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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-01-27"><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, Jan 27</span>
</h2>
<section class="introduction" id="notes-01-27-2"><div class="para" id="notes-01-27-2-1">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-01-27-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><section class="subsection" id="subsec-continuous-2"><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">Continuous Distributions</span>
</h3>
<article class="example example-like" id="subsec-continuous-2-2"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">55</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="subsec-continuous-2-2-1-1">
<div class="para">Let <span class="process-math">\(X\in [0, 1]\)</span> with <span class="process-math">\(f(x) = kx^{3/2}\text{.}\)</span> Find <span class="process-math">\(k\text{.}\)</span>
</div>
<div class="displaymath process-math" id="subsec-continuous-2-2-1-1-4">
\begin{align*}
1 = \int_0^1 f(x)\ dx \amp = \int_0^1 kx^{3/2} dx = \frac{2kx^{5/2}}{5}\bigg|_0^1 = \frac{2k}{5} - 0 \\
\Rightarrow \quad 1 \amp = \frac{2k}{5} \quad \Rightarrow \quad k = \frac{5}{2}
\end{align*}
</div>
<div class="para">Then, we can calculate probabilities, e.g.:</div>
<div class="displaymath process-math" id="subsec-continuous-2-2-1-1-5">
\begin{gather*}
\Pr\left(X \lt \frac{1}{2}\right) = \int_0^{1/2} \frac{5}{2}x^{3/2}\ dx = \frac{5}{2}\cdot \frac{2x^{5/2}}{5}\bigg|_0^{1/2} = \left(\frac{1}{2}\right)^{5/2} \approx 0.177.
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-continuous-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 55"><a href="#subsec-continuous-2-2" title="Copy heading and permalink for Example 55" aria-label="Copy heading and permalink for Example 55">🔗</a></div></article><div class="para" id="subsec-continuous-2-3">Note: a pdf outputs probability densities, not probabilities. To get probabilities, we must integrate.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-continuous-2-3" 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-cdf"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">56</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="def-cdf-1-1">
<div class="para">Let <span class="process-math">\(X\)</span> be a continuous random variable with values in <span class="process-math">\([a, b]\text{.}\)</span> The <dfn class="terminology">cumulative distribution function (cdf)</dfn> is:</div>
<div class="displaymath process-math" id="def-cdf-1-1-4">
\begin{gather*}
F(x) = \Pr(X \leq x).
\end{gather*}
</div>
<div class="para">To calculate it:</div>
<div class="displaymath process-math" id="def-cdf-1-1-5">
\begin{gather*}
F(x) = \int_a^x f(t)\ dt.
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#def-cdf-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 56"><a href="#def-cdf" title="Copy heading and permalink for Definition 56" aria-label="Copy heading and permalink for Definition 56">🔗</a></div></article><article class="example example-like" id="subsec-continuous-2-5"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">57</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="subsec-continuous-2-5-1-1">
<div class="para">Let <span class="process-math">\(X \in [0, 1], f(x) = \frac{5}{2}x^{3/2}\text{.}\)</span> Then:</div>
<div class="displaymath process-math" id="subsec-continuous-2-5-1-1-2">
\begin{align*}
F(x) = \int_0^x f(t)\ dt \amp = \int_0^x \frac{5}{2}t^{3/2}\ dt = \frac{5}{2}\cdot \frac{2t^{5/2}}{5}\bigg|_0^x \\
F(x) \amp = x^{5/2}
\end{align*}
</div>
<div class="para">Then, we can calculate probabilities, e.g.:</div>
<div class="displaymath process-math" id="subsec-continuous-2-5-1-1-3">
\begin{align*}
\Pr\left(X \lt \frac{1}{2}\right) \amp = F\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^{5/2} \approx 0.177 \\
\Pr(0.2 \leq X \leq 0.6) \amp = F(0.6) - F(0.2) = (0.6)^{5/2} - (0.2)^{5/2} \approx 0.261
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-continuous-2-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 57"><a href="#subsec-continuous-2-5" title="Copy heading and permalink for Example 57" aria-label="Copy heading and permalink for Example 57">🔗</a></div></article><div class="para logical" id="subsec-continuous-2-6">
<div class="para">Given <span class="process-math">\(f(x)\text{,}\)</span> we can find <span class="process-math">\(F(x)\)</span> by calculating:</div>
<div class="displaymath process-math" id="subsec-continuous-2-6-3">
\begin{gather*}
F(x) = \int_a^x f(t)\ dt.
\end{gather*}
</div>
<div class="para">Given <span class="process-math">\(F(x)\text{,}\)</span> we can find <span class="process-math">\(f(x)\)</span> by calculating:</div>
<div class="displaymath process-math" id="subsec-continuous-2-6-6">
\begin{gather*}
f(x) = F'(x).
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-continuous-2-6" 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-continuous-2-7"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">58</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="subsec-continuous-2-7-1-1">Suppose a machine needs repairs on average twice per month. Let <span class="process-math">\(T\)</span> be the time until repair. This is a Poisson process (<span class="process-math">\(\lambda = 2\)</span> per month).<div class="autopermalink" data-description="Paragraph"><a href="#subsec-continuous-2-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 58"><a href="#subsec-continuous-2-7" title="Copy heading and permalink for Example 58" aria-label="Copy heading and permalink for Example 58">🔗</a></div></article><article class="definition definition-like" id="subsec-continuous-2-8"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">59</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="subsec-continuous-2-8-1-1">
<div class="para">
<span class="process-math">\(T\)</span> is said to have the <dfn class="terminology">exponential distribution</dfn> with parameter <span class="process-math">\(\lambda\text{.}\)</span> We’ll write <span class="process-math">\(T \sim \Exp(\lambda)\text{.}\)</span> The <dfn class="terminology">exponential density function</dfn> is:</div>
<div class="displaymath process-math" id="subsec-continuous-2-8-1-1-6">
\begin{gather*}
f(t) = \lambda e^{-\lambda t}, \quad t \geq 0.
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-continuous-2-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Definition 59"><a href="#subsec-continuous-2-8" title="Copy heading and permalink for Definition 59" aria-label="Copy heading and permalink for Definition 59">🔗</a></div></article><article class="example example-like" id="subsec-continuous-2-9"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">60</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="subsec-continuous-2-9-1-1">
<div class="para">Let <span class="process-math">\(\lambda = 2, f(t) = 2e^{-2t}, t\geq 0\text{.}\)</span> Then:</div>
<div class="displaymath process-math" id="subsec-continuous-2-9-1-1-2">
\begin{align*}
F(t) \amp = \int_0^t f(x)\ dx = \int_0^t 2e^{-2x}\ dx = -e^{-2x}\bigg|_0^x \\
\amp = -e^{-2t} - (-e^0) = 1 - e^{-2t}.
\end{align*}
</div>
<div class="para">So, e.g.:</div>
<div class="displaymath process-math" id="subsec-continuous-2-9-1-1-3">
\begin{align*}
\Pr\left(T \leq \frac{3}{4}\right) \amp = F\left(\frac{3}{4}\right) = 1 - e^{-3/2} \approx 0.777 \\
\Pr(T \gt 1) \amp = 1 - \Pr(T \leq 1) = 1 - F(1) \\
\amp = 1 - (1 - e^{-2}) = e^{-2} \approx 0.135.
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-continuous-2-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 60"><a href="#subsec-continuous-2-9" title="Copy heading and permalink for Example 60" aria-label="Copy heading and permalink for Example 60">🔗</a></div></article><article class="remark remark-like" id="subsec-continuous-2-10"><h4 class="heading">
<span class="type">Remark</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">61</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="subsec-continuous-2-10-1">The distributions Bin, Geom, Poiss, and Exp are conceptually linked.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-continuous-2-10-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<figure class="table table-like" id="subsec-continuous-2-10-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">62<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Relationship of common distributions<div class="autopermalink" data-description="Table 62: Relationship of common distributions"><a href="#subsec-continuous-2-10-2" title="Copy heading and permalink for Table 62: Relationship of common distributions" aria-label="Copy heading and permalink for Table 62: Relationship of common distributions">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="l m b0 r0 l0 t0 lines"></td>
<td class="l m b1 r0 l0 t0 lines">Counting Events</td>
<td class="l m b1 r0 l0 t0 lines">Time Until</td>
</tr>
<tr>
<td class="l m b0 r1 l0 t0 lines">Discrete Time</td>
<td class="l m b0 r0 l0 t0 lines">Bin</td>
<td class="l m b0 r0 l0 t0 lines">Geom</td>
</tr>
<tr>
<td class="l m b0 r1 l0 t0 lines">Continuous Time</td>
<td class="l m b0 r0 l0 t0 lines">Poiss</td>
<td class="l m b0 r0 l0 t0 lines">Exp</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Remark 61"><a href="#subsec-continuous-2-10" title="Copy heading and permalink for Remark 61" aria-label="Copy heading and permalink for Remark 61">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Continuous Distributions"><a href="#subsec-continuous-2" title="Copy heading and permalink for Subsection: Continuous Distributions" aria-label="Copy heading and permalink for Subsection: Continuous Distributions">🔗</a></div></section><section class="subsection" id="subsec-Joint-Distributions"><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">Joint Distributions</span>
</h3>
<article class="definition definition-like" id="subsec-Joint-Distributions-2"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">63</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="subsec-Joint-Distributions-2-1-1">Let <span class="process-math">\(X\)</span> take values <span class="process-math">\(x_1, x_2, \dotsc, x_n\)</span> and <span class="process-math">\(Y\)</span> take values <span class="process-math">\(y_1, y_2, \dotsc, y_m\text{.}\)</span> The <dfn class="terminology">joint distribution</dfn> of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> is the collection of all values <span class="process-math">\(\Pr(X = x_i, Y = y_j)\)</span> for every <span class="process-math">\(i, j\)</span> combination. The separarte distributions for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> are called <dfn class="terminology">marginal distributions</dfn>.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Joint-Distributions-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 63"><a href="#subsec-Joint-Distributions-2" title="Copy heading and permalink for Definition 63" aria-label="Copy heading and permalink for Definition 63">🔗</a></div></article><article class="example example-like" id="subsec-Joint-Distributions-3"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">64</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="subsec-Joint-Distributions-3-1-1">Suppose <span class="process-math">\(X \in \{1, 2, 3\}, Y \in \{0, 1\}\text{,}\)</span> with joint distribution below.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Joint-Distributions-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="subsec-Joint-Distributions-3-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">65<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Example Joint Distribution<div class="autopermalink" data-description="Table 65: Example Joint Distribution"><a href="#subsec-Joint-Distributions-3-1-2" title="Copy heading and permalink for Table 65: Example Joint Distribution" aria-label="Copy heading and permalink for Table 65: Example 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 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 2\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 3\)</span></td>
</tr>
<tr>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">0.1</td>
<td class="c m b1 r1 l0 t0 lines">0.15</td>
<td class="c m b1 r0 l0 t0 lines">0.05</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 r1 l0 t0 lines">0.2</td>
<td class="c m b0 r1 l0 t0 lines">0.2</td>
<td class="c m b0 r0 l0 t0 lines">0.3</td>
</tr>
</table></div></figure> <div class="para logical" id="subsec-Joint-Distributions-3-1-3">
<div class="para">We find the marginal distribution for <span class="process-math">\(X\)</span> by summing along the columns:</div>
<div class="displaymath process-math" id="subsec-Joint-Distributions-3-1-3-2">
\begin{align*}
\Pr(X = 1) \amp = 0.3 \\
\Pr(X = 2) \amp = 0.35 \\
\Pr(X = 3) \amp = 0.35
\end{align*}
</div>
<div class="para">We find the marginal distribution for <span class="process-math">\(Y\)</span> by summing along the rows:</div>
<div class="displaymath process-math" id="subsec-Joint-Distributions-3-1-3-4">
\begin{align*}
\Pr(Y = 0) \amp = 0.3 \\
\Pr(Y = 1) \amp = 0.7
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Joint-Distributions-3-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 64"><a href="#subsec-Joint-Distributions-3" title="Copy heading and permalink for Example 64" aria-label="Copy heading and permalink for Example 64">🔗</a></div></article><article class="definition definition-like" id="subsec-Joint-Distributions-4"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">66</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="subsec-Joint-Distributions-4-1-1">
<div class="para">Random variables <span class="process-math">\(X, Y\)</span> are <dfn class="terminology">independent</dfn> if:</div>
<div class="displaymath process-math" id="subsec-Joint-Distributions-4-1-1-3">
\begin{gather*}
\Pr(X = x_i, Y = y_j) = \Pr(X = x_i)\Pr(Y = y_j)
\end{gather*}
</div>
<div class="para">for every <span class="process-math">\(i, j\)</span> combination.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Joint-Distributions-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="Definition 66"><a href="#subsec-Joint-Distributions-4" title="Copy heading and permalink for Definition 66" aria-label="Copy heading and permalink for Definition 66">🔗</a></div></article><article class="example example-like" id="subsec-Joint-Distributions-5"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">67</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="subsec-Joint-Distributions-5-1-1">
<div class="para">In the previous example:</div>
<div class="displaymath process-math" id="subsec-Joint-Distributions-5-1-1-1">
\begin{gather*}
\Pr(X = 2, Y = 1) = 0.2 \neq (0.35)(0.7) = \Pr(X = 2)\Pr(Y = 1),
\end{gather*}
</div>
<div class="para">so <span class="process-math">\(X, Y\)</span> are not independent.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Joint-Distributions-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 67"><a href="#subsec-Joint-Distributions-5" title="Copy heading and permalink for Example 67" aria-label="Copy heading and permalink for Example 67">🔗</a></div></article><article class="example example-like" id="subsec-Joint-Distributions-6"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">68</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="subsec-Joint-Distributions-6-1-1">Let <span class="process-math">\(X, Y\)</span> indicate heads on the first and second flip, respectively, of a fair coin. Then:<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Joint-Distributions-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="subsec-Joint-Distributions-6-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">69<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 for Indicator Random Variables<div class="autopermalink" data-description="Table 69: Joint Distribution for Indicator Random Variables"><a href="#subsec-Joint-Distributions-6-1-2" title="Copy heading and permalink for Table 69: Joint Distribution for Indicator Random Variables" aria-label="Copy heading and permalink for Table 69: Joint Distribution for Indicator Random Variables">🔗</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 r1 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 b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">1/4</td>
<td class="c m b1 r0 l0 t0 lines">1/4</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 r1 l0 t0 lines">1/4</td>
<td class="c m b0 r0 l0 t0 lines">1/4</td>
</tr>
</table></div></figure> <div class="para logical" id="subsec-Joint-Distributions-6-1-3">
<div class="para">Then the marginal distributions are:</div>
<div class="displaymath process-math" id="subsec-Joint-Distributions-6-1-3-1">
\begin{align*}
\Pr(X = 0) \amp = \frac{1}{2} \amp \Pr(Y = 0) \amp = \frac{1}{2} \\
\Pr(X = 1) \amp = \frac{1}{2} \amp \Pr(Y = 1) \amp = \frac{1}{2}
\end{align*}
</div>
<div class="para">Then, for any <span class="process-math">\(a, b\text{:}\)</span>
</div>
<div class="displaymath process-math" id="subsec-Joint-Distributions-6-1-3-3">
\begin{gather*}
\Pr(X = a, Y = b) = \frac{1}{4} = \frac{1}{2}\cdot \frac{1}{2} = \Pr(X = a)\Pr(Y = b).
\end{gather*}
</div>
<div class="para">So <span class="process-math">\(X, Y\)</span> are independent.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Joint-Distributions-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 68"><a href="#subsec-Joint-Distributions-6" title="Copy heading and permalink for Example 68" aria-label="Copy heading and permalink for Example 68">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Joint Distributions"><a href="#subsec-Joint-Distributions" title="Copy heading and permalink for Subsection: Joint Distributions" aria-label="Copy heading and permalink for Subsection: Joint Distributions">🔗</a></div></section><div class="autopermalink" data-description="Section: Thursday, Jan 27"><a href="#notes-01-27" title="Copy heading and permalink for Section: Thursday, Jan 27" aria-label="Copy heading and permalink for Section: Thursday, Jan 27">🔗</a></div></section></div>
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