andy/Probability-and-Statistics-Notes
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity
Files
gh-pages
Probability-and-Statistics-…/sec-Covariance.html
T
andy 331f133dfe Latest build deployed.
2026-08-01 15:00:14 +00:00

497 lines
50 KiB
HTML
Raw Permalink Blame History

This file contains invisible Unicode characters
This file contains invisible Unicode characters that are indistinguishable to humans but may be processed differently by a computer. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.
This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.
<!DOCTYPE html>
<!--********************************************-->
<!--* Generated from PreTeXt source *-->
<!--* *-->
<!--* https://pretextbook.org *-->
<!--* *-->
<!--********************************************-->
<html lang="en-US" dir="ltr">
<!--******************************************-->
<!--* Authored with PreTeXt *-->
<!--* pretextbook.org *-->
<!--* Theme: boulder *-->
<!--* Palette: *-->
<!--******************************************-->
<head xmlns:og="http://ogp.me/ns#" xmlns:book="https://ogp.me/ns/book#">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Covariance</title>
<link rel="preconnect" href="https://fonts.googleapis.com">
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin="">
<link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Material+Symbols+Outlined:opsz,wght,FILL,GRAD@24,400,0,0">
<link href="_static/pretext/css/theme.css" rel="stylesheet" type="text/css">
<script type="module">import { startMathJax } from './_static/pretext/js/dist/mathjax_startup.js';
startMathJax({
hasWebworkReps: false,
hasSage: false,
isReact: false,
htmlPresentation: false,
});
</script><script defer src="https://cdn.jsdelivr.net/npm/mathjax@4/tex-mml-chtml.js"></script><meta name="Keywords" content="Authored in PreTeXt">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta property="og:type" content="book">
<meta name="twitter:title" content="Math 1044 Notes">
<meta property="og:title" content="Math 1044 Notes">
<meta property="book:title" content="Math 1044 Notes">
<meta name="twitter:card" content="summary">
<script src="_static/pretext/js/jquery.min.js"></script><script src="_static/pretext/js/dist/pretext-core.js?v=2.45"></script><!--** eBookCongig is necessary to configure interactive **-->
<!--** Runestone components to run locally in reader's browser **-->
<!--** No external communication: **-->
<!--** log level is 0, Runestone Services are disabled **-->
<script type="text/javascript">
eBookConfig = {};
eBookConfig.useRunestoneServices = false;
eBookConfig.host = 'http://127.0.0.1:8000';
eBookConfig.course = 'PTX_Course_Title_Here';
eBookConfig.basecourse = 'PTX_Base_Course';
eBookConfig.isLoggedIn = false;
eBookConfig.email = '';
eBookConfig.isInstructor = false;
eBookConfig.logLevel = 0;
eBookConfig.username = '';
eBookConfig.readings = null;
eBookConfig.activities = null;
eBookConfig.downloadsEnabled = false;
eBookConfig.allow_pairs = false;
eBookConfig.enableScratchAC = false;
eBookConfig.build_info = "";
eBookConfig.python3 = null;
eBookConfig.runestone_version = '8.2.0';
eBookConfig.jobehost = '';
eBookConfig.proxyuri_runs = '';
eBookConfig.proxyuri_files = '';
eBookConfig.enable_chatcodes = false;
</script>
<!--*** Runestone Services ***-->
<script src="_static/prefix-runtime.82b0f3334d1d4a05.bundle.js"></script><script src="_static/prefix-926.5c7038c8ef173faf.bundle.js"></script><script src="_static/prefix-runestone.5555907f139e2014.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-926.229729d09512acc9.css">
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.8086b62682b271d1.css">
<script src="_static/pretext/js/dist/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/dist/pretext_search.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
</script>
</head>
<body class="pretext book ignore-math">
<a class="assistive" href="#ptx-content">Skip to main content</a><header id="ptx-masthead" class="ptx-masthead" role="banner"><div class="ptx-banner"><div class="title-container">
<div class="heading ptx-banner-heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></div>
<p class="byline">Andy Eisenberg (testing)</p>
</div></div></header><nav id="ptx-navbar" class="ptx-navbar navbar"><div class="ptx-navbar-contents">
<button id="ptx-toc-toggle" type="button" aria-controls="ptx-sidebar" class="toc-toggle button" title="Contents"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5d2;</span><span class="name">Contents</span></button><div class="ptx-search-box">
<div class="ptx-search-widget"><button id="ptx-search-button" type="button" class="ptx-search-button button" title="Search Book"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe8b6;</span><span class="name">Search Book</span></button></div>
<dialog id="ptx-search-dialog" class="ptx-dialog ptx-search-dialog"><div class="ptx-search-dialog-controls">
<input aria-label="Search term" id="ptx-search-terms" class="ptx-search-terms" type="text" name="terms" placeholder="Search terms"><button aria-label="Close" id="ptx-search-close" class="ptx-search-close"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cd;</span></button>
</div>
<div id="ptx-search-status" class="ptx-search-status" aria-live="polite" aria-atomic="true"></div>
<h2 class="heading ptx-search-results-heading">Search Results: </h2>
<div id="ptx-search-empty" class="ptx-search-empty"><span>No results.</span></div>
<ol id="ptx-search-results" class="ptx-search-results"></ol></dialog>
</div>
<span class="nav-other-controls"><button id="ptx-embed-button" class="ptx-embed-button button" title="Embed this page"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe86f;</span><span class="name">Embed</span></button><dialog class="ptx-dialog ptx-embed-popup" id="ptx-embed-popup"><div class="ptx-embed-popup-controls">
<p>Copy the code below to embed this page in your own website or LMS page.</p>
<button class="ptx-embed-close-button button" id="ptx-embed-close-button" title="Close"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cd;</span></button>
</div>
<div class="ptx-embed-code-container">
<textarea class="ptx-embed-code-textbox" id="ptx-embed-code-textbox" readonly aria-label="Embed"><iframe src="https://example.com/embed" width="100%" height="1000"></iframe></textarea><button autofocus="autofocus" class="ptx-embed-copy-button button" id="ptx-embed-copy-button" title="Copy"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe14d;</span><span class="name">Copy</span></button>
</div></dialog><button id="ptx-readability-options-button" class="ptx-readability-options-button button" title="Readability settings"><span class="icon material-symbols-outlined" aria-hidden="true">&#xf6f1;</span><span class="name">Readability settings</span></button><dialog class="ptx-dialog ptx-readability-options-popup" id="ptx-readability-options-popup"><div class="ptx-readability-options-popup-controls">
<h2 class="heading">Readability settings</h2>
<button class="ptx-readability-options-close-button button" id="ptx-readability-options-close-button" title="Close"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cd;</span></button>
</div>
<div class="ptx-readability-options-group">
<label class="ptx-readability-slider-label" for="ptx-readability-line-height"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe240;</span><span>Line height</span><output id="ptx-readability-line-height-value" for="ptx-readability-line-height">1.35</output></label><input type="range" id="ptx-readability-line-height" min="1.10" max="3" step="0.05" value="1.35">
</div>
<div class="ptx-readability-options-group">
<label class="ptx-readability-slider-label" for="ptx-readability-font-size"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe245;</span><span>Content font size</span><output id="ptx-readability-font-size-value" for="ptx-readability-font-size">100%</output></label><input type="range" id="ptx-readability-font-size" min="0.8" max="1.5" step="0.05" value="1">
</div>
<fieldset class="ptx-readability-options-group ptx-readability-options-group-theme">
<legend>Theme</legend>
<label for="ptx-readability-theme-system"><input type="radio" name="ptx-readability-theme" id="ptx-readability-theme-system" value="system"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe1ab;</span><span>System</span></label><label for="ptx-readability-theme-light"><input type="radio" name="ptx-readability-theme" id="ptx-readability-theme-light" value="light"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe518;</span><span>Light</span></label><label for="ptx-readability-theme-dark"><input type="radio" name="ptx-readability-theme" id="ptx-readability-theme-dark" value="dark"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe51c;</span><span>Dark</span></label>
</fieldset>
<fieldset class="ptx-readability-options-group">
<legend>Assistive technologies</legend>
<label for="ptx-readability-accessible-permalinks"><input type="checkbox" id="ptx-readability-accessible-permalinks"><span>🔗</span><span>Expose permalinks</span></label>
</fieldset>
<div class="ptx-readability-options-group"><button type="button" class="ptx-readability-reset-button button" id="ptx-readability-reset-button"><span class="icon material-symbols-outlined" aria-hidden="true">&#xf053;</span><span class="name">Reset All</span></button></div></dialog></span><span class="treebuttons"><a class="previous-button button" href="sec-Variance.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cb;</span><span class="name">Prev</span></a><a class="up-button button" href="ch-Expected-Value.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5ce;</span><span class="name">Up</span></a><a class="next-button button" href="ch-Confidence-Intervals.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cc;</span></a></span>
</div></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\DeclareMathOperator{\Bin}{Bin}
\DeclareMathOperator{\Geom}{Geom}
\DeclareMathOperator{\Poiss}{Poiss}
\DeclareMathOperator{\Exp}{Exp}
\DeclareMathOperator{\Norm}{N}
\DeclareMathOperator{\E}{E}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\Cov}{Cov}
\renewcommand{\L}{\mathcal{L}}
\newcommand{\est}{\widehat}
\newcommand{\lt}{&lt;}
\newcommand{\gt}{&gt;}
\newcommand{\amp}{&amp;}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)</span></div>
<div class="ptx-page">
<div id="ptx-sidebar" role="complementary" class="ptx-sidebar"><nav id="ptx-toc" tabindex="-1" aria-label="Contents" class="ptx-toc depth2"><ul class="structural toc-item-list contains-active">
<li class="toc-item toc-frontmatter"><div class="toc-title-box"><a href="frontmatter.html" class="internal"><span class="title">Front Matter</span></a></div></li>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="ch-Introduction.html" class="internal"><span class="codenumber">0</span> <span class="title">Introduction</span></a></div>
<ul id="ptx-toc-group-ch-Introduction" class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Introduction.html" class="internal"><span class="codenumber">0.1</span> <span class="title">Introduction</span></a></div></li></ul>
</li>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="ch-Probability.html" class="internal"><span class="codenumber">1</span> <span class="title">Probability Theory</span></a></div>
<ul id="ptx-toc-group-ch-Probability" class="structural toc-item-list">
<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Probability-2.html" class="internal"><span class="title">Introduction</span></a></div></li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Set-Theory.html" class="internal"><span class="codenumber">1.1</span> <span class="title">Set Theory</span></a></div>
<ul id="ptx-toc-group-sec-Set-Theory" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Set-Theory.html#exercises-Set-Theory" class="internal"><span class="codenumber">1.1</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</span></a></div>
<ul id="ptx-toc-group-sec-Probability" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Probability.html#exercises-Probability" class="internal"><span class="codenumber">1.2</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
<ul id="ptx-toc-group-sec-Conditional-Probability" class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-conditional-probability" class="internal"><span class="codenumber">1.3.1</span> <span class="title">Conditional Probability</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-diagnostic-testing" class="internal"><span class="codenumber">1.3.2</span> <span class="title">Diagnostic Testing</span></a></div></li>
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#exercises-Conditional-Probability" class="internal"><span class="codenumber">1.3.3</span> <span class="title">Exercises</span></a></div></li>
</ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Independent-Events.html" class="internal"><span class="codenumber">1.4</span> <span class="title">Independent Events</span></a></div>
<ul id="ptx-toc-group-sec-Independent-Events" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Independent-Events.html#exercises-Independent-Events" class="internal"><span class="codenumber">1.4</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div>
<ul id="ptx-toc-group-ch-Random-Variables" class="structural toc-item-list">
<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Random-Variables-2.html" class="internal"><span class="title">Introduction</span></a></div></li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Discrete-RVs.html" class="internal"><span class="codenumber">2.1</span> <span class="title">Discrete Random Variables</span></a></div>
<ul id="ptx-toc-group-sec-Discrete-RVs" class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-Discrete-RVs" class="internal"><span class="codenumber">2.1.1</span> <span class="title">Discrete Random Variables</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-geometric-distribution" class="internal"><span class="codenumber">2.1.3</span> <span class="title">Geometric Distribution</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
</ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
<ul id="ptx-toc-group-sec-Continuous-RVs" class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-pdfs" class="internal"><span class="codenumber">2.2.1</span> <span class="title">Probability Density Functions</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-cdf" class="internal"><span class="codenumber">2.2.2</span> <span class="title">Cumulative Distribution Functions</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-normal-distribution" class="internal"><span class="codenumber">2.2.4</span> <span class="title">Normal Distribution</span></a></div></li>
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
</ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Joint-Distributions.html" class="internal"><span class="codenumber">2.3</span> <span class="title">Joint Distributions</span></a></div>
<ul id="ptx-toc-group-sec-Joint-Distributions" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Joint-Distributions.html#exercises-Joint-Distributions" class="internal"><span class="codenumber">2.3</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-chapter contains-active">
<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
<ul class="structural toc-item-list contains-active" id="ptx-toc-group-ch-Expected-Value">
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
<ul id="ptx-toc-group-sec-Expected-Value" class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-discrete-EV" class="internal"><span class="codenumber">3.1.1</span> <span class="title">Discrete Expected Value</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-continuous-EV" class="internal"><span class="codenumber">3.1.2</span> <span class="title">Continuous Expected Value</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-linearity-EV" class="internal"><span class="codenumber">3.1.3</span> <span class="title">Linearity of Expected Value</span></a></div></li>
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Expected-Value.html#exercises-Expected-Value" class="internal"><span class="codenumber">3.1.4</span> <span class="title">Exercises</span></a></div></li>
</ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
<ul id="ptx-toc-group-sec-Variance" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section contains-active">
<div class="toc-title-box"><a href="sec-Covariance.html" class="internal"><span class="codenumber">3.3</span> <span class="title">Covariance</span></a></div>
<ul class="structural toc-item-list active" id="ptx-toc-group-sec-Covariance"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Covariance.html#exercises-Covariance" class="internal"><span class="codenumber">3.3</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
<ul id="ptx-toc-group-ch-Confidence-Intervals" class="structural toc-item-list">
<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Confidence-Intervals-2.html" class="internal"><span class="title">Introduction</span></a></div></li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Likelihood.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Likelihood</span></a></div>
<ul id="ptx-toc-group-sec-Likelihood" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Likelihood.html#sec-Likelihood-20" class="internal"><span class="codenumber">4.1</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Central Limit Theorem</span></a></div>
<ul id="ptx-toc-group-sec-CLT" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-CLT.html#exercises-CLT" class="internal"><span class="codenumber">4.2</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.3</span> <span class="title">Confidence Intervals</span></a></div>
<ul id="ptx-toc-group-sec-Confidence-Intervals" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html#exercises-Confidence-Intervals" class="internal"><span class="codenumber">4.3</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
<ul id="ptx-toc-group-ch-Hypothesis-Testing" class="structural toc-item-list">
<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Hypothesis-Testing-2.html" class="internal"><span class="title">Introduction</span></a></div></li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div>
<ul id="ptx-toc-group-sec-One-Sample-Tests" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html#exercises-One-Sample-Tests" class="internal"><span class="codenumber">5.1</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div>
<ul id="ptx-toc-group-sec-Two-Sample-Tests" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html#exercises-Two-Sample-Tests" class="internal"><span class="codenumber">5.2</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div>
<ul id="ptx-toc-group-sec-Power" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Power.html#exercises-Power" class="internal"><span class="codenumber">5.3</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div>
<ul id="ptx-toc-group-sec-Chi-Squared" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Chi-Squared.html#exercises-Chi-Squared" class="internal"><span class="codenumber">5.4</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
<ul id="ptx-toc-group-ch-Linear-Regression" class="structural toc-item-list">
<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Linear-Regression-2.html" class="internal"><span class="title">Introduction</span></a></div></li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div>
<ul id="ptx-toc-group-sec-Correlation" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Correlation.html#exercises-Correlation" class="internal"><span class="codenumber">6.1</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div>
<ul id="ptx-toc-group-sec-Linear-Regression" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Linear-Regression.html#exercises-Linear-Regression" class="internal"><span class="codenumber">6.2</span> <span class="title">Exercises</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-backmatter">
<div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div>
<ul id="ptx-toc-group-backmatter" class="structural toc-item-list">
<li class="toc-item toc-appendix">
<div class="toc-title-box"><a href="backmatter-2.html" class="internal"><span class="codenumber">A</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table</span></a></div>
<ul id="ptx-toc-group-backmatter-2" class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Phi-table.html" class="internal"><span class="codenumber">A.1</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table of Values</span></a></div></li></ul>
</li>
<li class="toc-item toc-appendix">
<div class="toc-title-box"><a href="backmatter-3.html" class="internal"><span class="codenumber">B</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values Table</span></a></div>
<ul id="ptx-toc-group-backmatter-3" class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Chi-squared-table.html" class="internal"><span class="codenumber">B.1</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values</span></a></div></li></ul>
</li>
<li class="toc-item toc-colophon"><div class="toc-title-box"><a href="backmatter-4.html" class="internal"><span class="title">Colophon</span></a></div></li>
</ul>
</li>
</ul></nav></div>
<main class="ptx-main"><div id="ptx-content" class="ptx-content">
<section class="section" id="sec-Covariance">
<h1 class="heading hide-type">
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.3</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Covariance</span>
</h1>
<article class="definition definition-like" id="def-covariance"><h2 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.3.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h2>
<div class="para logical" id="def-covariance-1-1">
<div class="para">Let <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> be random variables. The <dfn class="terminology">covariance</dfn> of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> is:</div>
<div class="displaymath process-math" id="def-covariance-1-1-6">
\begin{gather*}
\Cov(X, Y) = \E\left[\left(X - \E(X)\right) \left(Y - \E(Y)\right)\right].
\end{gather*}
</div>
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#def-covariance-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" aria-hidden="true" data-description="Definition 3.3.1"><a tabindex="-1" href="#def-covariance" title="Copy heading and permalink for Definition 3.3.1" aria-label="Copy heading and permalink for Definition 3.3.1">🔗</a></div>
</article>
<div class="para logical" id="sec-Covariance-3">
<div class="para">Let’s write <span class="process-math">\(\mu_X = \E(X)\)</span> and <span class="process-math">\(\mu_Y = \E(Y)\)</span> to make this is a bit cleaner to look at:</div>
<div class="displaymath process-math" id="sec-Covariance-3-3">
\begin{gather*}
\Cov(X, Y) = \E\left[\left(X - \mu_X \right) \left(Y - \mu_Y\right)\right].
\end{gather*}
</div>
<div class="para">This looks like a pretty natural generalization of the variance. In fact:</div>
<div class="displaymath process-math" id="sec-Covariance-3-4">
\begin{align*}
\Cov(X, X) \amp = \E\left[ (X - \mu_X)(X - \mu_X)\right] = \E\left[\left(X - \mu_X\right)^2\right] = \Var(X).
\end{align*}
</div>
<div class="para">One difference worth noting is that <span class="process-math">\(\Var(X)\text{,}\)</span> as the average of squared values, must always be nonnegative. The expression <span class="process-math">\((X - \mu_X)(X - \mu_X)\)</span> can be negative for combinations of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> values where one is greater than its average and the other is smaller. If this is the average behavior, then <span class="process-math">\(\Cov(X, Y)\)</span> can be negative overall. This is useful information: a negative <span class="process-math">\(\Cov(X, Y)\)</span> would imply that (on average, not necessarily always) when <span class="process-math">\(X\)</span> is larger than typical, <span class="process-math">\(Y\)</span> is smaller than typical, and vice versa.</div>
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Covariance-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="sec-Covariance-4">Also as with variance, the covariance definition formula gives us some useful insight, but it can be unwieldy to work with. There is an alternative formula, analogous to <a href="sec-Variance.html#thm-variance-formula" class="xref" data-knowl="./knowl/xref/thm-variance-formula.html" data-reveal-label="Reveal" data-close-label="Close" title="Theorem 3.2.2">Theorem 3.2.2</a>:<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Covariance-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="theorem theorem-like" id="thm-covariance-formula"><h2 class="heading">
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.3.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h2>
<div class="para logical" id="thm-covariance-formula-1-1">
<div class="para">Let <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> be random variables. Then:</div>
<div class="displaymath process-math" id="thm-covariance-formula-1-1-3">
\begin{gather*}
\Cov(X, Y) = \E\left(XY\right) - \E(X)\E(Y).
\end{gather*}
</div>
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#thm-covariance-formula-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" aria-hidden="true" data-description="Theorem 3.3.2"><a tabindex="-1" href="#thm-covariance-formula" title="Copy heading and permalink for Theorem 3.3.2" aria-label="Copy heading and permalink for Theorem 3.3.2">🔗</a></div>
</article>
<div class="para" id="sec-Covariance-6">To make use of this formula, it’s helpful to have both a joint distribution and marginal distributions. The marginal distributions let you find <span class="process-math">\(\E(X)\)</span> and <span class="process-math">\(\E(Y)\text{.}\)</span> The joint distribution lets you find <span class="process-math">\(\E(XY)\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Covariance-6" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="sec-Covariance-7"><h2 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.3.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h2>
<div class="para" id="sec-Covariance-7-1-1">Let <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> be indicator random variables with joint and marginal distributions below.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Covariance-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="sidebyside">
<div class="sbsrow" style="margin-left:0%;margin-right:0%;grid-template-columns:33.3333333333333% 33.3333333333333% 33.3333333333333%;column-gap:0%;">
<div class="sbspanel fixed-width sbspanel--top top" style="">
<figure class="table table-like" id="sec-Covariance-7-1-2-1"><table class="tabular">
<tr>
<td class="c m b1 r1 l0 t0 lines"></td>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
</tr>
<tr>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">0.2</td>
<td class="c m b1 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r1 l0 t0 lines">0.2</td>
<td class="c m b0 r0 l0 t0 lines">0.5</td>
</tr>
</table>
<figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.3.4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" aria-hidden="true" data-description="Table 3.3.4: "><a tabindex="-1" href="#sec-Covariance-7-1-2-1" title="Copy heading and permalink for Table 3.3.4: " aria-label="Copy heading and permalink for Table 3.3.4: ">🔗</a></div></figcaption>
</figure>
</div>
<div class="sbspanel fixed-width sbspanel--top top" style="">
<figure class="table table-like" id="sec-Covariance-7-1-2-2"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">0</td>
<td class="c m b0 r0 l0 t0 lines">0.4</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines">0.6</td>
</tr>
</table>
<figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.3.5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" aria-hidden="true" data-description="Table 3.3.5: "><a tabindex="-1" href="#sec-Covariance-7-1-2-2" title="Copy heading and permalink for Table 3.3.5: " aria-label="Copy heading and permalink for Table 3.3.5: ">🔗</a></div></figcaption>
</figure>
</div>
<div class="sbspanel fixed-width sbspanel--top top" style="">
<figure class="table table-like" id="sec-Covariance-7-1-2-3"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(y\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(Y = y)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">0</td>
<td class="c m b0 r0 l0 t0 lines">0.3</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines">0.7</td>
</tr>
</table>
<figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.3.6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" aria-hidden="true" data-description="Table 3.3.6: "><a tabindex="-1" href="#sec-Covariance-7-1-2-3" title="Copy heading and permalink for Table 3.3.6: " aria-label="Copy heading and permalink for Table 3.3.6: ">🔗</a></div></figcaption>
</figure>
</div>
</div>
</div>
<div class="para logical" id="sec-Covariance-7-1-3">
<div class="para">Since they’re indicator random variables, we can immediately see that <span class="process-math">\(\E(X) = 0.6\)</span> and <span class="process-math">\(\E(Y) = 0.7\)</span> (see <a href="sec-Expected-Value.html#example-indicator-EV" class="xref" data-knowl="./knowl/xref/example-indicator-EV.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.1.5">Example 3.1.5</a>). For <span class="process-math">\(\E(XY)\text{,}\)</span> we’ll traverse cell by cell through the joint distribution table, constructing products of the form <span class="process-math">\(X\)</span>-value times <span class="process-math">\(Y\)</span>-value times probability:</div>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-indicator-EV.html" id="sec-Covariance-7-1-3-7">
\begin{align*}
\E(XY) =\, \amp (0)(0)(0.2) + (1)(0)(0.1) \\
\amp (0)(1)(0.2) + (1)(1)(0.5) \\
=\, \amp 0.5.
\end{align*}
</div>
<div class="para">Finally:</div>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-indicator-EV.html" id="sec-Covariance-7-1-3-8">
\begin{gather*}
\Cov(X, Y) = \E(XY) - \E(X)\E(Y) = 0.5 - (0.6)(0.7) = 0.08.
\end{gather*}
</div>
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Covariance-7-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" aria-hidden="true" data-description="Example 3.3.3"><a tabindex="-1" href="#sec-Covariance-7" title="Copy heading and permalink for Example 3.3.3" aria-label="Copy heading and permalink for Example 3.3.3">🔗</a></div>
</article>
<section class="exercises" id="exercises-Covariance">
<h2 class="heading hide-type">
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
</h2>
<article class="exercise exercise-like" id="exercises-Covariance-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para" id="exercises-Covariance-1-1-1">Calculate the covariance of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{:}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Covariance-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<figure class="table table-like" id="exercises-Covariance-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.3.7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" aria-hidden="true" data-description="Table 3.3.7: "><a tabindex="-1" href="#exercises-Covariance-1-1-2" title="Copy heading and permalink for Table 3.3.7: " aria-label="Copy heading and permalink for Table 3.3.7: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r1 l0 t0 lines"></th>
<th scope="col" class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 2\)</span></th>
</tr>
<tr>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">0.12</td>
<td class="c m b1 r0 l0 t0 lines">0.24</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r1 l0 t0 lines">0.3</td>
<td class="c m b0 r0 l0 t0 lines">34</td>
</tr>
</table></div>
</figure><div class="autopermalink" aria-hidden="true" data-description="Exercise 3.3.1"><a tabindex="-1" href="#exercises-Covariance-1" title="Copy heading and permalink for Exercise 3.3.1" aria-label="Copy heading and permalink for Exercise 3.3.1">🔗</a></div>
</article>
<article class="exercise exercise-like" id="exercises-Covariance-2"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para" id="exercises-Covariance-2-1-1">Calculate the covariance of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{:}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Covariance-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<figure class="table table-like" id="exercises-Covariance-2-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.3.8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" aria-hidden="true" data-description="Table 3.3.8: "><a tabindex="-1" href="#exercises-Covariance-2-1-2" title="Copy heading and permalink for Table 3.3.8: " aria-label="Copy heading and permalink for Table 3.3.8: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r1 l0 t0 lines"></th>
<th scope="col" class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></th>
<th scope="col" class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 2\)</span></th>
</tr>
<tr>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">0.08</td>
<td class="c m b1 r1 l0 t0 lines">0.16</td>
<td class="c m b1 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r1 l0 t0 lines">0.14</td>
<td class="c m b0 r1 l0 t0 lines">0.2</td>
<td class="c m b0 r0 l0 t0 lines">0.32</td>
</tr>
</table></div>
</figure><div class="autopermalink" aria-hidden="true" data-description="Exercise 3.3.2"><a tabindex="-1" href="#exercises-Covariance-2" title="Copy heading and permalink for Exercise 3.3.2" aria-label="Copy heading and permalink for Exercise 3.3.2">🔗</a></div>
</article>
<article class="exercise exercise-like" id="exercises-Covariance-3"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para" id="exercises-Covariance-3-1-1">Suppose we roll a fair, 4-sided die two times. Let <span class="process-math">\(X\)</span> be the sum of the rolls, and let <span class="process-math">\(Y\)</span> be the product of the rolls. Find the covariance of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Covariance-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="exercises-Covariance-3-1-2">[Note: Somewhat tedious.]<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Covariance-3-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" aria-hidden="true" data-description="Exercise 3.3.3"><a tabindex="-1" href="#exercises-Covariance-3" title="Copy heading and permalink for Exercise 3.3.3" aria-label="Copy heading and permalink for Exercise 3.3.3">🔗</a></div>
</article><div class="autopermalink" aria-hidden="true" data-description="Exercises 3.3"><a tabindex="-1" href="#exercises-Covariance" title="Copy heading and permalink for Exercises 3.3" aria-label="Copy heading and permalink for Exercises 3.3">🔗</a></div>
</section><div class="autopermalink" aria-hidden="true" data-description="Section 3.3: Covariance"><a tabindex="-1" href="#sec-Covariance" title="Copy heading and permalink for Section 3.3: Covariance" aria-label="Copy heading and permalink for Section 3.3: Covariance">🔗</a></div>
</section>
</div>
<div id="ptx-content-footer" class="ptx-content-footer">
<a class="previous-button button" href="sec-Variance.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cb;</span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5ce;</span><span class="name">Top</span></a><a class="next-button button" href="ch-Confidence-Intervals.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cc;</span></a>
</div></main>
</div>
<div id="ptx-page-footer" class="ptx-page-footer">
<a class="pretext-link" href="https://pretextbook.org" title="PreTeXt"><div class="logo"><svg xmlns="http://www.w3.org/2000/svg" height="100%" viewBox="338 3000 8772 6866" role="img"><title>PreTeXt logo</title><g style="stroke-width:.025in; stroke:currentColor; fill:none"><polyline points="472,3590 472,9732 " style="stroke-width:174; stroke-linejoin:miter; stroke-linecap:round; "></polyline><path style="stroke-width:126;stroke-linecap:butt;" d="M 4724,9448 A 4660 4660 0 0 1 8598 9259"></path><path style="stroke-width:174;stroke-linecap:butt;" d="M 4488,9685 A 4228 4228 0 0 0 472 9732"></path><path style="stroke-width:126;stroke-linecap:butt;" d="M 4724,3590 A 4241 4241 0 0 1 8598 3496"></path><path style="stroke-width:126;stroke-linecap:round;" d="M 850,3496 A 4241 4241 0 0 1 4724 3590"></path><path style="stroke-width:126;stroke-linecap:round;" d="M 850,9259 A 4507 4507 0 0 1 4724 9448"></path><polyline points="5385,4299 4062,8125" style="stroke-width:300; stroke-linejoin:miter; stroke-linecap:round;"></polyline><polyline points="8598,3496 8598,9259" style="stroke-width:126; stroke-linejoin:miter; stroke-linecap:round;"></polyline><polyline points="850,3496 850,9259" style="stroke-width:126; stroke-linejoin:miter; stroke-linecap:round;"></polyline><polyline points="4960,9685 4488,9685" style="stroke-width:174; stroke-linejoin:miter; stroke-linecap:round;"></polyline><polyline points="3070,4582 1889,6141 3070,7700" style="stroke-width:300; stroke-linejoin:miter; stroke-linecap:round;"></polyline><polyline points="6418,4582 7600,6141 6418,7700" style="stroke-width:300; stroke-linejoin:miter; stroke-linecap:round;"></polyline><polyline points="8976,3590 8976,9732" style="stroke-width:174; stroke-linejoin:miter; stroke-linecap:round;"></polyline><path style="stroke-width:174;stroke-linecap:butt;" d="M 4960,9685 A 4228 4228 0 0 1 8976 9732"></path></g></svg></div></a><a class="runestone-link" href="https://runestone.academy" title="Runestone Academy"><img class="logo" src="https://runestone.academy/runestone/static/images/RAIcon_cropped.png" alt="Runstone Academy logo"></a><a class="mathjax-link" href="https://www.mathjax.org" title="MathJax"><img class="logo" src="https://www.mathjax.org/badge/badge-square-2.png" alt="MathJax logo"></a>
</div>
</body>
</html>
Reference in New Issue View Git Blame Copy Permalink