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@@ -192,7 +192,10 @@ eBookConfig.enable_chatcodes = false;
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<ul class="structural toc-item-list"><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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<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>
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<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>
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@@ -192,7 +192,10 @@ eBookConfig.enable_chatcodes = false;
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<ul class="structural toc-item-list"><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 class="structural toc-item-list">
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<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>
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<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>
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@@ -192,7 +192,10 @@ eBookConfig.enable_chatcodes = false;
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<ul class="structural toc-item-list"><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 class="structural toc-item-list">
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<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>
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<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>
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<html>
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<!--* Generated from PreTeXt source *-->
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<!--* on 2026-01-22T10:14:43-05:00 *-->
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<!--* on 2026-01-22T10:53:20-05:00 *-->
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<!--* A recent stable commit (2022-07-01): *-->
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@@ -313,7 +313,7 @@ var ptx_lunr_docs = [
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"type": "Section",
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"number": "",
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"title": "Tuesday, Jan 20",
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"body": " Tuesday, Jan 20 This is an outline of the topics we covered in class. These notes are not 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. Sec 2.1: Random Variables A random variable is a function . The idea is that is a variable representing a real number value which depends on the outcome of an experiment. An experiment consists of planting 50 seeds in a garden, then growing them for 3 months. Let be the height of plant . Let be the number of seeds that didn't sprout. Let A random variable has its own probability distribution. Roll a fair D6 twice. Let be the sum of the rolls. Then has 36 elements. Since the die is fair, the distribution on is uniform, i.e., for any . takes on the values , with probabilities: Distribution for 2 1\/36 3 2\/36 4 3\/36 7 6\/36 8 5\/36 12 1\/36 Note that the distribution on is uniform, but the distribution on is not. Let be a sample space and an event. Let is called an indicator random variable , and we say \" indicates \". The distribution on is: Suppose we flip a coin times. Let indicate heads on flip . Let be the total number of heads in all flips. Then: If is the probability of the coin coming up heads on a flip, then has the binomial distribution with parameters . We'll use the notation and: Suppose the coin has and we flip it times. Find . The relevant flip sequences are: Each individual sequence has a probability of . So the total probability is: That is, (number of flip sequences)(probability of each sequence). "
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"body": " Tuesday, Jan 20 This is an outline of the topics we covered in class. These notes are not 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. HW 1 Q5 Write for the event that there's a fire and for the event that there's visible smoke. Then the information we're given can be interpreted as: In this case, we can use the simpler version of Bayes' Theorem: Unlike our usual diagnostic testing examples, we do have access to the denominator probability here. Sec 2.1: Random Variables A random variable is a function . The idea is that is a variable representing a real number value which depends on the outcome of an experiment. An experiment consists of planting 50 seeds in a garden, then growing them for 3 months. Let be the height of plant . Let be the number of seeds that didn't sprout. Let A random variable has its own probability distribution. Roll a fair D6 twice. Let be the sum of the rolls. Then has 36 elements. Since the die is fair, the distribution on is uniform, i.e., for any . takes on the values , with probabilities: Distribution for 2 1\/36 3 2\/36 4 3\/36 7 6\/36 8 5\/36 12 1\/36 Note that the distribution on is uniform, but the distribution on is not. Let be a sample space and an event. Let is called an indicator random variable , and we say \" indicates \". The distribution on is: Suppose we flip a coin times. Let indicate heads on flip . Let be the total number of heads in all flips. Then: If is the probability of the coin coming up heads on a flip, then has the binomial distribution with parameters . We'll use the notation and: Suppose the coin has and we flip it times. Find . The relevant flip sequences are: Each individual sequence has a probability of . So the total probability is: That is, (number of flip sequences)(probability of each sequence). "
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},
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{
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"id": "def-RV",
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@@ -192,7 +192,10 @@ eBookConfig.enable_chatcodes = false;
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</li>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<ul class="structural toc-item-list"><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 class="structural toc-item-list">
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<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>
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<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>
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</ul>
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</ul>
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@@ -192,7 +192,10 @@ eBookConfig.enable_chatcodes = false;
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</li>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<ul class="structural toc-item-list"><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 class="structural toc-item-list">
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<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>
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<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>
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@@ -192,7 +192,10 @@ eBookConfig.enable_chatcodes = false;
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<li class="toc-item toc-section contains-active">
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<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>
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<ul class="structural toc-item-list active">
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<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>
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<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>
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@@ -209,7 +212,28 @@ eBookConfig.enable_chatcodes = false;
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<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>
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</h2>
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<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>
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</div></section><section class="subsection" id="subsec-Discrete-Random-Variables"><h3 class="heading hide-type">
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</div></section><section class="subsection" id="subsec-hw1-review"><h3 class="heading hide-type">
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<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>
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</h3>
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<div class="para logical" id="subsec-hw1-review-2">
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<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>
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<div class="displaymath process-math" id="subsec-hw1-review-2-3">
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\begin{align*}
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\Pr(F) \amp = 0.01 \\
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\Pr(S) \amp = 0.1 \\
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\Pr(S\mid F) \amp = 0.9
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\end{align*}
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</div>
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<div class="para">In this case, we can use the simpler version of Bayes’ Theorem:</div>
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<div class="displaymath process-math" id="subsec-hw1-review-2-4">
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\begin{gather*}
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\Pr(F \mid S) = \frac{\Pr(S \mid F)\Pr(F)}{\Pr(S)} = \dotsb
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\end{gather*}
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</div>
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<div class="para">Unlike our usual diagnostic testing examples, we do have access to the denominator probability here.</div>
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<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>
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</div>
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<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">
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<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>
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</h3>
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<article class="definition definition-like" id="def-RV"><h4 class="heading">
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<ul class="structural toc-item-list"><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 class="structural toc-item-list">
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<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>
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<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>
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<ul class="structural toc-item-list"><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 class="structural toc-item-list">
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<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>
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<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>
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<ul class="structural toc-item-list"><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 class="structural toc-item-list">
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<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>
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<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>
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@@ -192,7 +192,10 @@ eBookConfig.enable_chatcodes = false;
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<ul class="structural toc-item-list"><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 class="structural toc-item-list">
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<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>
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<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>
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@@ -192,7 +192,10 @@ eBookConfig.enable_chatcodes = false;
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<ul class="structural toc-item-list"><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 class="structural toc-item-list">
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<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>
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<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>
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</ul>
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@@ -192,7 +192,10 @@ eBookConfig.enable_chatcodes = false;
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</li>
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<ul class="structural toc-item-list"><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 class="structural toc-item-list">
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<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>
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||||
<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>
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@@ -192,7 +192,10 @@ eBookConfig.enable_chatcodes = false;
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<ul class="structural toc-item-list"><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 class="structural toc-item-list">
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<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>
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<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>
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</ul>
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@@ -192,7 +192,10 @@ eBookConfig.enable_chatcodes = false;
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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||||
<ul class="structural toc-item-list"><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 class="structural toc-item-list">
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<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>
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<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>
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</ul>
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</li>
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</ul>
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</li>
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Block a user