Latest build deployed.

course/.mapping.json

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+{"source/main.ptx": ["frontmatter", "course-documents", "ch-notes", "quizzes", "recitations"], "source/syllabus/syllabus.ptx": ["syllabus"], "source/notes/1-13.ptx": ["notes-01-13"], "source/notes/1-15.ptx": ["notes-01-15"], "source/notes/1-20.ptx": ["notes-01-20"], "source/notes/1-22.ptx": ["notes-01-22"], "source/notes/1-27.ptx": ["notes-01-27"], "source/quizzes/quiz-01.ptx": ["quiz-01"], "source/recitations/calculus-review.ptx": ["recitation-calculus-review"]}

course/ch-notes.html

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 <li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-continuous-distributions" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
 </ul>
 </li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
+</ul>
+</li>
 </ul>
 </li>
 <li class="toc-item toc-chapter">
@@ -223,6 +230,7 @@ eBookConfig.enable_chatcodes =  false;
 <li><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></li>
 <li><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></li>
 <li><a href="notes-01-22.html" class="internal"><span class="title">Thursday, Jan 22</span></a></li>
+<li><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></li>
 </ul></nav><div class="autopermalink" data-description="Chapter: Class Notes"><a href="#ch-notes" title="Copy heading and permalink for Chapter: Class Notes" aria-label="Copy heading and permalink for Chapter: Class Notes">🔗</a></div></section></div>
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 <a class="previous-button button" href="syllabus.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cb;</span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5ce;</span><span class="name">Top</span></a><a class="next-button button" href="notes-01-13.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cc;</span></a>

course/course-documents.html

@@ -204,6 +204,13 @@ eBookConfig.enable_chatcodes =  false;
 <li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-continuous-distributions" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
 </ul>
 </li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
+</ul>
+</li>
 </ul>
 </li>
 <li class="toc-item toc-chapter">

course/frontmatter.html

@@ -204,6 +204,13 @@ eBookConfig.enable_chatcodes =  false;
 <li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-continuous-distributions" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
 </ul>
 </li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
+</ul>
+</li>
 </ul>
 </li>
 <li class="toc-item toc-chapter">

course/index.html

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course/lunr-pretext-search-index.js

@@ -495,6 +495,123 @@ var ptx_lunr_docs = [
   "title": "",
   "body": "  Pick uniformly. Because of the uniform assumption, must be a constant function for some . To find : More generally, if is chosen uniformly, then the pdf would be:    "
 },
+{
+  "id": "notes-01-27",
+  "level": "1",
+  "url": "notes-01-27.html",
+  "type": "Section",
+  "number": "",
+  "title": "Thursday, Jan 27",
+  "body": " Thursday, Jan 27   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.    Continuous Distributions    Let with . Find . Then, we can calculate probabilities, e.g.:     Note: a pdf outputs probability densities, not probabilities. To get probabilities, we must integrate.    Let be a continuous random variable with values in . The cumulative distribution function (cdf) is: To calculate it:       Let . Then: Then, we can calculate probabilities, e.g.:     Given , we can find by calculating: Given , we can find by calculating:     Suppose a machine needs repairs on average twice per month. Let be the time until repair. This is a Poisson process ( per month).       is said to have the exponential distribution with parameter . We'll write . The exponential density function is:       Let . Then: So, e.g.:      The distributions Bin, Geom, Poiss, and Exp are conceptually linked.   Relationship of common distributions     Counting Events  Time Until    Discrete Time  Bin  Geom    Continuous Time  Poiss  Exp        Joint Distributions    Let take values and take values . The joint distribution of and is the collection of all values for every combination. The separarte distributions for and are called marginal distributions .      Suppose , with joint distribution below.   Example Joint Distribution           0.1  0.15  0.05     0.2  0.2  0.3     We find the marginal distribution for by summing along the columns: We find the marginal distribution for by summing along the rows:       Random variables are independent if: for every combination.      In the previous example: so are not independent.      Let indicate heads on the first and second flip, respectively, of a fair coin. Then:   Joint Distribution for Indicator Random Variables          1\/4  1\/4     1\/4  1\/4     Then the marginal distributions are: Then, for any : So are independent.     "
+},
+{
+  "id": "subsec-continuous-2-2",
+  "level": "2",
+  "url": "notes-01-27.html#subsec-continuous-2-2",
+  "type": "Example",
+  "number": "55",
+  "title": "",
+  "body": "  Let with . Find . Then, we can calculate probabilities, e.g.:    "
+},
+{
+  "id": "def-cdf",
+  "level": "2",
+  "url": "notes-01-27.html#def-cdf",
+  "type": "Definition",
+  "number": "56",
+  "title": "",
+  "body": "  Let be a continuous random variable with values in . The cumulative distribution function (cdf) is: To calculate it:    "
+},
+{
+  "id": "subsec-continuous-2-5",
+  "level": "2",
+  "url": "notes-01-27.html#subsec-continuous-2-5",
+  "type": "Example",
+  "number": "57",
+  "title": "",
+  "body": "  Let . Then: Then, we can calculate probabilities, e.g.:    "
+},
+{
+  "id": "subsec-continuous-2-7",
+  "level": "2",
+  "url": "notes-01-27.html#subsec-continuous-2-7",
+  "type": "Example",
+  "number": "58",
+  "title": "",
+  "body": "  Suppose a machine needs repairs on average twice per month. Let be the time until repair. This is a Poisson process ( per month).   "
+},
+{
+  "id": "subsec-continuous-2-8",
+  "level": "2",
+  "url": "notes-01-27.html#subsec-continuous-2-8",
+  "type": "Definition",
+  "number": "59",
+  "title": "",
+  "body": "   is said to have the exponential distribution with parameter . We'll write . The exponential density function is:    "
+},
+{
+  "id": "subsec-continuous-2-9",
+  "level": "2",
+  "url": "notes-01-27.html#subsec-continuous-2-9",
+  "type": "Example",
+  "number": "60",
+  "title": "",
+  "body": "  Let . Then: So, e.g.:    "
+},
+{
+  "id": "subsec-continuous-2-10",
+  "level": "2",
+  "url": "notes-01-27.html#subsec-continuous-2-10",
+  "type": "Remark",
+  "number": "61",
+  "title": "",
+  "body": " The distributions Bin, Geom, Poiss, and Exp are conceptually linked.   Relationship of common distributions     Counting Events  Time Until    Discrete Time  Bin  Geom    Continuous Time  Poiss  Exp     "
+},
+{
+  "id": "subsec-Joint-Distributions-2",
+  "level": "2",
+  "url": "notes-01-27.html#subsec-Joint-Distributions-2",
+  "type": "Definition",
+  "number": "63",
+  "title": "",
+  "body": "  Let take values and take values . The joint distribution of and is the collection of all values for every combination. The separarte distributions for and are called marginal distributions .   "
+},
+{
+  "id": "subsec-Joint-Distributions-3",
+  "level": "2",
+  "url": "notes-01-27.html#subsec-Joint-Distributions-3",
+  "type": "Example",
+  "number": "64",
+  "title": "",
+  "body": "  Suppose , with joint distribution below.   Example Joint Distribution           0.1  0.15  0.05     0.2  0.2  0.3     We find the marginal distribution for by summing along the columns: We find the marginal distribution for by summing along the rows:    "
+},
+{
+  "id": "subsec-Joint-Distributions-4",
+  "level": "2",
+  "url": "notes-01-27.html#subsec-Joint-Distributions-4",
+  "type": "Definition",
+  "number": "66",
+  "title": "",
+  "body": "  Random variables are independent if: for every combination.   "
+},
+{
+  "id": "subsec-Joint-Distributions-5",
+  "level": "2",
+  "url": "notes-01-27.html#subsec-Joint-Distributions-5",
+  "type": "Example",
+  "number": "67",
+  "title": "",
+  "body": "  In the previous example: so are not independent.   "
+},
+{
+  "id": "subsec-Joint-Distributions-6",
+  "level": "2",
+  "url": "notes-01-27.html#subsec-Joint-Distributions-6",
+  "type": "Example",
+  "number": "68",
+  "title": "",
+  "body": "  Let indicate heads on the first and second flip, respectively, of a fair coin. Then:   Joint Distribution for Indicator Random Variables          1\/4  1\/4     1\/4  1\/4     Then the marginal distributions are: Then, for any : So are independent.   "
+},
 {
   "id": "quiz-01",
   "level": "1",
@@ -581,7 +698,7 @@ var ptx_lunr_docs = [
   "level": "2",
   "url": "recitation-calculus-review.html#thm-FTC",
   "type": "Theorem",
-  "number": "55",
+  "number": "70",
   "title": "Fundamental Theorem of Calculus.",
   "body": " Fundamental Theorem of Calculus   If is an antiderivative of i.e., if then:    "
 },

course/notes-01-13.html

@@ -204,6 +204,13 @@ eBookConfig.enable_chatcodes =  false;
 <li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-continuous-distributions" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
 </ul>
 </li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
+</ul>
+</li>
 </ul>
 </li>
 <li class="toc-item toc-chapter">

course/notes-01-15.html

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 </ul>
 </li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
+</ul>
+</li>
 </ul>
 </li>
 <li class="toc-item toc-chapter">

course/notes-01-20.html

@@ -204,6 +204,13 @@ eBookConfig.enable_chatcodes =  false;
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 </ul>
 </li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
+</ul>
+</li>
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 </li>
 <li class="toc-item toc-chapter">

course/notes-01-22.html

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 </ul>
 </li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
+</ul>
+</li>
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 </li>
 <li class="toc-item toc-chapter">
@@ -489,7 +496,7 @@ f(x) = \frac{1}{b - a}, \quad a \leq x \leq b
 </div>
 <div class="autopermalink" data-description="Example 54"><a href="#subsec-continuous-distributions-6" title="Copy heading and permalink for Example 54" aria-label="Copy heading and permalink for Example 54">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Continuous Distributions"><a href="#subsec-continuous-distributions" title="Copy heading and permalink for Subsection: Continuous Distributions" aria-label="Copy heading and permalink for Subsection: Continuous Distributions">🔗</a></div></section><div class="autopermalink" data-description="Section: Thursday, Jan 22"><a href="#notes-01-22" title="Copy heading and permalink for Section: Thursday, Jan 22" aria-label="Copy heading and permalink for Section: Thursday, Jan 22">🔗</a></div></section></div>
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course/notes-01-27.html

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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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+<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
+</ul>
+</li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Independent-Events" class="internal"><span class="title">Sec 1.4: Independent Events</span></a></div></li>
+</ul>
+</li>
+<li class="toc-item toc-section">
+<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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+<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>
+</li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-22.html" class="internal"><span class="title">Thursday, Jan 22</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-discrete-distributions" class="internal"><span class="title">Discrete Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-continuous-distributions" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+</ul>
+</li>
+<li class="toc-item toc-section contains-active">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
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+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
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+</li>
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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>
+<div id="ptx-content-footer" class="ptx-content-footer">
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course/quiz-01-printable.html

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+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
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course/quiz-01.html

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+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
+</ul>
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course/quizzes.html

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+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
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+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
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course/recitation-calculus-review-printable.html

@@ -204,6 +204,13 @@ eBookConfig.enable_chatcodes =  false;
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 </ul>
 </li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
+</ul>
+</li>
 </ul>
 </li>
 <li class="toc-item toc-chapter">
@@ -281,7 +288,7 @@ eBookConfig.enable_chatcodes =  false;
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 <div class="autopermalink" data-description="Task 4.b"><a href="#recitation-calculus-review-3-4-3" title="Copy heading and permalink for Task 4.b" aria-label="Copy heading and permalink for Task 4.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#recitation-calculus-review-3-4" title="Copy heading and permalink for Worksheet Exercise 4" aria-label="Copy heading and permalink for Worksheet Exercise 4">🔗</a></div></article></section><section class="onepage" id="recitation-calculus-review-4"><article class="theorem theorem-like" id="thm-FTC"><h3 class="heading">
-<span class="type">Theorem</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><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
+<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">70</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
 </h3>
 <div class="para logical" id="thm-FTC-2-1">
 <div class="para">If <span class="process-math">\(F(x)\)</span> is an antiderivative of <span class="process-math">\(f(x)\)</span>—i.e., if <span class="process-math">\(F'(x) = f(x)\)</span>—then:</div>
@@ -292,7 +299,7 @@ eBookConfig.enable_chatcodes =  false;
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-<div class="autopermalink" data-description="Theorem 55: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 55: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 55: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
+<div class="autopermalink" data-description="Theorem 70: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 70: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 70: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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 <div class="introduction" id="recitation-calculus-review-4-3-1"><div class="para" id="recitation-calculus-review-4-3-1-1">Practice with the following integrals:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>

course/recitation-calculus-review.html

@@ -204,6 +204,13 @@ eBookConfig.enable_chatcodes =  false;
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 </ul>
 </li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
+</ul>
+</li>
 </ul>
 </li>
 <li class="toc-item toc-chapter">
@@ -266,7 +273,7 @@ eBookConfig.enable_chatcodes =  false;
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 <div class="autopermalink" data-description="Task 4.b"><a href="#recitation-calculus-review-3-4-3" title="Copy heading and permalink for Task 4.b" aria-label="Copy heading and permalink for Task 4.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#recitation-calculus-review-3-4" title="Copy heading and permalink for Worksheet Exercise 4" aria-label="Copy heading and permalink for Worksheet Exercise 4">🔗</a></div></article></section><section class="onepage" id="recitation-calculus-review-4"><article class="theorem theorem-like" id="thm-FTC"><h3 class="heading">
-<span class="type">Theorem</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><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
+<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">70</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
 </h3>
 <div class="para logical" id="thm-FTC-2-1">
 <div class="para">If <span class="process-math">\(F(x)\)</span> is an antiderivative of <span class="process-math">\(f(x)\)</span>—i.e., if <span class="process-math">\(F'(x) = f(x)\)</span>—then:</div>
@@ -277,7 +284,7 @@ eBookConfig.enable_chatcodes =  false;
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-<div class="autopermalink" data-description="Theorem 55: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 55: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 55: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
+<div class="autopermalink" data-description="Theorem 70: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 70: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 70: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
 </div>
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 <div class="introduction" id="recitation-calculus-review-4-3-1"><div class="para" id="recitation-calculus-review-4-3-1-1">Practice with the following integrals:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>

course/recitations.html

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 </ul>
 </li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
+</ul>
+</li>
 </ul>
 </li>
 <li class="toc-item toc-chapter">

course/root-1-2.html

@@ -204,6 +204,13 @@ eBookConfig.enable_chatcodes =  false;
 <li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-continuous-distributions" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
 </ul>
 </li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
+</ul>
+</li>
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 </li>
 <li class="toc-item toc-chapter">

course/syllabus.html

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 </ul>
 </li>
+<li class="toc-item toc-section">
+<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
+<ul class="structural toc-item-list">
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
+<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
+</ul>
+</li>
 </ul>
 </li>
 <li class="toc-item toc-chapter">