(760,0)\"><use data-c=\"28\" xlink:href=\"#MJX-521-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1149,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-521-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1794,0)\"><use data-c=\"29\" xlink:href=\"#MJX-521-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">C(S)<\/script><span>.<\/span><\/p><h4 id='example-51-power-constraint'><span>Example 5.1 Power Constraint<\/span><\/h4><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"36.78ex\" height=\"2.548ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 16256.5 1126.1\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.661ex;\"><defs><path id=\"MJX-492-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-492-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-492-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 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data-mml-node=\"mo\" transform=\"translate(13138.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-492-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13694.1,0)\"><use data-c=\"2211\" xlink:href=\"#MJX-492-TEX-SO-2211\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(14916.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-492-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(936.2,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-492-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-284.4) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-492-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>,<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>,<\/mo><msup><mi>s<\/mi><mi>n<\/mi><\/msup><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mi>n<\/mi><\/msup><mo>,<\/mo><msup><mi>y<\/mi><mi>n<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo data-mjx-texclass=\"OP\">\u2211<\/mo><msubsup><mi>X<\/mi><mi>i<\/mi><mn>2<\/mn><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">s(x,y)=x^2,s^n(x^n,y^n)=(1\/n)\\sum X_i^2<\/script><span>, this is the most common constraint.<\/span><\/p><h3 id='4-block-and-bit-error-probability'><span>(4) Block and Bit Error Probability<\/span><\/h3><p><span>We define the channel coding by requiring the <\/span><em><span>block error probability<\/span><\/em><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block 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transform=\"translate(0,-150.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mo>=<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">[<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>W<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>\u2260<\/mo><mi>W<\/mi><mo stretchy=\"false\">]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>to be small.<\/span><\/p><p><span>But we also want to minimize the <\/span><em><span>average bit error probability<\/span><\/em><\/p><div contenteditable=\"false\" spellcheck=\"false\" 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data-mml-node=\"mi\"><use data-c=\"1D449\" xlink:href=\"#MJX-122-TEX-I-1D449\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(616,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-122-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9583.7,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-122-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1743.3 1 2986.6\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,2.6)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>P<\/mi><mi>b<\/mi><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mi>b<\/mi><\/mfrac><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>b<\/mi><\/munderover><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">[<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>\u2260<\/mo><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><blockquote><h4 id='theorem-51-blockbit-error-probability'><span>Theorem 5.1 Block\/Bit Error Probability<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n617\" cid=\"n617\" mdtype=\"math_block\" 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xlink:href=\"#MJX-123-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(4971.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-123-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5400.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-123-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-123-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1152.1 1 1804.1\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-402.1)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>P<\/mi><mi>b<\/mi><\/msub><mover><mo>\u2264<\/mo><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mover><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mover><mo>\u2264<\/mo><mrow><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mover><mi>b<\/mi><msub><mi>P<\/mi><mi>b<\/mi><\/msub><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>With equality on the left if all bits in a block are wrong. And equlity on the right if <\/span><em><span>exaclty one bit<\/span><\/em><span> is wrong in each block.<\/span><\/p><\/blockquote><p><em><span>Proof. 1<\/span><\/em><\/p><p><span>Intuitively, <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n621\" cid=\"n621\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"35.3ex\" height=\"5.204ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1400 15602.6 2300\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.036ex;\"><defs><path id=\"MJX-124-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-124-TEX-N-31\" 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data-c=\"72\" xlink:href=\"#MJX-124-TEX-N-72\" transform=\"translate(6225,0)\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-124-TEX-N-6F\" transform=\"translate(6617,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-124-TEX-N-72\" transform=\"translate(7117,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15324.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-124-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><\/mtd><mtd><mi><\/mi><mo stretchy=\"false\">[<\/mo><mtext>1 bit error<\/mtext><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">\u21d2<\/mo><mo stretchy=\"false\">[<\/mo><mtext>1 block error<\/mtext><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo stretchy=\"false\">[<\/mo><mtext>1 block error<\/mtext><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">\u21d2<\/mo><mo stretchy=\"false\">[<\/mo><mtext>at-leat-1 bit error<\/mtext><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Therefore,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n623\" cid=\"n623\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"69.344ex\" height=\"5.204ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1400 30650.1 2300\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.036ex;\"><defs><path id=\"MJX-125-TEX-I-1D443\" 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xlink:href=\"#MJX-125-TEX-N-74\" transform=\"translate(2223,0)\"><\/use><use data-c=\"20\" xlink:href=\"#MJX-125-TEX-N-20\" transform=\"translate(2612,0)\"><\/use><use data-c=\"65\" xlink:href=\"#MJX-125-TEX-N-65\" transform=\"translate(2862,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-125-TEX-N-72\" transform=\"translate(3306,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-125-TEX-N-72\" transform=\"translate(3698,0)\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-125-TEX-N-6F\" transform=\"translate(4090,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-125-TEX-N-72\" transform=\"translate(4590,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18138.1,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-125-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18693.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-125-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(19749.7,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-125-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(20178.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-125-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-125-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><msub><mi>P<\/mi><mi>b<\/mi><\/msub><mo>=<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">[<\/mo><mtext>1 bit error<\/mtext><mo stretchy=\"false\">]<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">[<\/mo><mtext>1 block error<\/mtext><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><\/mtd><\/mtr><mtr><mtd><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mo>=<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">[<\/mo><mtext>1 block error<\/mtext><mo stretchy=\"false\">]<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">[<\/mo><mtext>at-leat-1 bit error<\/mtext><mo stretchy=\"false\">]<\/mo><mo>\u2264<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">[<\/mo><mtext>all-bit error<\/mtext><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>b<\/mi><msub><mi>P<\/mi><mi>b<\/mi><\/msub><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><em><span>Proof. 2<\/span><\/em><\/p><blockquote><h4 id='theorem-52-lower-bounds-on-\\np\\ne\\n-and-\\np\\nb'><span>Theorem 5.2 Lower Bounds on <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" 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225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-499-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D452\" xlink:href=\"#MJX-499-TEX-I-1D452\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>e<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_e<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.327ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1028.3 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-503-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 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transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-503-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>b<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_b<\/script><\/h4><p><strong><span>Rate-reliability Tradeoff<\/span><\/strong><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n628\" cid=\"n628\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"42.384ex\" height=\"12.728ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -3062.9 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xlink:href=\"#MJX-126-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3379,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-126-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-126-TEX-I-1D44F\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4407.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-126-TEX-N-29\"><\/use><\/g><\/g><rect width=\"4996.3\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(6847.7,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-126-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(7903.5,0)\"><g data-mml-node=\"mrow\" transform=\"translate(1938.2,676)\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-126-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(600,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-126-TEX-I-1D436\"><\/use><\/g><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-710)\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-126-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(722.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-126-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1722.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-126-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(864,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-126-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2990,0)\"><use data-c=\"28\" xlink:href=\"#MJX-126-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3379,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-126-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-126-TEX-I-1D44F\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4407.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-126-TEX-N-29\"><\/use><\/g><\/g><rect width=\"4996.3\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><mi>n<\/mi><mi>R<\/mi><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mfrac><mrow><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo>;<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>W<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><\/mrow><\/mfrac><mo>\u2264<\/mo><mfrac><mrow><mi>n<\/mi><mi>C<\/mi><mo>+<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><mi>n<\/mi><mi>R<\/mi><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mfrac><mrow><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo>;<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>W<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>b<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>\u2264<\/mo><mfrac><mrow><mi>n<\/mi><mi>C<\/mi><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>b<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"19.825ex\" height=\"2.276ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 8762.5 1006.1\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.579ex;\"><defs><path id=\"MJX-495-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-495-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-495-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-495-TEX-N-61\" d=\"M137 305T115 305T78 320T63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59Q481 29 455 12T400 -6T353 12T329 54V58L327 55Q325 52 322 49T314 40T302 29T287 17T269 6T247 -2T221 -8T190 -11Q130 -11 82 20T34 107Q34 128 41 147T68 188T116 225T194 253T304 268H318V290Q318 324 312 340Q290 411 215 411Q197 411 181 410T156 406T148 403Q170 388 170 359Q170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175Q318 233 317 233Q309 233 296 232T251 223T193 203T147 166T126 106Z\"><\/path><path id=\"MJX-495-TEX-N-78\" d=\"M201 0Q189 3 102 3Q26 3 17 0H11V46H25Q48 47 67 52T96 61T121 78T139 96T160 122T180 150L226 210L168 288Q159 301 149 315T133 336T122 351T113 363T107 370T100 376T94 379T88 381T80 383Q74 383 44 385H16V431H23Q59 429 126 429Q219 429 229 431H237V385Q201 381 201 369Q201 367 211 353T239 315T268 274L272 270L297 304Q329 345 329 358Q329 364 327 369T322 376T317 380T310 384L307 385H302V431H309Q324 428 408 428Q487 428 493 431H499V385H492Q443 385 411 368Q394 360 377 341T312 257L296 236L358 151Q424 61 429 57T446 50Q464 46 499 46H516V0H510H502Q494 1 482 1T457 2T432 2T414 3Q403 3 377 3T327 1L304 0H295V46H298Q309 46 320 51T331 63Q331 65 291 120L250 175Q249 174 219 133T185 88Q181 83 181 74Q181 63 188 55T206 46Q208 46 208 23V0H201Z\"><\/path><path id=\"MJX-495-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-495-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-495-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-495-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-495-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-495-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-495-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-495-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-495-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1037.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-495-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"munder\" transform=\"translate(2093.6,0)\"><g data-mml-node=\"mo\"><use data-c=\"6D\" xlink:href=\"#MJX-495-TEX-N-6D\"><\/use><use data-c=\"61\" xlink:href=\"#MJX-495-TEX-N-61\" transform=\"translate(833,0)\"><\/use><use data-c=\"78\" xlink:href=\"#MJX-495-TEX-N-78\" transform=\"translate(1333,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(1894,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-495-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-495-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(5142.9,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-495-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5646.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-495-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6035.9,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-495-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6887.9,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-495-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7332.5,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-495-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8095.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-495-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8484.5,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-495-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>C<\/mi><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><mrow data-mjx-texclass=\"ORD\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><\/mrow><\/munder><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">C=\\max_{P_X}I(X;Y).<\/script><\/p><p><span>We see that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.075ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1359 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.048ex;\"><defs><path id=\"MJX-496-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-496-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-496-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(600,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-496-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mi>R<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">nR<\/script><span> is at most <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.14ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 504 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-497-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-497-TEX-I-1D43C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I<\/script><span> if reliability is good(<\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.386ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1054.5 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-499-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-499-TEX-I-1D452\" d=\"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-499-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D452\" xlink:href=\"#MJX-499-TEX-I-1D452\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>e<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_e<\/script><span> is small).<\/span><\/p><\/blockquote><p><em><span>Proof. of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.386ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1054.5 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-499-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-499-TEX-I-1D452\" d=\"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-499-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D452\" xlink:href=\"#MJX-499-TEX-I-1D452\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>e<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_e<\/script><\/em><\/p><p><span>Using Fano&#39;s inequality and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.139ex\" height=\"2.478ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 4481.5 1095.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.564ex;\"><defs><path id=\"MJX-500-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-500-TEX-I-1D44A\" d=\"M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 628T831 637Q817 637 817 647Q817 650 819 660Q823 676 825 679T839 682Q842 682 856 682T895 682T949 681Q1015 681 1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 154 304T131 622Q129 631 121 633T82 637H58Q51 644 51 648Q52 671 64 683H76Q118 680 176 680Q301 680 313 683H323Q329 677 329 674T327 656Q322 641 318 637H297Q236 634 232 620Q262 160 266 136L501 550L499 587Q496 629 489 632Q483 636 447 637Q428 637 422 639T416 648Q416 650 418 660Q419 664 420 669T421 676T424 680T428 682T436 683Z\"><\/path><path id=\"MJX-500-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-500-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-500-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-500-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-500-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(278,0)\"><use data-c=\"1D44A\" xlink:href=\"#MJX-500-TEX-I-1D44A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1326,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-500-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1881.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-500-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(2937.6,0)\"><g data-mml-node=\"mn\"><use data-c=\"32\" xlink:href=\"#MJX-500-TEX-N-32\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(533,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-500-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(600,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-500-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>W<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>=<\/mo><msup><mn>2<\/mn><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mi>R<\/mi><\/mrow><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">|W|=2^{nR}<\/script><span>, we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n633\" cid=\"n633\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"37.865ex\" height=\"5.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1550.5 16736.2 2601\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.377ex;\"><defs><path id=\"MJX-127-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-127-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-127-TEX-I-1D44A\" d=\"M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 628T831 637Q817 637 817 647Q817 650 819 660Q823 676 825 679T839 682Q842 682 856 682T895 682T949 681Q1015 681 1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 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388 355 386Q355 385 359 381T368 369T379 351T388 325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z\"><\/path><path id=\"MJX-127-TEX-N-2061\" d=\"\"><\/path><path id=\"MJX-127-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-127-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 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612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\"><g data-mml-node=\"mtr\" transform=\"translate(0,499.5)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" 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display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>W<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>W<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mi>n<\/mi><mi>R<\/mi><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>We suppose source messages are equal-probable and get<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n635\" cid=\"n635\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"31.581ex\" height=\"16.079ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -3803.4 13958.7 7106.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 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xlink:href=\"#MJX-508-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3807.6,0)\"><use data-c=\"1D44A\" xlink:href=\"#MJX-508-TEX-I-1D44A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4855.6,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-508-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(5300.2,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44A\" xlink:href=\"#MJX-508-TEX-I-1D44A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(550,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-508-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6348.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-508-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mi>R<\/mi><mo stretchy=\"false\">\u2192<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo>;<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>W<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">nR\\rightarrow I(W;\\hat{W})<\/script><span>. That means, for reliable transmission, the number of bits we transmit over any channel is at most the mutual information.<\/span><\/p><p><em><span>Proof. of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.327ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1028.3 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-503-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 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data-c=\"1D43B\" xlink:href=\"#MJX-129-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5014.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-129-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5403.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D449\" xlink:href=\"#MJX-129-TEX-I-1D449\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(616,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-129-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6313.3,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-129-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(6591.3,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D449\" xlink:href=\"#MJX-129-TEX-I-1D449\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(616,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-129-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(455,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-129-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7501.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-129-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>b<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mfrac><mn>1<\/mn><mi>b<\/mi><\/mfrac><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>b<\/mi><\/munderover><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">[<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>\u2260<\/mo><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mover><mo>\u2265<\/mo><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mover><mfrac><mn>1<\/mn><mi>b<\/mi><\/mfrac><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>b<\/mi><\/munderover><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">[<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>\u2260<\/mo><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mover><mo>\u2265<\/mo><mrow><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mover><mfrac><mn>1<\/mn><mi>b<\/mi><\/mfrac><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>b<\/mi><\/munderover><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.957ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1307 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-520-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-520-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-520-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-520-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-520-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(918,0)\"><use data-c=\"29\" xlink:href=\"#MJX-520-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml 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stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H_2(\\cdot)<\/script><span>, and (b) follows by Fano&#39;s inequality in case of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.334ex\" height=\"2.26ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -749.5 3241.6 999\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.564ex;\"><defs><path id=\"MJX-506-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-506-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 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xlink:href=\"#MJX-130-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15759.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-130-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16426,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-130-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(17481.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-130-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(864,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-130-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(18749.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-130-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(19138.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-130-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D452\" xlink:href=\"#MJX-130-TEX-I-1D452\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(20192.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-130-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -900.5 1 1301\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-150.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>e<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Using the chain rule<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n642\" cid=\"n642\" mdtype=\"math_block\" data-math-tag-before=\"0\" 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scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-131-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3520.5,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-131-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3798.5,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D449\" xlink:href=\"#MJX-131-TEX-I-1D449\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(431,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-131-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(861.3,413) scale(0.707)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-131-TEX-I-1D44F\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5013.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-131-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-1751.5)\"><g data-mml-node=\"mtd\" transform=\"translate(3763.9,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(3763.9,0)\"><g data-mml-node=\"mi\"><\/g><g data-mml-node=\"mo\" transform=\"translate(277.8,0)\"><use data-c=\"2265\" xlink:href=\"#MJX-131-TEX-N-2265\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1333.6,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-131-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2221.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-131-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2610.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D449\" xlink:href=\"#MJX-131-TEX-I-1D449\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(616,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-131-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3520.5,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-131-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3798.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D449\" xlink:href=\"#MJX-131-TEX-I-1D449\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(861.3,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-131-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(345,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-131-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1123,0)\"><use data-c=\"31\" xlink:href=\"#MJX-131-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msup\" transform=\"translate(5857.4,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D449\" xlink:href=\"#MJX-131-TEX-I-1D449\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(431,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-131-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(861.3,413) scale(0.707)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-131-TEX-I-1D44F\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7072.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-131-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2265<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mrow data-mjx-texclass=\"ORD\"><mover><msub><mi>V<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><msub><mi>V<\/mi><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/msub><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2265<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mover><mi>V<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2265<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>V<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow data-mjx-texclass=\"ORD\"><mover><mi>V<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>we get the following bound <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n644\" cid=\"n644\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"32.544ex\" height=\"41.022ex\" 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50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-132-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-132-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 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transform=\"translate(864,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-132-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2990,0)\"><use data-c=\"28\" xlink:href=\"#MJX-132-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3379,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-132-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-132-TEX-I-1D44F\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4407.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-132-TEX-N-29\"><\/use><\/g><\/g><rect width=\"4996.3\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>b<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2265<\/mo><mfrac><mn>1<\/mn><mi>b<\/mi><\/mfrac><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>b<\/mi><\/munderover><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>V<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mi>V<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow data-mjx-texclass=\"ORD\"><mover><mi>V<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mi>b<\/mi><\/mfrac><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>V<\/mi><mi>b<\/mi><\/msup><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mover><mi>V<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mi>b<\/mi><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>V<\/mi><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>V<\/mi><mi>b<\/mi><\/msup><mo>;<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mover><mi>V<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mi>b<\/mi><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo>;<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>W<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mi>b<\/mi><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo>\u2212<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo>;<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>W<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mfrac><mrow><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo>;<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>W<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mi>R<\/mi><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><mi>n<\/mi><mi>R<\/mi><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mfrac><mrow><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo>;<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>W<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>b<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.977ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3083.9 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-507-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-507-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path id=\"MJX-507-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-507-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-507-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-507-TEX-I-1D44F\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1306.1,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-507-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2583.9,0)\"><use data-c=\"30\" xlink:href=\"#MJX-507-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>b<\/mi><\/msub><mo stretchy=\"false\">\u2192<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_b\\rightarrow 0<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"15.243ex\" height=\"2.943ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1051 6737.2 1301\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-508-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-508-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 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id=\"MJX-508-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-508-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-508-TEX-I-1D44A\" d=\"M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 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8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-508-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><path id=\"MJX-508-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-508-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(600,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-508-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1636.8,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-508-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2914.6,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-508-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3418.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-508-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3807.6,0)\"><use data-c=\"1D44A\" xlink:href=\"#MJX-508-TEX-I-1D44A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4855.6,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-508-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(5300.2,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D44A\" xlink:href=\"#MJX-508-TEX-I-1D44A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(550,257) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-508-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6348.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-508-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mi>R<\/mi><mo stretchy=\"false\">\u2192<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo>;<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>W<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">nR\\rightarrow I(W;\\hat{W})<\/script><span>. That means, for reliable transmission, the number of bits we transmit over any channel is at most the mutual information.<\/span><\/p><h3 id='5-random-coding'><span>(5) Random Coding<\/span><\/h3><blockquote><h4 id='capacity-cost-function'><span>Capacity-Cost Function<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n649\" cid=\"n649\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 31.493ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"3.926ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.397ex; min-width: 31.493ex;\"><defs><path 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xlink:href=\"#MJX-133-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9871.4,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-133-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10723.4,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-133-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11168,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-133-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11931,0)\"><use data-c=\"29\" xlink:href=\"#MJX-133-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1117.6 1 1735.1\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,367.6)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><mrow data-mjx-texclass=\"ORD\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo>:<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>,<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>\u2264<\/mo><mi>S<\/mi><\/mrow><\/munder><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p>&nbsp;<\/p><\/blockquote><h3 id='6-concavity-and-converse'><span>(6) Concavity and Converse<\/span><\/h3><blockquote><h4 id='theorem'><span>Theorem<\/span><\/h4><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.939ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2183 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-521-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-521-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-521-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-521-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-521-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(760,0)\"><use data-c=\"28\" xlink:href=\"#MJX-521-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1149,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-521-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1794,0)\"><use data-c=\"29\" xlink:href=\"#MJX-521-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">C(S)<\/script><span> is concave in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.459ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 645 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-512-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-512-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>S<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">S<\/script><span>.<\/span><\/p><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.939ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2183 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-521-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-521-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-521-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-521-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-521-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(760,0)\"><use data-c=\"28\" xlink:href=\"#MJX-521-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1149,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-521-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1794,0)\"><use data-c=\"29\" xlink:href=\"#MJX-521-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">C(S)<\/script><span> is non-decreasing in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.459ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 645 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-512-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-512-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>S<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">S<\/script><span>.<\/span><\/p><\/blockquote><p><em><span>Proof.<\/span><\/em><\/p><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"23.635ex\" height=\"2.943ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1051 10446.5 1301\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-513-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-513-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-513-TEX-I-1D44A\" d=\"M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 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579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path id=\"MJX-513-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-513-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path 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data-mml-node=\"msub\" transform=\"translate(8871.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-513-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(614,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-513-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9779.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-513-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10168.5,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-513-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo>;<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>W<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>;<\/mo><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I(W;\\hat{W})\\le \\sum I(X_i;Y_i).<\/script><span> Since the mutual information about <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.613ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1155 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-516-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 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135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-516-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-516-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>X<\/mi><mi>i<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X_i<\/script><span> is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.31ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5883.1 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-515-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-515-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 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239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-515-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(504,0)\"><use data-c=\"28\" xlink:href=\"#MJX-515-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(893,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-515-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-515-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2048,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-515-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2492.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-515-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(614,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-515-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3400.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-515-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4067.3,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-515-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5123.1,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-515-TEX-I-1D436\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>;<\/mo><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mi>C<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I(X_i;Y_i)\\le C<\/script><span>, and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.613ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1155 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-516-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 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48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-517-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-517-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-517-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-517-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-517-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-517-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-517-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-517-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-517-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-517-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-517-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-517-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1511,0)\"><use data-c=\"28\" xlink:href=\"#MJX-517-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1900,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-517-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-517-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3055,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-517-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3499.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-517-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(614,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-517-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4407.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-517-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4796.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-517-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>,<\/mo><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[s(X_i,Y_i)]<\/script><span>, we can rewrite <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.719ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 760 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-518-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-518-TEX-I-1D436\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>C<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">C<\/script><span> as <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.961ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6612.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-519-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-519-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-519-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-519-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-519-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-519-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-519-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-519-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-519-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-519-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-519-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-519-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(760,0)\"><use data-c=\"28\" xlink:href=\"#MJX-519-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1149,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-519-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1913,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-519-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2191,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-519-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2660,0)\"><use data-c=\"28\" xlink:href=\"#MJX-519-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3049,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-519-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-519-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4204,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-519-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4648.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-519-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(614,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-519-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5556.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-519-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5945.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-519-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6223.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-519-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>,<\/mo><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">C(E[s(X_i,Y_i)])<\/script><span> and get <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n658\" cid=\"n658\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"34.358ex\" height=\"31.062ex\" role=\"img\" focusable=\"false\" 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data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>,<\/mo><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>n<\/mi><msub><mi>E<\/mi><mi>n<\/mi><\/msub><mo stretchy=\"false\">[<\/mo><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>,<\/mo><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mover><mo>\u2264<\/mo><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mover><mi>n<\/mi><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>E<\/mi><mi>n<\/mi><\/msub><mo stretchy=\"false\">[<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>,<\/mo><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>n<\/mi><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mfrac><mn>1<\/mn><mi>n<\/mi><\/mfrac><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>,<\/mo><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mi>n<\/mi><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.957ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1307 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-520-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-520-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-520-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-520-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-520-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(918,0)\"><use data-c=\"29\" xlink:href=\"#MJX-520-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(a)<\/script><span> follows from the concavity of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.939ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2183 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-521-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-521-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-521-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-521-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-521-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(760,0)\"><use data-c=\"28\" xlink:href=\"#MJX-521-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1149,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-521-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1794,0)\"><use data-c=\"29\" xlink:href=\"#MJX-521-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">C(S)<\/script><span> and Jensen&#39;s Inequality <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"18.868ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 8339.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-522-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-522-TEX-N-5B\" 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transform=\"translate(2191,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-522-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2836,0)\"><use data-c=\"29\" xlink:href=\"#MJX-522-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3225,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-522-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3780.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-522-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4836.6,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-522-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5596.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-522-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5985.6,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-522-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6749.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-522-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7027.6,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-522-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7672.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-522-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7950.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-522-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>\u2264<\/mo><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>S<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[C(S)]\\le C(E[S])<\/script><span>.<\/span><\/p><blockquote><p><strong><span>Rate-reliability Tradeoff<\/span><\/strong><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n662\" cid=\"n662\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"45.604ex\" height=\"12.728ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -3062.9 20156.8 5625.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -5.798ex;\"><defs><path id=\"MJX-135-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 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stretchy=\"false\">(<\/mo><msub><mi>P<\/mi><mi>b<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><h3 id='7-discrete-alphabet-examples'><span>(7) Discrete Alphabet Examples<\/span><\/h3><h4 id='1-binary-symmetric-channel'><span>&lt;1&gt; Binary Symmetric Channel<\/span><\/h4><p><span>The best choice for <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.003ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1327.5 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-538-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-538-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-538-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-538-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X<\/script><span> is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"21.217ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 9378 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-524-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-524-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-524-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-524-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-524-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-524-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-524-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-524-TEX-N-2F\" d=\"M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z\"><\/path><path id=\"MJX-524-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-524-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-524-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1327.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-524-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1716.5,0)\"><use data-c=\"30\" xlink:href=\"#MJX-524-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2216.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-524-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2883.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-524-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3939,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-524-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-524-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5266.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-524-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(5655.5,0)\"><use data-c=\"31\" xlink:href=\"#MJX-524-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6155.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-524-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6822.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-524-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(7878,0)\"><use data-c=\"31\" xlink:href=\"#MJX-524-TEX-N-31\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(8378,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-524-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"mn\" transform=\"translate(8878,0)\"><use data-c=\"32\" xlink:href=\"#MJX-524-TEX-N-32\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mn>2<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X(0)=P_X(1)=1\/2<\/script><span> so that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.399ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6364.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-525-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-525-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-525-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-525-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-525-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-525-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-525-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-525-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-525-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-525-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1037.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-525-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2093.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-525-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2815.8,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-525-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3816,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-525-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(864,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-525-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5083.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-525-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5472.6,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-525-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5975.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-525-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>C<\/mi><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>p<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">C=1-H_2(p)<\/script><span>, where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.138ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 503 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-526-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-526-TEX-I-1D45D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>p<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">p<\/script><span> is the crossover probability.<\/span><\/p><h4 id='2-binary-erasure-channel'><span>&lt;2&gt; Binary Erasure Channel<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n667\" cid=\"n667\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"72.219ex\" height=\"16.968ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -4000 31920.7 7500\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -7.919ex;\"><defs><path id=\"MJX-136-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 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data-c=\"2B\" xlink:href=\"#MJX-136-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4778.4,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-136-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5281.4,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-136-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6169.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-136-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6558.4,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-136-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7410.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-136-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mi>p<\/mi><mo>+<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mi>p<\/mi><mo>=<\/mo><mi>p<\/mi><\/mtd><\/mtr><mtr><mtd><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>P<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><\/mtr><mtr><mtd><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo>=<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo>=<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>P<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mo stretchy=\"false\">)<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo>=<\/mo><mi mathvariant=\"normal\">\u25b3<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mn>0<\/mn><mo>+<\/mo><mn>0<\/mn><mo>+<\/mo><mi>p<\/mi><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Therefore, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"23.905ex\" height=\"2.347ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 10566 1037.2\" 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198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-527-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-527-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-527-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 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transform=\"translate(1894,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-527-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-527-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(5044.6,0)\"><use data-c=\"1D43B\" xlink:href=\"#MJX-527-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5932.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-527-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6321.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-527-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7173.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-527-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7562.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-527-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(7951.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-527-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8673.8,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-527-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9674,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-527-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10177,0)\"><use data-c=\"29\" xlink:href=\"#MJX-527-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>C<\/mi><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><mrow data-mjx-texclass=\"ORD\"><msub><mi>p<\/mi><mi>X<\/mi><\/msub><\/mrow><\/munder><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">C=\\max_{p_X}H(X)(1-p)<\/script><span>, choosing <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.003ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1327.5 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-538-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-538-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-538-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(675,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-538-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mi>X<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_X<\/script><span> to be coin-flipping, we get <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.771ex\" height=\"2.034ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 4319 899\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-529-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-529-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-529-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-529-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-529-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-529-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1037.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-529-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2093.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-529-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2815.8,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-529-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3816,0)\"><use data-c=\"1D45D\" xlink:href=\"#MJX-529-TEX-I-1D45D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>C<\/mi><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">C=1-p<\/script><span>.<\/span><\/p><h4 id='3-strongly-symmetric-channels'><span>&lt;3&gt; Strongly Symmetric Channels<\/span><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n670\" cid=\"n670\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 36.438ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"11.765ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -5.317ex; min-width: 36.438ex;\"><defs><path id=\"MJX-137-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-137-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-137-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-137-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-137-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-137-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-137-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-137-TEX-S4-239B\" d=\"M837 1154Q843 1148 843 1145Q843 1141 818 1106T753 1002T667 841T574 604T494 299Q417 -84 417 -609Q417 -641 416 -647T411 -654Q409 -655 366 -655Q299 -655 297 -654Q292 -652 292 -643T291 -583Q293 -400 304 -242T347 110T432 470T574 813T785 1136Q787 1139 790 1142T794 1147T796 1150T799 1152T802 1153T807 1154T813 1154H819H837Z\"><\/path><path id=\"MJX-137-TEX-S4-239D\" d=\"M843 -635Q843 -638 837 -644H820Q801 -644 800 -643Q792 -635 785 -626Q684 -503 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xlink:href=\"#MJX-137-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(1500,0)\"><g data-mml-node=\"mo\"><use data-c=\"2217\" xlink:href=\"#MJX-137-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(3000,0)\"><g data-mml-node=\"mo\"><use data-c=\"2217\" xlink:href=\"#MJX-137-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(4500,0)\"><g data-mml-node=\"mo\"><use data-c=\"2217\" xlink:href=\"#MJX-137-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(6000,0)\"><g data-mml-node=\"mo\"><use data-c=\"2217\" xlink:href=\"#MJX-137-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(7500,0)\"><g data-mml-node=\"mo\"><use data-c=\"2217\" xlink:href=\"#MJX-137-TEX-N-2217\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-2100)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mo\"><use data-c=\"2217\" xlink:href=\"#MJX-137-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(1500,0)\"><g data-mml-node=\"mo\"><use data-c=\"2217\" xlink:href=\"#MJX-137-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(3000,0)\"><g data-mml-node=\"mo\"><use data-c=\"2217\" xlink:href=\"#MJX-137-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(4500,0)\"><g data-mml-node=\"mo\"><use data-c=\"2217\" xlink:href=\"#MJX-137-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(6000,0)\"><g data-mml-node=\"mo\"><use data-c=\"2217\" xlink:href=\"#MJX-137-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(7500,0)\"><g data-mml-node=\"mo\"><use data-c=\"2217\" xlink:href=\"#MJX-137-TEX-N-2217\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8875,0)\"><use data-c=\"239E\" xlink:href=\"#MJX-137-TEX-S4-239E\" transform=\"translate(0,1696)\"><\/use><use data-c=\"23A0\" xlink:href=\"#MJX-137-TEX-S4-23A0\" transform=\"translate(0,-1706)\"><\/use><svg width=\"875\" height=\"1782\" y=\"-641\" x=\"0\" viewBox=\"0 402.4 875 1782\"><use data-c=\"239F\" xlink:href=\"#MJX-137-TEX-S4-239F\" transform=\"scale(1,4.311)\"><\/use><\/svg><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -2850 1 5200\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><\/mtr><mtr><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><\/mtr><mtr><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><\/mtr><mtr><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><mtd><mo>\u2217<\/mo><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Notice the entires in one row add up to <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.507ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 500 666\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-530-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-530-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">1<\/script><span>, while the entries in one column don&#39;t necessarily add up to <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.76ex\" height=\"1.507ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 778 666\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-531-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-531-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-531-TEX-N-31\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-531-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>1.<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">1.<\/script><\/p><blockquote><p><strong><span>Uniformly dispersive<\/span><\/strong><span>: (entries in rows are the same, their order may change)<\/span><\/p><p><span>for every input letter <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.294ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 572 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-532-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-532-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x<\/script><span>, the list of probabilities <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"15.36ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6789.2 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-533-TEX-N-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path id=\"MJX-533-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-533-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-533-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-533-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-533-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-533-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-533-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-533-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-533-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-533-TEX-N-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"7B\" xlink:href=\"#MJX-533-TEX-N-7B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(500,0)\"><use data-c=\"1D443\" xlink:href=\"#MJX-533-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1251,0)\"><use data-c=\"28\" xlink:href=\"#MJX-533-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1640,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-533-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2130,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-533-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2408,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-533-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2980,0)\"><use data-c=\"29\" xlink:href=\"#MJX-533-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3369,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-533-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3813.7,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-533-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4581.4,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-533-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5526.2,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-533-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6289.2,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-533-TEX-N-7D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo fence=\"false\" stretchy=\"false\">{<\/mo><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><mi>y<\/mi><mo>\u2208<\/mo><mi>Y<\/mi><mo fence=\"false\" stretchy=\"false\">}<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\{P(y|x),y\\in Y\\}<\/script><span> is the same. Therefore, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.363ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5464.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-534-TEX-I-1D43B\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-534-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-534-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-534-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-534-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-534-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-534-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-534-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43B\" xlink:href=\"#MJX-534-TEX-I-1D43B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-534-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1277,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-534-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2040,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-534-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2318,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-534-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3447.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-534-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4503.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-534-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5075.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-534-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo>=<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">H(Y|X=x)<\/script><span> is the same for all <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.988ex\" height=\"1.636ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2646.6 723\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-535-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-535-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-535-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-535-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(849.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-535-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1794.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-535-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>\u2208<\/mo><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x\\in X<\/script><span>.<\/span><\/p><p><strong><span>Uniformly focusing<\/span><\/strong><span>: (entries in columns are the same, their  order may change )<\/span><\/p><p><span>for every output letter <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.109ex\" height=\"1.464ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 490 647\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-536-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-536-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">y<\/script><span>, the list of probabilities <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"15.747ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6960.2 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-537-TEX-N-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 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52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-537-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-537-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 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xlink:href=\"#MJX-537-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1251,0)\"><use data-c=\"28\" xlink:href=\"#MJX-537-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1640,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-537-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2130,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-537-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2408,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-537-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2980,0)\"><use data-c=\"29\" xlink:href=\"#MJX-537-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3369,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-537-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3813.7,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-537-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4663.4,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-537-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5608.2,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-537-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6460.2,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-537-TEX-N-7D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo fence=\"false\" stretchy=\"false\">{<\/mo><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><mi>x<\/mi><mo>\u2208<\/mo><mi>X<\/mi><mo fence=\"false\" stretchy=\"false\">}<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\{P(y|x),x\\in X\\}<\/script><span> is the same. Uniform input in a u.f channel implies uniform output.<\/span><\/p><p><strong><span>Strongly symmetric<\/span><\/strong><span>:<\/span><\/p><p><span>A channel is said to be strongly symmetric if it both uniformly dispersive and uniformly focusing.<\/span><\/p><\/blockquote><p><span>BEC is u.d. but not u.f. 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xlink:href=\"#MJX-138-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1397.5 1 2294.9\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,447.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>C<\/mi><mo>=<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>H<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2212<\/mo><munder><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><mo>\u2208<\/mo><mi>Y<\/mi><\/mrow><\/munder><mo>\u2212<\/mo><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Capacity is achieved by a uniform <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.003ex\" height=\"1.885ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1327.5 833\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-538-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-538-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 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columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><\/mtd><mtd><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mtable columnalign=\"center\" columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mtable columnalign=\"center center center\" columnspacing=\"1em\" rowspacing=\"4pt\" columnlines=\"none solid\"><mtr><mtd><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mi>p<\/mi><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><\/mtd><mtd><mi>p<\/mi><\/mtd><\/mtr><\/mtable><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><\/mtd><mtd><mtext>&nbsp;subchannel 1:&nbsp;<\/mtext><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mtable columnalign=\"center\" columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><mtext>&nbsp;is strongly symmetric<\/mtext><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mtext>&nbsp;subchannel 2:&nbsp;<\/mtext><mi>p<\/mi><mo>\u22c5<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mtable columnalign=\"center\" columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>1<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><mtext>&nbsp;is strongly symmetric<\/mtext><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><\/mtd><mtd><mi>C<\/mi><mo>=<\/mo><msub><mi>q<\/mi><mn>1<\/mn><\/msub><msub><mi>C<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>q<\/mi><mn>2<\/mn><\/msub><msub><mi>C<\/mi><mn>2<\/mn><\/msub><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>p<\/mi><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><blockquote><p><span>Capacity of Symmetric DMC<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n689\" cid=\"n689\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 16.107ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"6.74ex\" role=\"img\" focusable=\"false\" 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xlink:href=\"#MJX-140-TEX-I-1D43F\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(3704.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45E\" xlink:href=\"#MJX-140-TEX-I-1D45E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(479,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-140-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(4477.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-140-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(748,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-140-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1739.4 1 2978.9\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,6.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>C<\/mi><mo>=<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>L<\/mi><\/munderover><msub><mi>q<\/mi><mi>i<\/mi><\/msub><msub><mi>C<\/mi><mi>i<\/mi><\/msub><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><h4 id='example-52-bpsk-over-awgn'><span>Example 5.2 BPSK over AWGN<\/span><\/h4><p><span>We use a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.507ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 500 666\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-539-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"32\" xlink:href=\"#MJX-539-TEX-N-32\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>2<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">2<\/script><span>-bit quantizer:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n693\" cid=\"n693\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"84.476ex\" height=\"28.045ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -6448 37338.2 12396\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -13.457ex;\"><defs><path id=\"MJX-141-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 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columnalign=\"center center center center\" columnspacing=\"1em\" rowspacing=\"4pt\" columnlines=\"solid none solid\"><mtr><mtd><msub><mi>p<\/mi><mn>0<\/mn><\/msub><\/mtd><mtd><msub><mi>p<\/mi><mn>1<\/mn><\/msub><\/mtd><mtd><msub><mi>p<\/mi><mn>2<\/mn><\/msub><\/mtd><mtd><msub><mi>p<\/mi><mn>3<\/mn><\/msub><\/mtd><\/mtr><mtr><mtd><msub><mi>p<\/mi><mn>3<\/mn><\/msub><\/mtd><mtd><msub><mi>p<\/mi><mn>2<\/mn><\/msub><\/mtd><mtd><msub><mi>p<\/mi><mn>1<\/mn><\/msub><\/mtd><mtd><msub><mi>p<\/mi><mn>0<\/mn><\/msub><\/mtd><\/mtr><\/mtable><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><\/mtd><mtd><mtext>&nbsp;subchannel 1:&nbsp;<\/mtext><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mn>0<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>3<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mtable columnalign=\"center\" columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msub><mi>p<\/mi><mn>0<\/mn><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mn>0<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>3<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><msub><mi>p<\/mi><mn>3<\/mn><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mn>0<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>3<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><msub><mi>p<\/mi><mn>3<\/mn><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mn>0<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>3<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><msub><mi>p<\/mi><mn>0<\/mn><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mn>0<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>3<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><mtext>&nbsp;is BSC(strongly symmetric)<\/mtext><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mtext>&nbsp;subchannel 2:&nbsp;<\/mtext><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mtable columnalign=\"center\" columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><msub><mi>p<\/mi><mn>2<\/mn><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><msub><mi>p<\/mi><mn>2<\/mn><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><mtext>&nbsp;is BSC(strongly symmetric)<\/mtext><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><\/mtd><mtd><mtext>&nbsp;<\/mtext><mi>C<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mn>0<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>3<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mfrac><msub><mi>p<\/mi><mn>0<\/mn><\/msub><mrow><msub><mi>p<\/mi><mn>0<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>3<\/mn><\/msub><\/mrow><\/mfrac><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><msub><mi>H<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mfrac><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mrow><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>p<\/mi><mn>2<\/mn><\/msub><\/mrow><\/mfrac><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='8-continuous-alphabet-examples'><span>(8) Continuous Alphabet Examples<\/span><\/h3><p><span>Consider the additive white Gaussian noise channel with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.735ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 5628.9 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-540-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-540-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-540-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-540-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-540-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-540-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-540-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(614,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-540-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1185.7,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-540-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2241.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-540-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-540-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3618.7,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-540-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4618.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44D\" xlink:href=\"#MJX-540-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(716,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-540-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><mo>=<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>+<\/mo><msub><mi>Z<\/mi><mi>i<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y_i=X_i+Z_i<\/script><span> where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.694ex\" height=\"2.452ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 6052.7 1083.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-541-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-541-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-541-TEX-N-223C\" d=\"M55 166Q55 241 101 304T222 367Q260 367 296 349T362 304T421 252T484 208T554 189Q616 189 655 236T694 338Q694 350 698 358T708 367Q722 367 722 334Q722 260 677 197T562 134H554Q517 134 481 152T414 196T355 248T292 293T223 311Q179 311 145 286Q109 257 96 218T80 156T69 133Q55 133 55 166Z\"><\/path><path id=\"MJX-541-TEX-C-4E\" d=\"M343 705Q358 705 358 698Q360 696 370 658T411 524T484 319Q536 174 590 82L595 73L615 152Q646 274 683 407Q729 571 752 637T799 727Q852 780 937 788Q939 788 947 788T958 789H962Q979 789 979 765Q979 722 951 692Q942 683 924 683Q888 681 859 672T818 654T803 639Q784 608 708 322T631 15Q631 14 630 15Q630 17 629 15Q628 14 628 12Q621 -4 601 -17T560 -31Q550 -31 546 -28T530 -7Q484 67 458 123T398 272Q352 392 314 514L306 535V534Q306 533 296 488T272 379T234 239T185 100T127 -7T61 -50Q34 -50 4 -34T-27 8Q-27 33 -12 61T18 90Q21 90 36 77T87 57H92Q109 57 123 78T162 173Q206 299 232 417T265 599T276 667Q284 681 304 693T343 705Z\"><\/path><path id=\"MJX-541-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-541-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-541-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-541-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-541-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-541-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44D\" xlink:href=\"#MJX-541-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(716,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-541-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1287.7,0)\"><use data-c=\"223C\" xlink:href=\"#MJX-541-TEX-N-223C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2343.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"4E\" xlink:href=\"#MJX-541-TEX-C-4E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3322.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-541-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3711.5,0)\"><use data-c=\"30\" xlink:href=\"#MJX-541-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4211.5,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-541-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(4656.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-541-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-541-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5663.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-541-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>Z<\/mi><mi>i<\/mi><\/msub><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo>,<\/mo><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Z_i\\sim\\mathcal{N}(0,\\sigma^2)<\/script><span>, we have power constraint <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.529ex\" height=\"2.452ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 5095.8 1083.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-542-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 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data-mml-node=\"mn\" transform=\"translate(604,413) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-142-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9823.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-142-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>P<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo>:<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msup><mi>X<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">]<\/mo><mo>\u2264<\/mo><mi>P<\/mi><\/mrow><\/munder><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>;<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>Z<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>Z<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mi>e<\/mi><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>We now maximize <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.79ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2117 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-544-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-544-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-544-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-544-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g 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stretchy=\"false\">]<\/mo><mo>,<\/mo><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><msup><mi>X<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">]<\/mo><mo>+<\/mo><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><msup><mi>X<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">]<\/mo><mo>+<\/mo><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msup><mi>X<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">]<\/mo><mo>+<\/mo><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><\/mtd><\/mtr><mtr><mtd><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi>Y<\/mi><mo stretchy=\"false\">]<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mi>P<\/mi><mo>+<\/mo><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>Y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>\u2264<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mi>e<\/mi><mo stretchy=\"false\">(<\/mo><mi>P<\/mi><mo>+<\/mo><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>with equality iff. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.385ex\" height=\"2.351ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -789 5916.2 1039\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-545-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 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58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-545-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-545-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-545-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-545-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-545-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-545-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1129.8,0)\"><use data-c=\"223C\" xlink:href=\"#MJX-545-TEX-N-223C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2185.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"4E\" xlink:href=\"#MJX-545-TEX-C-4E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3164.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-545-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3553.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-545-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4053.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-545-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4498.2,0)\"><use data-c=\"1D443\" xlink:href=\"#MJX-545-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5249.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-545-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5638.2,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-545-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo>,<\/mo><mi>P<\/mi><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X\\sim\\mathcal{N}(0,P).<\/script><\/p><p><span>Finally, we get <\/span><\/p><blockquote><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n702\" cid=\"n702\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 33.722ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"4.728ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.799ex; min-width: 33.722ex;\"><defs><path id=\"MJX-144-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-144-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-144-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-144-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-144-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-144-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-144-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-144-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-144-TEX-N-6C\" d=\"M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z\"><\/path><path id=\"MJX-144-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-144-TEX-N-67\" d=\"M329 409Q373 453 429 453Q459 453 472 434T485 396Q485 382 476 371T449 360Q416 360 412 390Q410 404 415 411Q415 412 416 414V415Q388 412 363 393Q355 388 355 386Q355 385 359 381T368 369T379 351T388 325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z\"><\/path><path id=\"MJX-144-TEX-N-2061\" d=\"\"><\/path><path id=\"MJX-144-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-144-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019382,-0.019382) translate(0, -1295)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1295) matrix(1 0 0 -1 0 0) scale(51.6)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"6652.6 -1295 1 2089.9\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-64)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-144-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(760,0)\"><use data-c=\"28\" xlink:href=\"#MJX-144-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1149,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-144-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1794,0)\"><use data-c=\"29\" xlink:href=\"#MJX-144-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2460.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-144-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3516.6,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-144-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4276.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-144-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4665.6,0)\"><use data-c=\"1D443\" xlink:href=\"#MJX-144-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5416.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-144-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6083.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-144-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(7139.1,0)\"><g data-mml-node=\"mn\" transform=\"translate(220,676)\"><use data-c=\"31\" xlink:href=\"#MJX-144-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(220,-686)\"><use data-c=\"32\" xlink:href=\"#MJX-144-TEX-N-32\"><\/use><\/g><rect width=\"700\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mi\" transform=\"translate(8079.1,0)\"><use data-c=\"6C\" xlink:href=\"#MJX-144-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-144-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-144-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9357.1,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-144-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9357.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-144-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(9746.1,0)\"><use data-c=\"31\" xlink:href=\"#MJX-144-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10468.3,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-144-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(11468.6,0)\"><g data-mml-node=\"mi\" transform=\"translate(348.3,676)\"><use data-c=\"1D443\" xlink:href=\"#MJX-144-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(220,-719.9)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-144-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,289) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-144-TEX-N-32\"><\/use><\/g><\/g><rect width=\"1207.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(12916.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-144-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1295 1 2089.9\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-64)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>C<\/mi><mo stretchy=\"false\">(<\/mo><mi>P<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mfrac><mi>P<\/mi><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/blockquote><p>&nbsp;<\/p><h4 id='example-53-awgn-channel-with-bpsk'><span>Example 5.3 AWGN channel with BPSK<\/span><\/h4><p><span>We transmit symbols <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.345ex\" height=\"2.758ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -969 5456.6 1219\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-546-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-546-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-546-TEX-N-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path id=\"MJX-546-TEX-N-B1\" d=\"M56 320T56 333T70 353H369V502Q369 651 371 655Q376 666 388 666Q402 666 405 654T409 596V500V353H707Q722 345 722 333Q722 320 707 313H409V40H707Q722 32 722 20T707 0H70Q56 7 56 20T70 40H369V313H70Q56 320 56 333Z\"><\/path><path id=\"MJX-546-TEX-N-221A\" d=\"M95 178Q89 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