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2026-01-22 10:53:53 -05:00
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@@ -192,7 +192,10 @@ eBookConfig.enable_chatcodes = false;
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<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-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li></ul>
<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>
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</ul>
</li>
@@ -209,7 +212,28 @@ eBookConfig.enable_chatcodes = false;
<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-Discrete-Random-Variables"><h3 class="heading hide-type">
</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">