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<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>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-02-19"><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">Thursday, Feb 19</span>
</h2>
<section class="introduction" id="notes-02-19-2"><div class="para" id="notes-02-19-2-1">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-02-19-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><section class="subsection" id="notes-02-19-3"><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">Central Limit Theorem</span>
</h3>
<article class="definition definition-like" id="notes-02-19-3-2"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">116</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="notes-02-19-3-2-1-1">
<div class="para">A continuous random variable <span class="process-math">\(X\)</span> has the <dfn class="terminology">normal distribution</dfn> with mean <span class="process-math">\(\mu\)</span> and variance <span class="process-math">\(\sigma^2\)</span> if it has pdf:</div>
<div class="displaymath process-math" id="notes-02-19-3-2-1-1-5">
\begin{gather*}
f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{-(x - \mu)^2}{2\sigma^2}}
\end{gather*}
</div>
<div class="para">We’ll write <span class="process-math">\(X \sim \Norm(\mu, \sigma^2)\text{.}\)</span>
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-19-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para logical" id="notes-02-19-3-2-1-2">
<div class="para">The specific distribution <span class="process-math">\(\Norm(0, 1)\)</span> is called the <dfn class="terminology">standard normal distribution</dfn>, and its pdf has the notation:</div>
<div class="displaymath process-math" id="notes-02-19-3-2-1-2-3">
\begin{gather*}
f(x; 0, 1) = \phi(x).
\end{gather*}
</div>
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</div>
<div class="autopermalink" data-description="Definition 116"><a href="#notes-02-19-3-2" title="Copy heading and permalink for Definition 116" aria-label="Copy heading and permalink for Definition 116">🔗</a></div></article><div class="para logical" id="notes-02-19-3-3">
<div class="para">If <span class="process-math">\(X \sim \Norm(\mu, \sigma^2)\text{,}\)</span> then:</div>
<div class="displaymath process-math" id="notes-02-19-3-3-2">
\begin{gather*}
\Pr(a \leq X \leq b) = \int_a^b f(x; \mu, \sigma^2)\ dx.
\end{gather*}
</div>
<div class="para">Unfortunately, <span class="process-math">\(e^{-x^2}\)</span> has no elementary antiderivative. But we know <span class="process-math">\(\int_a^b f(x)\ dx\)</span> represents the area under the graph <span class="process-math">\(y = f(x; \mu, \sigma^2)\text{,}\)</span> and this area can be approximated to arbitrary precision.</div>
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<article class="theorem theorem-like" id="thm-CLT"><h4 class="heading">
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">117</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="thm-CLT-1-1">
<div class="para">Suppose <span class="process-math">\(X_1, \dotsc, X_n\)</span> are <dfn class="terminology">independent and identically distributed</dfn> (or <dfn class="terminology">i.i.d.</dfn>) with finite expected value <span class="process-math">\(\mu\)</span> and finite variance <span class="process-math">\(\sigma^2\text{.}\)</span> Define:</div>
<div class="displaymath process-math" id="thm-CLT-1-1-6">
\begin{align*}
S_n \amp = X_1 + X_2 + \dotsb + X_n \\
A_n \amp = \frac{X_1 + X_2 + \dotsb + X_n}{n} = \frac{S_n}{n}
\end{align*}
</div>
<div class="para">Then, for large enough <span class="process-math">\(n\text{:}\)</span>
</div>
<div class="displaymath process-math" id="thm-CLT-1-1-8">
\begin{align*}
S_n \amp \approx \Norm(n\mu, n\sigma^2) \\
A_n \amp \approx \Norm\left(\mu, \frac{\sigma^2}{n}\right)
\end{align*}
</div>
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<div class="autopermalink" data-description="Theorem 117"><a href="#thm-CLT" title="Copy heading and permalink for Theorem 117" aria-label="Copy heading and permalink for Theorem 117">🔗</a></div></article><div class="para logical" id="notes-02-19-3-5">
<div class="para">We’ll use two general rules of thumb for determining whether the number of measurements <span class="process-math">\(n\)</span> is large enough for the approximation to be a good one:</div>
<ol class="decimal" id="notes-02-19-3-5-2">
<li id="notes-02-19-3-5-2-1">
<div class="para" id="notes-02-19-3-5-2-1-1">
<span class="process-math">\(\displaystyle n \geq 30\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-19-3-5-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
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</li>
<li id="notes-02-19-3-5-2-2">
<div class="para" id="notes-02-19-3-5-2-2-1">If <span class="process-math">\(S\sim \Bin(n, p)\text{,}\)</span> then <span class="process-math">\(n\)</span> should be large enough so that there are at least 5 heads and 5 tails.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-19-3-5-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
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</li>
</ol>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-19-3-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para logical" id="notes-02-19-3-6">
<div class="para">We’ll omit a proof of <a href="notes-02-19.html#thm-CLT" class="xref" data-knowl="./knowl/xref/thm-CLT.html" data-reveal-label="Reveal" data-close-label="Close" title="Theorem 117">Theorem 117</a>, but we can at least check that the expected values and variances of <span class="process-math">\(S_n, A_n\)</span> are correct. Given <span class="process-math">\(\E(X_i) = \mu, \Var(X_i) = \sigma^2\text{:}\)</span>
</div>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/thm-CLT.html" id="notes-02-19-3-6-4">
\begin{align*}
\E(S_n) \amp = \E(X_1 + \dotsb + X_n) \\
\amp = \E(X_1) + \dotsb + \E(X_n) \\
\amp = \mu + \dotsb + \mu \\
\amp = n \mu. \checkmark \\
\Var(S_n) \amp = \Var(X_1 + \dotsb + X_n) \\
\amp = \Var(X_1) + \dotsb + \Var(X_n) \\
\amp = \sigma^2 + \dotsb + \sigma^2 \\
\amp = n \sigma^2. \checkmark \\
\E(A_n) \amp = \E\left(\frac{S_n}{n}\right) \\
\amp = \frac{1}{n} \E(S_n) \\
\amp = \frac{1}{n} n \mu \\
\amp = \frac{1}{n} n \mu \\
\amp = \mu. \checkmark \\
\Var(A_n) \amp = \Var\left(\frac{S_n}{n}\right) \\
\amp = \frac{1}{n^2} \Var(S_n) \\
\amp = \frac{1}{n^2} n \sigma^2 \\
\amp = \frac{\sigma^2}{n}. \checkmark
\end{align*}
</div>
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<article class="example example-like" id="notes-02-19-3-7"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">118</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="notes-02-19-3-7-1-1">Suppose we flip a fair coin 100 times, and let <span class="process-math">\(S\)</span> be the number of heads. Estimate <span class="process-math">\(\Pr(40 \leq S \leq 60)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-19-3-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para logical" id="notes-02-19-3-7-1-2">
<div class="para">Since <span class="process-math">\(S \sim \Bin(n, p)\text{,}\)</span> we know:</div>
<div class="displaymath process-math" id="notes-02-19-3-7-1-2-2">
\begin{align*}
\E(S) \amp = np = (100)(0.5) = 50 \\
\Var(S) \amp = np(1-p) = (100)(0.5)(1 - 0.5) = 25
\end{align*}
</div>
<div class="para">So <span class="process-math">\(S \approx \Norm(50, 25)\text{.}\)</span> Then:</div>
<div class="displaymath process-math" id="notes-02-19-3-7-1-2-4">
\begin{gather*}
\Pr(40 \leq S \leq 60) \approx \int_{40}^{60} f(x; 50, 25)\ dx,
\end{gather*}
</div>
<div class="para">but we can’t compute this integral.</div>
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</div> <div class="para logical" id="notes-02-19-3-7-1-3">
<div class="para">Instead, we can <dfn class="terminology">standardize</dfn> by shifting and scaling:</div>
<div class="displaymath process-math" id="notes-02-19-3-7-1-3-2">
\begin{gather*}
Z = \frac{S - 50}{\sqrt{25}} = \frac{S - 50}{5}.
\end{gather*}
</div>
<div class="para">Then <span class="process-math">\(Z \sim \Norm(0, 1)\text{.}\)</span> Values of <span class="process-math">\(Z\)</span> are called <dfn class="terminology"><span class="process-math">\(z\)</span>-scores</dfn>, sometimes indicated by an asterisk. (I.e., if <span class="process-math">\(a\)</span> is a value of <span class="process-math">\(S\text{,}\)</span> then <span class="process-math">\(a^*\)</span> is the corresponding <span class="process-math">\(z\)</span>-score.) Now:</div>
<div class="displaymath process-math" id="notes-02-19-3-7-1-3-10">
\begin{align*}
\Pr(40 \leq S \leq 60) \amp \approx \Pr\left(\frac{40 - 50}{5} \leq Z \leq \frac{60 - 50}{5}\right) \\
\amp = \Pr(-2 \leq Z \leq 2) \\
\amp = \int_{-2}^2 \phi(x)\ dx.
\end{align*}
</div>
<div class="para">Now, we still can’t compute the value of the integral. However, there’s a huge benefit to translating to the standard normal distribution, regardless of <em class="emphasis">which</em> normal distribution we used to approximate <span class="process-math">\(S\text{.}\)</span> We can approximate integrals like <span class="process-math">\(\int_a^b \phi(x)\ dx\)</span> to abritrary precision and record the results in a table. Then, we can look up the values when needed. We don’t need to redo our approximations for different normal distributions, we just standardize whatever normal distribution we come across.</div>
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</div> <div class="para logical" id="notes-02-19-3-7-1-4">
<div class="para">So:</div>
<div class="displaymath process-math" id="notes-02-19-3-7-1-4-1">
\begin{align*}
\Pr(40 \leq S \leq 60) \amp \approx \Pr(-2 \leq Z \leq 2) \\
\amp = \Phi(2) - \Phi(-2) \\
\amp \approx 0.9772 - 0.0228 \\
\amp \approx 0.9544.
\end{align*}
</div>
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<div class="autopermalink" data-description="Example 118"><a href="#notes-02-19-3-7" title="Copy heading and permalink for Example 118" aria-label="Copy heading and permalink for Example 118">🔗</a></div></article><div class="para" id="notes-02-19-3-8">TODO: complex image introducing the idea of the <dfn class="terminology">continuity correction</dfn>.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-19-3-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para logical" id="notes-02-19-3-9">
<div class="para">We can make the approximation more accurate by extending the range of <span class="process-math">\(S\)</span>-values by half a unit in each direction:</div>
<div class="displaymath process-math" id="notes-02-19-3-9-2">
\begin{align*}
\Pr(40 \leq S \leq 60) \amp \approx \Pr\left(\frac{39.5 - 50}{5} \leq Z \leq \frac{60.5 - 50}{5}\right) \\
\amp = \Pr(-2.1 \leq Z \leq 2.1) \\
\amp = \Phi(2.1) - \Phi(-2.1) \\
\amp \approx 0.9821 - 0.0179 \\
\amp \approx 0.9642.
\end{align*}
</div>
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<div class="autopermalink" data-description="Subsection: Central Limit Theorem"><a href="#notes-02-19-3" title="Copy heading and permalink for Subsection: Central Limit Theorem" aria-label="Copy heading and permalink for Subsection: Central Limit Theorem">🔗</a></div></section><div class="autopermalink" data-description="Section: Thursday, Feb 19"><a href="#notes-02-19" title="Copy heading and permalink for Section: Thursday, Feb 19" aria-label="Copy heading and permalink for Section: Thursday, Feb 19">🔗</a></div></section></div>
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