Files
Fall-2026-Math-1044/course/notes-01-20.html
T
2026-03-26 09:52:49 -04:00

442 lines
37 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, Jan 20</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"
}
},
"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/jquery.min.js"></script><script src="_static/pretext/js/pretext.js"></script><script src="_static/pretext/js/pretext_add_on.js?x=1"></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 = '7.11.19';
eBookConfig.jobehost = '';
eBookConfig.proxyuri_runs = '';
eBookConfig.proxyuri_files = '';
eBookConfig.enable_chatcodes = false;
</script>
<!--*** Runestone Services ***-->
<script src="_static/prefix-runtime.998c9d9ee2a5e7ac.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.98ef3b0c170cd180.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.b12a7b3f0773ebcd.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/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-01-15.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="notes-01-22.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">\(\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\DeclareMathOperator{\Bin}{Bin}
\DeclareMathOperator{\Geom}{Geom}
\DeclareMathOperator{\Poiss}{Poiss}
\DeclareMathOperator{\Exp}{Exp}
\DeclareMathOperator{\Norm}{N}
\DeclareMathOperator{\E}{E}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\Cov}{Cov}
\newcommand{\L}{\mathcal{L}}
\newcommand{\lt}{&lt;}
\newcommand{\gt}{&gt;}
\newcommand{\amp}{&amp;}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)</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 contains-active">
<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 active">
<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">
<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"><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>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
<ul class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
</ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-02-19.html" class="internal"><span class="title">Thursday, Feb 19</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-19.html#notes-02-19-3" class="internal"><span class="title">Central Limit Theorem</span></a></div></li></ul>
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="notes-02-24.html" class="internal"><span class="title">Tuesday, Feb 24</span></a></div>
<ul class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-24.html#notes-02-24-3" class="internal"><span class="title">More Central Limit Theorem</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-24.html#notes-02-24-4" class="internal"><span class="title">Confidence Intervals</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>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li>
</ul>
</li>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="exams.html" class="internal"><span class="title">Exams</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="exam-01.html" class="internal"><span class="title">Exam 1</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>
<li class="toc-item toc-chapter">
<div class="toc-title-box"><a href="fun.html" class="internal"><span class="title">Just for Fun</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="boxes.html" class="internal"><span class="title">Boxes</span></a></div></li></ul>
</li>
</ul></nav></div>
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-01-20"><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, Jan 20</span>
</h2>
<section class="introduction" id="notes-01-20-2"><div class="para" id="notes-01-20-2-1">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-01-20-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><section class="subsection" id="subsec-hw1-review"><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">HW 1 Q5</span>
</h3>
<div class="para logical" id="subsec-hw1-review-2">
<div class="para">Write <span class="process-math">\(F\)</span> for the event that there’s a fire and <span class="process-math">\(S\)</span> for the event that there’s visible smoke. Then the information we’re given can be interpreted as:</div>
<div class="displaymath process-math" id="subsec-hw1-review-2-3">
\begin{align*}
\Pr(F) \amp = 0.01 \\
\Pr(S) \amp = 0.1 \\
\Pr(S\mid F) \amp = 0.9
\end{align*}
</div>
<div class="para">In this case, we can use the simpler version of Bayes’ Theorem:</div>
<div class="displaymath process-math" id="subsec-hw1-review-2-4">
\begin{gather*}
\Pr(F \mid S) = \frac{\Pr(S \mid F)\Pr(F)}{\Pr(S)} = \dotsb
\end{gather*}
</div>
<div class="para">Unlike our usual diagnostic testing examples, we do have access to the denominator probability here.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-hw1-review-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Subsection: HW 1 Q5"><a href="#subsec-hw1-review" title="Copy heading and permalink for Subsection: HW 1 Q5" aria-label="Copy heading and permalink for Subsection: HW 1 Q5">🔗</a></div></section><section class="subsection" id="subsec-Discrete-Random-Variables"><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">Sec 2.1: Random Variables</span>
</h3>
<article class="definition definition-like" id="def-RV"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">34</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="def-RV-1-1">A <dfn class="terminology">random variable</dfn> is a function <span class="process-math">\(X \colon \Omega \to \R\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-RV-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 34"><a href="#def-RV" title="Copy heading and permalink for Definition 34" aria-label="Copy heading and permalink for Definition 34">🔗</a></div></article><div class="para" id="subsec-Discrete-Random-Variables-3">The idea is that <span class="process-math">\(X\)</span> is a variable representing a real number value which depends on the outcome of an experiment.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="subsec-Discrete-Random-Variables-4"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">35</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="subsec-Discrete-Random-Variables-4-1-1">
<div class="para">An experiment consists of planting 50 seeds in a garden, then growing them for 3 months. Let <span class="process-math">\(H_i\)</span> be the height of plant <span class="process-math">\(i\text{.}\)</span> Let <span class="process-math">\(D\)</span> be the number of seeds that didn’t sprout. Let</div>
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-4-1-1-4">
\begin{align*}
A \amp = \text{avg height of all 50 plants} \\
\amp = \frac{H_1 + H_2 + \dotsb + H_{50}}{50}
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 35"><a href="#subsec-Discrete-Random-Variables-4" title="Copy heading and permalink for Example 35" aria-label="Copy heading and permalink for Example 35">🔗</a></div></article><div class="para" id="subsec-Discrete-Random-Variables-5">A random variable has its own probability distribution.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="subsec-Discrete-Random-Variables-6"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">36</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="subsec-Discrete-Random-Variables-6-1-1">Roll a fair D6 twice. Let <span class="process-math">\(S\)</span> be the sum of the rolls. Then <span class="process-math">\(\Omega = \{(1, 1), (1, 2), \dotsc, (6, 6)\}\)</span> has 36 elements. Since the die is fair, the distribution on <span class="process-math">\(\Omega\)</span> is uniform, i.e., <span class="process-math">\(\Pr(\omega) = \frac{1}{36}\)</span> for any <span class="process-math">\(\omega \in \Omega\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para" id="subsec-Discrete-Random-Variables-6-1-2">
<span class="process-math">\(S\)</span> takes on the values <span class="process-math">\(2, 3, 4, \dotsc, 12\text{,}\)</span> with probabilities:<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-6-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="subsec-Discrete-Random-Variables-6-1-3"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">37<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for <span class="process-math">\(S\)</span><div class="autopermalink" data-description="Table 37: Distribution for S"><a href="#subsec-Discrete-Random-Variables-6-1-3" title="Copy heading and permalink for Table 37: Distribution for S" aria-label="Copy heading and permalink for Table 37: Distribution for S">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\Pr(S = x)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">2</td>
<td class="c m b0 r0 l0 t0 lines">1/36</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines">2/36</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines">3/36</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">7</td>
<td class="c m b0 r0 l0 t0 lines">6/36</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">8</td>
<td class="c m b0 r0 l0 t0 lines">5/36</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
<td class="c m b0 r0 l0 t0 lines"></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">12</td>
<td class="c m b0 r0 l0 t0 lines">1/36</td>
</tr>
</table></div></figure> <div class="para" id="subsec-Discrete-Random-Variables-6-1-4">Note that the distribution on <span class="process-math">\(\Omega\)</span> is uniform, but the distribution on <span class="process-math">\(S\)</span> is not.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-6-1-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 36"><a href="#subsec-Discrete-Random-Variables-6" title="Copy heading and permalink for Example 36" aria-label="Copy heading and permalink for Example 36">🔗</a></div></article><article class="example example-like" id="subsec-Discrete-Random-Variables-7"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">38</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="subsec-Discrete-Random-Variables-7-1-1">
<div class="para">Let <span class="process-math">\(\Omega\)</span> be a sample space and <span class="process-math">\(A \subset \Omega\)</span> an event. Let</div>
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-7-1-1-3">
\begin{align*}
X = \begin{cases} 1 \amp x \in A \\ 0 \amp x \notin A \end{cases}
\end{align*}
</div>
<div class="para">
<span class="process-math">\(X\)</span> is called an <dfn class="terminology">indicator random variable</dfn>, and we say "<span class="process-math">\(X\)</span> <dfn class="terminology">indicates</dfn> <span class="process-math">\(A\)</span>".</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para logical" id="subsec-Discrete-Random-Variables-7-1-2">
<div class="para">The distribution on <span class="process-math">\(X\)</span> is:</div>
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-7-1-2-2">
\begin{align*}
\Pr(X = 1) \amp = \Pr(\{ x \mid x \in A\}) = \Pr(A) \\
\Pr(X = 0) \amp = 1 - \Pr(A)
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-7-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 38"><a href="#subsec-Discrete-Random-Variables-7" title="Copy heading and permalink for Example 38" aria-label="Copy heading and permalink for Example 38">🔗</a></div></article><article class="example example-like" id="subsec-Discrete-Random-Variables-8"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">39</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="subsec-Discrete-Random-Variables-8-1-1">
<div class="para">Suppose we flip a coin <span class="process-math">\(n\)</span> times. Let <span class="process-math">\(H_i\)</span> indicate heads on flip <span class="process-math">\(i\text{.}\)</span> Let <span class="process-math">\(S\)</span> be the total number of heads in all flips. Then:</div>
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-8-1-1-5">
\begin{gather*}
S = H_1 + H_2 + \dotsb + H_{n}
\end{gather*}
</div>
<div class="para">If <span class="process-math">\(p\)</span> is the probability of the coin coming up heads on a flip, then <span class="process-math">\(S\)</span> has the <dfn class="terminology">binomial distribution</dfn> with parameters <span class="process-math">\(n, p\text{.}\)</span> We’ll use the notation <span class="process-math">\(S \sim \Bin(n, p)\)</span> and:</div>
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-8-1-1-11">
\begin{gather*}
b(k) = b(k; n, p) = \Pr(S = k).
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para" id="subsec-Discrete-Random-Variables-8-1-2">Suppose the coin has <span class="process-math">\(p = 0.3\)</span> and we flip it <span class="process-math">\(n = 4\)</span> times. Find <span class="process-math">\(b(2) = b(2; 4, 0.3)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-8-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para logical" id="subsec-Discrete-Random-Variables-8-1-3">
<div class="para">The relevant flip sequences are:</div>
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-8-1-3-1">
\begin{gather*}
THHT, HHTT, TTHH, THTH, HTHT, HTTH
\end{gather*}
</div>
<div class="para">Each individual sequence has a probability of <span class="process-math">\((0.3)(0.3)(0.7)(0.7) = 0.0441\text{.}\)</span> So the total probability is:</div>
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-8-1-3-3">
\begin{gather*}
b(2; 4, 0.3) = (6)(0.0441) = 0.2646.
\end{gather*}
</div>
<div class="para">That is, (number of flip sequences)(probability of each sequence).</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-8-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 39"><a href="#subsec-Discrete-Random-Variables-8" title="Copy heading and permalink for Example 39" aria-label="Copy heading and permalink for Example 39">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Sec 2.1: Random Variables"><a href="#subsec-Discrete-Random-Variables" title="Copy heading and permalink for Subsection: Sec 2.1: Random Variables" aria-label="Copy heading and permalink for Subsection: Sec 2.1: Random Variables">🔗</a></div></section><div class="autopermalink" data-description="Section: Tuesday, Jan 20"><a href="#notes-01-20" title="Copy heading and permalink for Section: Tuesday, Jan 20" aria-label="Copy heading and permalink for Section: Tuesday, Jan 20">🔗</a></div></section></div>
<div id="ptx-content-footer" class="ptx-content-footer">
<a class="previous-button button" href="notes-01-15.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="notes-01-22.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>