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@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
eBookConfig.enableScratchAC = false;
eBookConfig.build_info = "";
eBookConfig.python3 = null;
eBookConfig.runestone_version = '7.11.13';
eBookConfig.runestone_version = '7.11.14';
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">
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.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>
@@ -135,7 +135,7 @@ eBookConfig.enable_chatcodes = false;
<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>
<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="notes-02-05.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}
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
\DeclareMathOperator{\E}{E}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\Cov}{Cov}
\newcommand{\L}{\mathcal{L}}
\newcommand{\lt}{&lt;}
\newcommand{\gt}{&gt;}
\newcommand{\amp}{&amp;}
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
<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>
<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>
</ul>
</li>
<li class="toc-item toc-chapter">
@@ -274,9 +286,212 @@ eBookConfig.enable_chatcodes = false;
<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 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="para logical" id="notes-02-03-3-4">
<div class="para">We can write an alternative formula here:</div>
<div class="displaymath process-math" id="notes-02-03-3-4-1">
\begin{align*}
\Var(X) \amp = \E\left[ (X - \mu)^2 \right] \\
\amp = \E\left[ X^2 - 2\mu X + \mu^2 \right] \\
\amp = \E(X^2) - 2\mu \E(X) + \E(\mu^2) \\
\amp = \E(X^2) - 2\mu^2 + \mu^2 \\
\amp = \E(X^2) - \mu^2.
\end{align*}
</div>
<div class="para">This formula is generally more useful for performing computations.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="notes-02-03-3-5"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">91</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="notes-02-03-3-5-1-1">Let <span class="process-math">\(X\)</span> indicate event <span class="process-math">\(A\)</span> with <span class="process-math">\(\Pr(A) = p\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-5-1-1" 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:10%;margin-right:10%;grid-template-columns:37.5% 37.5%;column-gap:20%;">
<div class="sbspanel fixed-width sbspanel--top top" style=""><figure class="table table-like" id="notes-02-03-3-5-1-2-1"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = k)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1 - p\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
</tr>
</table>
<figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">92<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">\(X\)</span><div class="autopermalink" data-description="Table 92: Distribution for X"><a href="#notes-02-03-3-5-1-2-1" title="Copy heading and permalink for Table 92: Distribution for X" aria-label="Copy heading and permalink for Table 92: Distribution for X">🔗</a></div></figcaption></figure></div>
<div class="sbspanel fixed-width sbspanel--top top" style=""><figure class="table table-like" id="notes-02-03-3-5-1-2-2"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X^2 = k)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0^2\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1 - p\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1^2\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
</tr>
</table>
<figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">93<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">\(X^2\)</span><div class="autopermalink" data-description="Table 93: Distribution for X^2"><a href="#notes-02-03-3-5-1-2-2" title="Copy heading and permalink for Table 93: Distribution for X^2" aria-label="Copy heading and permalink for Table 93: Distribution for X^2">🔗</a></div></figcaption></figure></div>
</div></div> <div class="para logical" id="notes-02-03-3-5-1-3">
<div class="para">Then <span class="process-math">\(\E(X) = p\)</span> and <span class="process-math">\(\E(X^2) = p\text{,}\)</span> so:</div>
<div class="displaymath process-math" id="notes-02-03-3-5-1-3-3">
\begin{gather*}
\Var(X) = \E(X^2) - \left(\E(X)\right)^2 = p - p^2 = p(1 - p).
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-5-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 91"><a href="#notes-02-03-3-5" title="Copy heading and permalink for Example 91" aria-label="Copy heading and permalink for Example 91">🔗</a></div></article><article class="example example-like" id="notes-02-03-3-6"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">94</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="notes-02-03-3-6-1-1">Let <span class="process-math">\(R\)</span> be the roll of a fair D6.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-6-1-1" 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:10%;margin-right:10%;grid-template-columns:37.5% 37.5%;column-gap:20%;">
<div class="sbspanel fixed-width sbspanel--top top" style=""><figure class="table table-like" id="notes-02-03-3-6-1-2-1"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(R = k)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
</table>
<figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">95<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">\(R\)</span><div class="autopermalink" data-description="Table 95: Distribution for R"><a href="#notes-02-03-3-6-1-2-1" title="Copy heading and permalink for Table 95: Distribution for R" aria-label="Copy heading and permalink for Table 95: Distribution for R">🔗</a></div></figcaption></figure></div>
<div class="sbspanel fixed-width sbspanel--top top" style=""><figure class="table table-like" id="notes-02-03-3-6-1-2-2"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(R^2 = k)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1^2\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2^2\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3^2\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4^2\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5^2\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6^2\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
</table>
<figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">96<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">\(R^2\)</span><div class="autopermalink" data-description="Table 96: Distribution for R^2"><a href="#notes-02-03-3-6-1-2-2" title="Copy heading and permalink for Table 96: Distribution for R^2" aria-label="Copy heading and permalink for Table 96: Distribution for R^2">🔗</a></div></figcaption></figure></div>
</div></div> <div class="para logical" id="notes-02-03-3-6-1-3">
<div class="para">Then <span class="process-math">\(\E(R) = \frac{7}{2}\)</span> and <span class="process-math">\(\E(R^2) = \frac{91}{6}\text{,}\)</span> so:</div>
<div class="displaymath process-math" id="notes-02-03-3-6-1-3-3">
\begin{gather*}
\Var(R) = \E(R^2) - \left(\E(R)\right)^2 = \frac{91}{6} - \frac{49}{4} = \frac{35}{12}.
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-6-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 94"><a href="#notes-02-03-3-6" title="Copy heading and permalink for Example 94" aria-label="Copy heading and permalink for Example 94">🔗</a></div></article><article class="example example-like" id="notes-02-03-3-7"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">97</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="notes-02-03-3-7-1-1">
<div class="para">Let <span class="process-math">\(X \in [0, 1]\)</span> with pdf <span class="process-math">\(f(x) = 2x\text{.}\)</span>
</div>
<div class="displaymath process-math" id="notes-02-03-3-7-1-1-3">
\begin{align*}
\E(X) \amp = \int_0^1 x \cdot 2x\ dx \\
\amp = \int_0^1 2x^2\ dx \\
\amp = \frac{2x^3}{3}\bigg|_0^1 \\
\amp = \frac{2}{3} - 0 \\
\amp = \frac{2}{3} \\
\E(X^2) \amp = \int_0^1 x^2\cdot 2x\ dx \\
\amp = \int_0^1 2x^3\ dx \\
\amp = \frac{2x^4}{4}\bigg|_0^1 \\
\amp = \frac{1}{2} - 0 \\
\amp = \frac{1}{2} \\
\Rightarrow \quad \Var(X) \amp = \E(X^2) - \left(\E(X)\right)^2 \\
\amp = \frac{1}{2} - \frac{4}{9} \\
\amp = \frac{1}{18}.
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 97"><a href="#notes-02-03-3-7" title="Copy heading and permalink for Example 97" aria-label="Copy heading and permalink for Example 97">🔗</a></div></article><div class="para logical" id="notes-02-03-3-8">
<div class="para">WARNING!!! In general, variance is <em class="emphasis">not</em> linear! That is,</div>
<div class="displaymath process-math" id="notes-02-03-3-8-2">
\begin{align*}
\Var(X + Y) \amp \neq \Var(X) + \Var(Y) \\
\Var(kX) \amp \neq k \Var(X)
\end{align*}
</div>
<div class="para">But there are still relevant properties we can state here:</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="theorem theorem-like" id="notes-02-03-3-9"><h4 class="heading">
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">98</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="notes-02-03-3-9-1-1">
<div class="para">Let <span class="process-math">\(X\)</span> be a random variable and <span class="process-math">\(k\in \R\text{.}\)</span> Then:</div>
<div class="displaymath process-math" id="notes-02-03-3-9-1-1-3">
\begin{align*}
\Var(X + k) \amp = \Var(X) \amp \Var(kX) \amp = k^2\Var(X).
\end{align*}
</div>
<div class="para">Moreover, if <span class="process-math">\(Y\)</span> is another random variable and <span class="process-math">\(X, Y\)</span> are independent, then:</div>
<div class="displaymath process-math" id="notes-02-03-3-9-1-1-6">
\begin{gather*}
\Var(X + Y) = \Var(X) + \Var(Y).
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Theorem 98"><a href="#notes-02-03-3-9" title="Copy heading and permalink for Theorem 98" aria-label="Copy heading and permalink for Theorem 98">🔗</a></div></article><article class="example example-like" id="notes-02-03-3-10"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">99</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="notes-02-03-3-10-1-1">
<div class="para">Let <span class="process-math">\(H_1, H_2, \dotsc, H_n\)</span> be indicators with parameter <span class="process-math">\(p\text{.}\)</span> Let <span class="process-math">\(S = H_1 + \dotsb + H_n\text{,}\)</span> so <span class="process-math">\(S \sim \Bin(n, p)\text{.}\)</span> We know <span class="process-math">\(\Var(H_i) = p(1-p)\text{,}\)</span> and <span class="process-math">\(H_1, \dotsc, H_n\)</span> are independent. So:</div>
<div class="displaymath process-math" id="notes-02-03-3-10-1-1-7">
\begin{align*}
\Var(S) \amp = \Var(H_1 + \dotsb + H_n) \\
\amp = \Var(H_1) + \dotsb + \Var(H_n) \\
\amp = p(1-p) + \dotsb + p(1-p) \\
\amp = np(1-p)
\end{align*}
</div>
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<div class="autopermalink" data-description="Example 99"><a href="#notes-02-03-3-10" title="Copy heading and permalink for Example 99" aria-label="Copy heading and permalink for Example 99">🔗</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>
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