Files
Fall-2026-Math-1041/course/notes-02-03.html
T
2026-02-10 10:56:08 -05:00

287 lines
23 KiB
HTML
Executable File
Raw Blame History

This file contains ambiguous Unicode characters
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>Tuesday, Feb 3</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>
var runestoneMathReady = new Promise((resolve) => window.rsMathReady = resolve);
window.MathJax = {
"tex": {
"inlineMath": [
[
"\\(",
"\\)"
]
],
"tags": "none",
"tagSide": "right",
"tagIndent": ".8em",
"packages": {
"[+]": [
"base",
"extpfeil",
"ams",
"amscd",
"color",
"newcommand",
"knowl"
]
}
},
"options": {
"ignoreHtmlClass": "tex2jax_ignore|ignore-math",
"processHtmlClass": "process-math"
},
"chtml": {
"scale": 0.98,
"mtextInheritFont": true
},
"loader": {
"load": [
"input/asciimath",
"[tex]/extpfeil",
"[tex]/amscd",
"[tex]/color",
"[tex]/newcommand",
"[pretext]/mathjaxknowl3.js"
],
"paths": {
"pretext": "_static/pretext/js/lib"
}
},
"startup": {
pageReady() {
return MathJax.startup.defaultPageReady().then(function () {
console.log("in ready function");
rsMathReady();
}
)}
}
};
</script><script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-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 property="book:title" content="Introduction to Probability and Statistics for the Life Sciences">
<meta property="book:author" content="Andy Eisenberg">
<script src="_static/pretext/js/lib/jquery.min.js"></script><script src="_static/pretext/js/lib/jquery.sticky.js"></script><script src="_static/pretext/js/lib/jquery.espy.min.js"></script><script src="_static/pretext/js/pretext.js"></script><script src="_static/pretext/js/pretext_add_on.js?x=1"></script><script src="_static/pretext/js/user_preferences.js"></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 = '7.11.13';
eBookConfig.jobehost = '';
eBookConfig.proxyuri_runs = '';
eBookConfig.proxyuri_files = '';
eBookConfig.enable_chatcodes = false;
</script>
<!--*** Runestone Services ***-->
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
<script src="_static/pretext/js/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/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.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"><div class="ptx-banner"><div class="title-container">
<h1 class="heading"><a href="root-1-2.html"><span class="title">Introduction to Probability and Statistics for the Life Sciences</span> <span class="subtitle">Spring 2026 MATH 1044</span></a></h1>
<p class="byline">Andy Eisenberg</p>
</div></div></header><nav id="ptx-navbar" class="ptx-navbar navbar"><div class="ptx-navbar-contents">
<button 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="searchbox">
<div class="searchwidget"><button id="searchbutton" class="searchbutton button" type="button" title="Search book"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe8b6;</span><span class="name">Search Book</span></button></div>
<div id="searchresultsplaceholder" class="searchresultsplaceholder" style="display: none">
<div class="search-results-controls">
<input aria-label="Search term" id="ptxsearch" class="ptxsearch" type="text" name="terms" placeholder="Search term"><button title="Close search" id="closesearchresults" class="closesearchresults"><span class="material-symbols-outlined">close</span></button>
</div>
<h2 class="search-results-heading">Search Results: </h2>
<div id="searchempty" class="searchempty"><span>No results.</span></div>
<ol id="searchresults" class="searchresults"></ol>
</div>
</div>
<span class="nav-other-controls"><button id="embed-button" class="embed-button button" title="Embed this page"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe86f;</span><span class="name">Embed</span></button><div class="embed-popup hidden" id="embed-popup">
<p>Copy the code below to embed this page in your own website or LMS page.</p>
<div class="embed-code-container">
<textarea class="embed-code-textbox" id="embed-code-textbox" readonly aria-label="textbox"><iframe src="https://example.com/embed" width="100%" height="1000"></iframe></textarea><button class="copy-embed-button button" id="copy-embed-button" title="Copy embed code"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe14d;</span><span class="name">Copy</span></button>
</div>
</div>
<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe51c;</span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="notes-01-29.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-notes.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="quizzes.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">\(\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\DeclareMathOperator{\Bin}{Bin}
\DeclareMathOperator{\Geom}{Geom}
\DeclareMathOperator{\Poiss}{Poiss}
\DeclareMathOperator{\Exp}{Exp}
\DeclareMathOperator{\E}{E}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\Cov}{Cov}
\newcommand{\lt}{&lt;}
\newcommand{\gt}{&gt;}
\newcommand{\amp}{&amp;}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)</span></div>
<div class="ptx-page">
<div id="ptx-sidebar" class="ptx-sidebar"><nav id="ptx-toc" 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="course-documents.html" class="internal"><span class="title">Course Documents</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-section">
<div class="toc-title-box"><a href="syllabus.html" class="internal"><span class="title">Syllabus</span></a></div>
<ul class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="syllabus.html#sec-course-info" class="internal"><span class="title">Course Information</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="syllabus.html#sec-materials" class="internal"><span class="title">Textbook and Other Required Materials</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="syllabus.html#sec-assessments" class="internal"><span class="title">Assessments</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="syllabus.html#sec-grading" class="internal"><span class="title">Grading</span></a></div></li>
</ul>
</li></ul>
</li>
<li class="toc-item toc-chapter contains-active">
<div class="toc-title-box"><a href="ch-notes.html" class="internal"><span class="title">Class Notes</span></a></div>
<ul class="structural toc-item-list contains-active">
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
<ul class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
</ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
<ul class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Independent-Events" class="internal"><span class="title">Sec 1.4: Independent Events</span></a></div></li>
</ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
<ul class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-hw1-review" class="internal"><span class="title">HW 1 Q5</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li>
</ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-01-22.html" class="internal"><span class="title">Thursday, Jan 22</span></a></div>
<ul class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-discrete-distributions" class="internal"><span class="title">Discrete Distributions</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-continuous-distributions" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
</ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Tuesday, Jan 27</span></a></div>
<ul class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
</ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
</li>
<li class="toc-item toc-section contains-active">
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
<ul class="structural toc-item-list active"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
</li>
</ul>
</li>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="quizzes.html" class="internal"><span class="title">Quizzes</span></a></div>
<ul class="structural toc-item-list">
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-01.html" class="internal"><span class="title">Quiz 1</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
</ul>
</li>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
</li>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="review.html" class="internal"><span class="title">Exam Review</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-section">
<div class="toc-title-box"><a href="Exam-1-Review.html" class="internal"><span class="title">Exam 1 Review</span></a></div>
<ul class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="Exam-1-Review.html#Exam-1-Review-3" class="internal"><span class="title">Allowed Materials</span></a></div></li>
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="Exam-1-Review.html#Exam-1-Review-4" class="internal"><span class="title">Exercises</span></a></div></li>
</ul>
</li></ul>
</li>
</ul></nav></div>
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-02-03"><h2 class="heading hide-type">
<span class="type">Section</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">Tuesday, Feb 3</span>
</h2>
<section class="introduction" id="notes-02-03-2"><div class="para" id="notes-02-03-2-1">This is an outline of the topics we covered in class. These notes are <em class="emphasis">not</em> a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><section class="subsection" id="notes-02-03-3"><h3 class="heading hide-type">
<span class="type">Subsection</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">Variance</span>
</h3>
<div class="para logical" id="notes-02-03-3-2">
<div class="para">Question: How spread out are <span class="process-math">\(X\)</span> values? One answer we might try is to measure the average distance from the average value:</div>
<div class="displaymath process-math" id="notes-02-03-3-2-2">
\begin{gather*}
\E\left[|X - \E(X)|\right]
\end{gather*}
</div>
<div class="para">To simplify notation, we’ll write <span class="process-math">\(\mu = \E(X)\text{.}\)</span> Also, it’s often usefull to square a term rather than take absolute value when we want to ensure a positive output, so we’ll define...</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="definition definition-like" id="def-variance"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">90</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="def-variance-1-1">
<div class="para">Let <span class="process-math">\(X\)</span> be a random variable with <span class="process-math">\(\E(X) = \mu\text{.}\)</span> The <dfn class="terminology">variance</dfn> of <span class="process-math">\(X\)</span> is:</div>
<div class="displaymath process-math" id="def-variance-1-1-5">
\begin{gather*}
\sigma^2 = \Var(x) = \E\left[ (X - \mu)^2\right]
\end{gather*}
</div>
<div class="para">
<span class="process-math">\(\sigma = \sqrt{\Var(X)}\)</span> is called the <dfn class="terminology">standard deviation</dfn>.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#def-variance-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Definition 90"><a href="#def-variance" title="Copy heading and permalink for Definition 90" aria-label="Copy heading and permalink for Definition 90">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Variance"><a href="#notes-02-03-3" title="Copy heading and permalink for Subsection: Variance" aria-label="Copy heading and permalink for Subsection: Variance">🔗</a></div></section><div class="autopermalink" data-description="Section: Tuesday, Feb 3"><a href="#notes-02-03" title="Copy heading and permalink for Section: Tuesday, Feb 3" aria-label="Copy heading and permalink for Section: Tuesday, Feb 3">🔗</a></div></section></div>
<div id="ptx-content-footer" class="ptx-content-footer">
<a class="previous-button button" href="notes-01-29.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="quizzes.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>