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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-02-05"><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 5</span>
</h2>
<section class="introduction" id="notes-02-05-2"><div class="para" id="notes-02-05-2-1">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-02-05-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><section class="subsection" id="notes-02-05-3"><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">Summary</span>
</h3>
<div class="para" id="notes-02-05-3-2">Here are the expected value and variance formulas for common distributions. Some of these, we’ve shown justification for. Others requires techniques beyond the scope of the class to justify.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-3-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<figure class="table table-like" id="notes-02-05-3-3"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">100<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Expected Value and Variance Formulas<div class="autopermalink" data-description="Table 100: Expected Value and Variance Formulas"><a href="#notes-02-05-3-3" title="Copy heading and permalink for Table 100: Expected Value and Variance Formulas" aria-label="Copy heading and permalink for Table 100: Expected Value and Variance Formulas">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines">Distribution</td>
<td class="c m b1 r0 l0 t0 lines">Parameters</td>
<td class="c m b1 r0 l0 t0 lines">Expected Value</td>
<td class="c m b1 r0 l0 t0 lines">Variance</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">Indicator</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(p(1-p)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">Binomial</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(n, p\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(np\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(np(1-p)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">Geometric</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\frac{1}{p}\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\frac{1-p}{p^2}\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">Poisson</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\lambda\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\lambda\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\lambda\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">Exponential</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\lambda\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\frac{1}{\lambda}\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\frac{1}{\lambda^2}\)</span></td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Subsection: Summary"><a href="#notes-02-05-3" title="Copy heading and permalink for Subsection: Summary" aria-label="Copy heading and permalink for Subsection: Summary">🔗</a></div></section><section class="subsection" id="notes-02-05-4"><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">Covariance</span>
</h3>
<article class="definition definition-like" id="notes-02-05-4-2"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">101</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="notes-02-05-4-2-1-1">
<div class="para">If <span class="process-math">\(X, Y\)</span> are random variables with expected values of <span class="process-math">\(\mu_X, \mu_Y\text{,}\)</span> then the <dfn class="terminology">covariance</dfn> of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> is:</div>
<div class="displaymath process-math" id="notes-02-05-4-2-1-1-6">
\begin{align*}
\Cov(X, Y) \amp = \E\left[ (X - \mu_X)(Y - \mu_Y)\right].
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Definition 101"><a href="#notes-02-05-4-2" title="Copy heading and permalink for Definition 101" aria-label="Copy heading and permalink for Definition 101">🔗</a></div></article><div class="para logical" id="notes-02-05-4-3">
<div class="para">Observe that:</div>
<div class="displaymath process-math" id="notes-02-05-4-3-1">
\begin{align*}
\Cov(X, X) \amp = \E\left[(X - \mu_X)(X - \mu_X)\right] \\
\amp = \E\left[(X - \mu_X)^2\right] \\
\amp = \Var(X),
\end{align*}
</div>
<div class="para">so covariance generalizes the variance formula to two variables. As with variance, there’s an alternative formula more suited to doing computations:</div>
<div class="displaymath process-math" id="notes-02-05-4-3-2">
\begin{align*}
\Var(X) \amp = \E\left[(X - \mu_X)(X - \mu_X)\right] = \E(X^2) - \mu_X^2 \\
\Cov(X, Y) \amp = \E\left[(X - \mu_X)(Y - \mu_Y)\right] = \E(XY) - \mu_X\mu_Y
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="notes-02-05-4-4"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">102</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="notes-02-05-4-4-1-1">Consider the joint distribution table:<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="notes-02-05-4-4-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">103<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution<div class="autopermalink" data-description="Table 103: Joint Distribution"><a href="#notes-02-05-4-4-1-2" title="Copy heading and permalink for Table 103: Joint Distribution" aria-label="Copy heading and permalink for Table 103: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r1 l0 t0 lines"></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b0 r0 l0 t0 lines">0.2</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r0 l0 t0 lines">0.05</td>
<td class="c m b0 r0 l0 t0 lines">0.65</td>
</tr>
</table></div></figure> <div class="para logical" id="notes-02-05-4-4-1-3">
<div class="para">From the table, we can calculate the marginal distributions:</div>
<div class="displaymath process-math" id="notes-02-05-4-4-1-3-1">
\begin{align*}
\Pr(X = 0) \amp = 0.25 \amp \Pr(Y = 0) \amp = 0.3 \\
\Pr(X = 1) \amp = 0.75 \amp \Pr(Y = 1) \amp = 0.7
\end{align*}
</div>
<div class="para">So <span class="process-math">\(\E(X) = \mu_X = 0.75\)</span> and <span class="process-math">\(\E(Y) = \mu_Y = 0.7\text{.}\)</span> Then:</div>
<div class="displaymath process-math" id="notes-02-05-4-4-1-3-4">
\begin{align*}
\E(XY) \amp = (0)(0)(0.2) + (1)(0)(0.1) + (0)(1)(0.05) + (1)(1)(0.65) \\
\amp = 0.65 \\
\Cov(X, Y) \amp = \E(XY) - \mu_X \mu_Y \\
\amp = 0.65 - (0.75)(0.7) \\
\amp = 0.125.
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-4-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 102"><a href="#notes-02-05-4-4" title="Copy heading and permalink for Example 102" aria-label="Copy heading and permalink for Example 102">🔗</a></div></article><div class="para" id="notes-02-05-4-5">Question: the formula <span class="process-math">\(\Var(X) = \E\left[(X - \mu_X)^2\right]\)</span> makes it clear that variance cannot be negative, since squares are nonnegative. What about <span class="process-math">\(\Cov(X, Y)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="notes-02-05-4-6"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">104</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="notes-02-05-4-6-1-1">In the previous example, since <span class="process-math">\(X, Y\)</span> were both indicator random variables, the variances for each were simply equal to the sum of the second row/column. Similarly, <span class="process-math">\(\E(XY)\)</span> was equal to the <span class="process-math">\(X = 1, Y = 1\)</span> entry in the table. Using two indicator random variables, significantly simplifies the covariance calculation, so we can vary the table and recalculate covariance quickly. Consider the following joint distribution:<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="notes-02-05-4-6-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">105<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution<div class="autopermalink" data-description="Table 105: Joint Distribution"><a href="#notes-02-05-4-6-1-2" title="Copy heading and permalink for Table 105: Joint Distribution" aria-label="Copy heading and permalink for Table 105: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r1 l0 t0 lines"></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
<td class="c m b0 r0 l0 t0 lines">0.4</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r0 l0 t0 lines">0.3</td>
<td class="c m b0 r0 l0 t0 lines">0.2</td>
</tr>
</table></div></figure> <div class="para logical" id="notes-02-05-4-6-1-3">
<div class="para">Then:</div>
<div class="displaymath process-math" id="notes-02-05-4-6-1-3-1">
\begin{gather*}
\Cov(X, Y) = 0.2 - (0.6)(0.5) = -0.1 \lt 0.
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-6-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 104"><a href="#notes-02-05-4-6" title="Copy heading and permalink for Example 104" aria-label="Copy heading and permalink for Example 104">🔗</a></div></article><div class="para" id="notes-02-05-4-7">Question: how do we interpret <span class="process-math">\(\Cov(X, Y)\text{?}\)</span> <span class="process-math">\(X - \mu_X\)</span> is positive when <span class="process-math">\(X \gt \mu_X\)</span> and negative when <span class="process-math">\(X \lt \mu_X\text{.}\)</span> <span class="process-math">\(Y - \mu_Y\)</span> is positive when <span class="process-math">\(Y \gt \mu_Y\)</span> and negative when <span class="process-math">\(Y \lt \mu_Y\text{.}\)</span> So the product <span class="process-math">\((X - \mu_X)(Y - \mu_Y)\)</span> is positive when <span class="process-math">\(X, Y\)</span> are both larger or both smaller than their expected values, and negative when one is larger and one is smaller. That is, covariance tries to quantify the tendency of <span class="process-math">\(X, Y\)</span> to get big/small at the same time.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Subsection: Covariance"><a href="#notes-02-05-4" title="Copy heading and permalink for Subsection: Covariance" aria-label="Copy heading and permalink for Subsection: Covariance">🔗</a></div></section><div class="autopermalink" data-description="Section: Thursday, Feb 5"><a href="#notes-02-05" title="Copy heading and permalink for Section: Thursday, Feb 5" aria-label="Copy heading and permalink for Section: Thursday, Feb 5">🔗</a></div></section></div>
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