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{"source/main.ptx": ["frontmatter", "course-documents", "ch-notes", "quizzes", "recitations", "review"], "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/notes/1-29.ptx": ["notes-01-29"], "source/notes/2-3.ptx": ["notes-02-03"], "source/quizzes/quiz-01.ptx": ["quiz-01"], "source/quizzes/quiz-02.ptx": ["quiz-02"], "source/recitations/calculus-review.ptx": ["recitation-calculus-review"], "source/review/Exam-1-Review.ptx": ["Exam-1-Review"]}
|
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{"source/main.ptx": ["frontmatter", "course-documents", "ch-notes", "quizzes", "recitations", "review"], "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/notes/1-29.ptx": ["notes-01-29"], "source/notes/2-3.ptx": ["notes-02-03"], "source/notes/2-5.ptx": ["notes-02-05"], "source/notes/2-10.ptx": ["notes-02-10"], "source/quizzes/quiz-01.ptx": ["quiz-01"], "source/quizzes/quiz-02.ptx": ["quiz-02"], "source/recitations/calculus-review.ptx": ["recitation-calculus-review"], "source/review/Exam-1-Review.ptx": ["Exam-1-Review"]}
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+22
-10
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
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eBookConfig.enableScratchAC = false;
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||||
eBookConfig.build_info = "";
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||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
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<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
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<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
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<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
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</script>
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@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
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\DeclareMathOperator{\E}{E}
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\DeclareMathOperator{\Var}{Var}
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\DeclareMathOperator{\Cov}{Cov}
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\newcommand{\L}{\mathcal{L}}
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\newcommand{\lt}{<}
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\newcommand{\gt}{>}
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\newcommand{\amp}{&}
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@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
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||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</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-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
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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-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
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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-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
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</ul>
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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-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</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-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
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</li>
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</ul>
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</li>
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<li class="toc-item toc-chapter">
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@@ -341,7 +353,7 @@ eBookConfig.enable_chatcodes = false;
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</div></details></div>
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<div class="autopermalink" data-description="Exercise 4"><a href="#Exam-1-Review-4-4" title="Copy heading and permalink for Exercise 4" aria-label="Copy heading and permalink for Exercise 4">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="Exam-1-Review-4-5-1-1">Consider the sample space <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6, 7, 8\}\)</span> with probability distribution below. Calculate the probabilities of <span class="process-math">\(A = \{1, 3, 7, 8\}\text{,}\)</span> <span class="process-math">\(B = \{2, 3, 6, 7\}\text{,}\)</span> <span class="process-math">\(A\cup B\text{,}\)</span> and <span class="process-math">\(A \cap B\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div> <figure class="table table-like" id="Exam-1-Review-4-5-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">93<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 93: "><a href="#Exam-1-Review-4-5-1-2" title="Copy heading and permalink for Table 93: " aria-label="Copy heading and permalink for Table 93: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
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||||
</div> <figure class="table table-like" id="Exam-1-Review-4-5-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">118<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 118: "><a href="#Exam-1-Review-4-5-1-2" title="Copy heading and permalink for Table 118: " aria-label="Copy heading and permalink for Table 118: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
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||||
<tr>
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||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
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||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
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||||
@@ -392,7 +404,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="para" id="Exam-1-Review-4-6-1-1">Suppose a die has the values <span class="process-math">\(1, 2, 3, 4, 5, 6\)</span> on the faces, but the die is not fair. Instead, the probabilities scale by the same amount as the face values. For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on. Write a probability distribution table for this die.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="Exam-1-Review-4-6-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<figure class="table table-like" id="Exam-1-Review-4-6-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">94<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Probability Distribution for a Linearly Scaled Die<div class="autopermalink" data-description="Table 94: Probability Distribution for a Linearly Scaled Die"><a href="#Exam-1-Review-4-6-2-1" title="Copy heading and permalink for Table 94: Probability Distribution for a Linearly Scaled Die" aria-label="Copy heading and permalink for Table 94: Probability Distribution for a Linearly Scaled Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<figure class="table table-like" id="Exam-1-Review-4-6-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">119<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Probability Distribution for a Linearly Scaled Die<div class="autopermalink" data-description="Table 119: Probability Distribution for a Linearly Scaled Die"><a href="#Exam-1-Review-4-6-2-1" title="Copy heading and permalink for Table 119: Probability Distribution for a Linearly Scaled Die" aria-label="Copy heading and permalink for Table 119: Probability Distribution for a Linearly Scaled Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
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||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
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||||
@@ -427,7 +439,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="para" id="Exam-1-Review-4-7-1-1">Suppose a die has the values <span class="process-math">\(1, 2, 3, 4, 5, 6\)</span> on the faces, but the die is not fair. Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll. Write a probability distribution table for this die.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="Exam-1-Review-4-7-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<figure class="table table-like" id="Exam-1-Review-4-7-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">95<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Probability Distribution for an Even-biased Die<div class="autopermalink" data-description="Table 95: Probability Distribution for an Even-biased Die"><a href="#Exam-1-Review-4-7-2-1" title="Copy heading and permalink for Table 95: Probability Distribution for an Even-biased Die" aria-label="Copy heading and permalink for Table 95: Probability Distribution for an Even-biased Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<figure class="table table-like" id="Exam-1-Review-4-7-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">120<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Probability Distribution for an Even-biased Die<div class="autopermalink" data-description="Table 120: Probability Distribution for an Even-biased Die"><a href="#Exam-1-Review-4-7-2-1" title="Copy heading and permalink for Table 120: Probability Distribution for an Even-biased Die" aria-label="Copy heading and permalink for Table 120: Probability Distribution for an Even-biased Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
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||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
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||||
@@ -621,7 +633,7 @@ A\cap B \amp = \{HHH, HHT\}
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||||
<div class="para" id="Exam-1-Review-4-13-1-1">Let <span class="process-math">\(A = \{1, 2, 3\}\)</span> and <span class="process-math">\(B = \{3, 4, 5\}\)</span> be events in the sample space <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> Create a probability distribution for <span class="process-math">\(\Omega\)</span> so that <span class="process-math">\(A, B\)</span> are independent.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-13-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
|
||||
<div class="solutions"><details id="Exam-1-Review-4-13-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
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||||
<figure class="table table-like" id="Exam-1-Review-4-13-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">96<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Example Distribution<div class="autopermalink" data-description="Table 96: Example Distribution"><a href="#Exam-1-Review-4-13-2-1" title="Copy heading and permalink for Table 96: Example Distribution" aria-label="Copy heading and permalink for Table 96: Example Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<figure class="table table-like" id="Exam-1-Review-4-13-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">121<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Example Distribution<div class="autopermalink" data-description="Table 121: Example Distribution"><a href="#Exam-1-Review-4-13-2-1" title="Copy heading and permalink for Table 121: Example Distribution" aria-label="Copy heading and permalink for Table 121: Example Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
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<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
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||||
@@ -746,7 +758,7 @@ A\cap B \amp = \{HHH, HHT\}
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||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Exercise 20"><a href="#Exam-1-Review-4-19" title="Copy heading and permalink for Exercise 20" aria-label="Copy heading and permalink for Exercise 20">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-20"><h4 class="heading"><span class="codenumber">21<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="Exam-1-Review-4-20-1-1">Consider <span class="process-math">\(X, Y\)</span> with the joint distribution table below. Are <span class="process-math">\(X, Y\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-20-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div> <figure class="table table-like" id="Exam-1-Review-4-20-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">97<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 <span class="process-math">\(X, Y\)</span><div class="autopermalink" data-description="Table 97: Joint distribution for X, Y"><a href="#Exam-1-Review-4-20-1-2" title="Copy heading and permalink for Table 97: Joint distribution for X, Y" aria-label="Copy heading and permalink for Table 97: Joint distribution for X, Y">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
</div> <figure class="table table-like" id="Exam-1-Review-4-20-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">122<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 <span class="process-math">\(X, Y\)</span><div class="autopermalink" data-description="Table 122: Joint distribution for X, Y"><a href="#Exam-1-Review-4-20-1-2" title="Copy heading and permalink for Table 122: Joint distribution for X, Y" aria-label="Copy heading and permalink for Table 122: Joint distribution for X, Y">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
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||||
<tr>
|
||||
<td class="c m b1 r1 l0 t0 lines"></td>
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<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
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@@ -781,7 +793,7 @@ A\cap B \amp = \{HHH, HHT\}
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||||
<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-21-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions"><details id="Exam-1-Review-4-21-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<figure class="table table-like" id="Exam-1-Review-4-21-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">98<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution<div class="autopermalink" data-description="Table 98: Joint Distribution"><a href="#Exam-1-Review-4-21-2-1" title="Copy heading and permalink for Table 98: Joint Distribution" aria-label="Copy heading and permalink for Table 98: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<figure class="table table-like" id="Exam-1-Review-4-21-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">123<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution<div class="autopermalink" data-description="Table 123: Joint Distribution"><a href="#Exam-1-Review-4-21-2-1" title="Copy heading and permalink for Table 123: Joint Distribution" aria-label="Copy heading and permalink for Table 123: 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>
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||||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
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||||
@@ -804,7 +816,7 @@ A\cap B \amp = \{HHH, HHT\}
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Exercise 22"><a href="#Exam-1-Review-4-21" title="Copy heading and permalink for Exercise 22" aria-label="Copy heading and permalink for Exercise 22">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-22"><h4 class="heading"><span class="codenumber">23<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="Exam-1-Review-4-22-1-1">Consider a random variable <span class="process-math">\(X\)</span> with probability distribution below. Find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-22-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="Exam-1-Review-4-22-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">99<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 99: "><a href="#Exam-1-Review-4-22-1-2" title="Copy heading and permalink for Table 99: " aria-label="Copy heading and permalink for Table 99: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
</div> <figure class="table table-like" id="Exam-1-Review-4-22-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">124<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 124: "><a href="#Exam-1-Review-4-22-1-2" title="Copy heading and permalink for Table 124: " aria-label="Copy heading and permalink for Table 124: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr class="header-horizontal">
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></th>
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th>
|
||||
@@ -913,7 +925,7 @@ A\cap B \amp = \{HHH, HHT\}
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Exercise 28"><a href="#Exam-1-Review-4-27" title="Copy heading and permalink for Exercise 28" aria-label="Copy heading and permalink for Exercise 28">🔗</a></div></article><article class="exercise exercise-like" id="Exam-1-Review-4-28"><h4 class="heading"><span class="codenumber">29<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="Exam-1-Review-4-28-1-1">Consider a random variable <span class="process-math">\(X\)</span> with probability distribution below. Find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-28-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="Exam-1-Review-4-28-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">100<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 100: "><a href="#Exam-1-Review-4-28-1-2" title="Copy heading and permalink for Table 100: " aria-label="Copy heading and permalink for Table 100: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
</div> <figure class="table table-like" id="Exam-1-Review-4-28-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">125<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 125: "><a href="#Exam-1-Review-4-28-1-2" title="Copy heading and permalink for Table 125: " aria-label="Copy heading and permalink for Table 125: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr class="header-horizontal">
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></th>
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th>
|
||||
|
||||
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|
||||
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|
||||
<all>
|
||||
<js type="list">
|
||||
<item type="str">prefix-runtime.0d5f811896ddeac1.bundle.js</item>
|
||||
<item type="str">prefix-runtime.d892923478694492.bundle.js</item>
|
||||
<item type="str">prefix-723.3e6434f80549315a.bundle.js</item>
|
||||
<item type="str">prefix-runestone.e2abc79debce15fd.bundle.js</item>
|
||||
<item type="str">prefix-runestone.1d6e4aba17c2c7f2.bundle.js</item>
|
||||
</js>
|
||||
<css type="list">
|
||||
<item type="str">prefix-723.3bccd435914aa0ff.css</item>
|
||||
<item type="str">prefix-runestone.6ec5960970d8b364.css</item>
|
||||
</css>
|
||||
<cdn-url type="str">https://runestone.academy/cdn/runestone/</cdn-url>
|
||||
<version type="str">7.11.13</version>
|
||||
<version type="str">7.11.14</version>
|
||||
</all>
|
||||
|
||||
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|
||||
/*!
|
||||
* BIDI embedding support for jQuery.i18n
|
||||
*
|
||||
* Copyright (C) 2015, David Chan
|
||||
*
|
||||
* This code is dual licensed GPLv2 or later and MIT. You don't have to do
|
||||
* anything special to choose one license or the other and you don't have to
|
||||
* notify anyone which license you are using. You are free to use this code
|
||||
* in commercial projects as long as the copyright header is left intact.
|
||||
* See files GPL-LICENSE and MIT-LICENSE for details.
|
||||
*
|
||||
* @licence GNU General Public Licence 2.0 or later
|
||||
* @licence MIT License
|
||||
*/
|
||||
|
||||
/*!
|
||||
* jQuery Internationalization library
|
||||
*
|
||||
* Copyright (C) 2011-2013 Santhosh Thottingal, Neil Kandalgaonkar
|
||||
*
|
||||
* jquery.i18n is dual licensed GPLv2 or later and MIT. You don't have to do
|
||||
* anything special to choose one license or the other and you don't have to
|
||||
* notify anyone which license you are using. You are free to use
|
||||
* UniversalLanguageSelector in commercial projects as long as the copyright
|
||||
* header is left intact. See files GPL-LICENSE and MIT-LICENSE for details.
|
||||
*
|
||||
* @licence GNU General Public Licence 2.0 or later
|
||||
* @licence MIT License
|
||||
*/
|
||||
|
||||
/*!
|
||||
* jQuery Internationalization library
|
||||
*
|
||||
* Copyright (C) 2012 Santhosh Thottingal
|
||||
*
|
||||
* jquery.i18n is dual licensed GPLv2 or later and MIT. You don't have to do
|
||||
* anything special to choose one license or the other and you don't have to
|
||||
* notify anyone which license you are using. You are free to use
|
||||
* UniversalLanguageSelector in commercial projects as long as the copyright
|
||||
* header is left intact. See files GPL-LICENSE and MIT-LICENSE for details.
|
||||
*
|
||||
* @licence GNU General Public Licence 2.0 or later
|
||||
* @licence MIT License
|
||||
*/
|
||||
|
||||
/*!
|
||||
* jQuery Internationalization library
|
||||
*
|
||||
* Copyright (C) 2012 Santhosh Thottingal
|
||||
*
|
||||
* jquery.i18n is dual licensed GPLv2 or later and MIT. You don't have to do anything special to
|
||||
* choose one license or the other and you don't have to notify anyone which license you are using.
|
||||
* You are free to use UniversalLanguageSelector in commercial projects as long as the copyright
|
||||
* header is left intact. See files GPL-LICENSE and MIT-LICENSE for details.
|
||||
*
|
||||
* @licence GNU General Public Licence 2.0 or later
|
||||
* @licence MIT License
|
||||
*/
|
||||
|
||||
/*!
|
||||
* jQuery Internationalization library - Message Store
|
||||
*
|
||||
* Copyright (C) 2012 Santhosh Thottingal
|
||||
*
|
||||
* jquery.i18n is dual licensed GPLv2 or later and MIT. You don't have to do anything special to
|
||||
* choose one license or the other and you don't have to notify anyone which license you are using.
|
||||
* You are free to use UniversalLanguageSelector in commercial projects as long as the copyright
|
||||
* header is left intact. See files GPL-LICENSE and MIT-LICENSE for details.
|
||||
*
|
||||
* @licence GNU General Public Licence 2.0 or later
|
||||
* @licence MIT License
|
||||
*/
|
||||
|
||||
/*!
|
||||
* jQuery idleTimer plugin
|
||||
* version 0.9.100511
|
||||
* by Paul Irish.
|
||||
* http://github.com/paulirish/yui-misc/tree/
|
||||
* MIT license
|
||||
|
||||
* adapted from YUI idle timer by nzakas:
|
||||
* http://github.com/nzakas/yui-misc/
|
||||
*/
|
||||
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@@ -1 +1 @@
|
||||
{"js":["prefix-runtime.0d5f811896ddeac1.bundle.js","prefix-723.3e6434f80549315a.bundle.js","prefix-runestone.e2abc79debce15fd.bundle.js"],"css":["prefix-723.3bccd435914aa0ff.css","prefix-runestone.6ec5960970d8b364.css"]}
|
||||
{"js":["prefix-runtime.d892923478694492.bundle.js","prefix-723.3e6434f80549315a.bundle.js","prefix-runestone.1d6e4aba17c2c7f2.bundle.js"],"css":["prefix-723.3bccd435914aa0ff.css","prefix-runestone.6ec5960970d8b364.css"]}
|
||||
Binary file not shown.
@@ -1,14 +1,14 @@
|
||||
<?xml version="1.0" ?>
|
||||
<all>
|
||||
<js type="list">
|
||||
<item type="str">prefix-runtime.0d5f811896ddeac1.bundle.js</item>
|
||||
<item type="str">prefix-runtime.d892923478694492.bundle.js</item>
|
||||
<item type="str">prefix-723.3e6434f80549315a.bundle.js</item>
|
||||
<item type="str">prefix-runestone.e2abc79debce15fd.bundle.js</item>
|
||||
<item type="str">prefix-runestone.1d6e4aba17c2c7f2.bundle.js</item>
|
||||
</js>
|
||||
<css type="list">
|
||||
<item type="str">prefix-723.3bccd435914aa0ff.css</item>
|
||||
<item type="str">prefix-runestone.6ec5960970d8b364.css</item>
|
||||
</css>
|
||||
<cdn-url type="str">https://runestone.academy/cdn/runestone/</cdn-url>
|
||||
<version type="str">7.11.13</version>
|
||||
<version type="str">7.11.14</version>
|
||||
</all>
|
||||
|
||||
+16
-2
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
@@ -254,6 +266,8 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li><a href="notes-01-27.html" class="internal"><span class="title">Tuesday, Jan 27</span></a></li>
|
||||
<li><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></li>
|
||||
<li><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></li>
|
||||
<li><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></li>
|
||||
<li><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</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>
|
||||
<div id="ptx-content-footer" class="ptx-content-footer">
|
||||
<a class="previous-button button" href="syllabus.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></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"></span></a>
|
||||
|
||||
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
+14
-2
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
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<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</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-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
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<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
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|
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
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<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
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"body": " Tuesday, Feb 3 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. Variance Question: How spread out are values? One answer we might try is to measure the average distance from the average value: To simplify notation, we'll write . Also, it's often usefull to square a term rather than take absolute value when we want to ensure a positive output, so we'll define... Let be a random variable with . The variance of is: is called the standard deviation . We can write an alternative formula here: This formula is generally more useful for performing computations. Let indicate event with . Distribution for Distribution for Then and , so: Let be the roll of a fair D6. Distribution for Distribution for Then and , so: Let with pdf . WARNING!!! In general, variance is not linear! That is, But there are still relevant properties we can state here: Let be a random variable and . Then: Moreover, if is another random variable and are independent, then: Let be indicators with parameter . Let , so . We know , and are independent. So: "
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"body": " Thursday, Feb 5 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. Summary Here are the expected value and variance formulas for common distributions. Some of these, we've shown justification for. Others requires techniques beyond the scope of the class to justify. Expected Value and Variance Formulas Distribution Parameters Expected Value Variance Indicator Binomial Geometric Poisson Exponential Covariance If are random variables with expected values of , then the covariance of and is: Observe that: so covariance generalizes the variance formula to two variables. As with variance, there's an alternative formula more suited to doing computations: Consider the joint distribution table: Joint Distribution 0.2 0.1 0.05 0.65 From the table, we can calculate the marginal distributions: So and . Then: Question: the formula makes it clear that variance cannot be negative, since squares are nonnegative. What about ? In the previous example, since were both indicator random variables, the variances for each were simply equal to the sum of the second row\/column. Similarly, was equal to the entry in the table. Using two indicator random variables, significantly simplifies the covariance calculation, so we can vary the table and recalculate covariance quickly. Consider the following joint distribution: Joint Distribution 0.1 0.4 0.3 0.2 Then: Question: how do we interpret ? is positive when and negative when . is positive when and negative when . So the product is positive when are both larger or both smaller than their expected values, and negative when one is larger and one is smaller. That is, covariance tries to quantify the tendency of to get big\/small at the same time. "
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||||
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"body": " Consider the joint distribution table: Joint Distribution 0.2 0.1 0.05 0.65 From the table, we can calculate the marginal distributions: So and . Then: "
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||||
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||||
"body": " In the previous example, since were both indicator random variables, the variances for each were simply equal to the sum of the second row\/column. Similarly, was equal to the entry in the table. Using two indicator random variables, significantly simplifies the covariance calculation, so we can vary the table and recalculate covariance quickly. Consider the following joint distribution: Joint Distribution 0.1 0.4 0.3 0.2 Then: "
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||||
"body": " Tuesday, Feb 10 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. Likelihood In probability theory, we start with some probability model and its parameter values, and we try to find the probabilities of seeing certain types of data. In statistics, we start with the collected data, and we try to find the most likely values of parameters for some underlying probability model. An estimator is a way of estimating a parameter value based on data collected. Suppose we flip a coin 100 times and see 52 heads. Let be the bias of the coin. Then we might estimate: (The notation is sometimes used to indicate an estimator for .) If we see heads in flips, then the estimator is called the common sense estimator for the binomial distribution parameter . An estimator is unbiased if . Flip a coin 100 times, and let be the number of heads. Then . Let . Then: So , i.e., the common sense estimator is unbiased. We perform an experiment and collect data. Let be an unknown parameter value. The likelihood function is: Let . Suppose we see 52 heads. Then: In the first line, the variable represents the data. In the second line, the variable represents the parameter value. Our goal, given the collected data, is to find the maximum likelihood estimation (MLE) for the parameter value. is a continuous function over a closed interval , so we use the Closed Interval Method. Now, we look for critical numbers in the interior of the interval: Finally, we test the critical numbers and the endpoints of the interval to find the max: Check Candidate Locations for Max So is the MLE. More generally, a similar calculation will show that, with heads in flips, the MLE will be . Suppose we observe a cell, measuring the time until a toxin molecule leaves the cell. Then for some , with pdf If we see a toxin molecule leave at 0.3 min, what's the MLE for ? We want to maximize over , an open interval. So we'll use the \"Open Interval Method\". Then when . Checking values to the left and the right: So (and therefore is increasing) on , and (and therefore is decreasing) on . Now we can conclude that is the location of a global (and not just local) maximum value. More generally, if the observed time is , then the MLE will be . What if we had more data points? For example, suppose two toxin molecules leave the cell at min and min? Waiting Times Molecule Time Rate Estimate How do we combine these data points? We could take the average of the rate estimates: Alternatively, we could average the times first, then create a new rate estimate from the average time: Both of these make some sense, but let's do a careful computation to be certain which way is correct (if either of them is!). Using the Open Interval Method: So when . Testing points to the left and right: Now ( is increasing) on , and ( is decreasing) on . Therefore, the MLE is . Tracing the values and throughout the calculation, we can see that the value will generally match the number of waiting times collected, and the value will be the sum of the waiting times. So, generally, with collected waiting times of , the MLE will be: "
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||||
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||||
"body": " If we see heads in flips, then the estimator is called the common sense estimator for the binomial distribution parameter . "
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"body": " Flip a coin 100 times, and let be the number of heads. Then . Let . Then: So , i.e., the common sense estimator is unbiased. "
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"body": " We perform an experiment and collect data. Let be an unknown parameter value. The likelihood function is: "
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||||
"body": " Let . Suppose we see 52 heads. Then: In the first line, the variable represents the data. In the second line, the variable represents the parameter value. Our goal, given the collected data, is to find the maximum likelihood estimation (MLE) for the parameter value. is a continuous function over a closed interval , so we use the Closed Interval Method. Now, we look for critical numbers in the interior of the interval: Finally, we test the critical numbers and the endpoints of the interval to find the max: Check Candidate Locations for Max So is the MLE. More generally, a similar calculation will show that, with heads in flips, the MLE will be . "
|
||||
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|
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||||
"body": " Suppose we observe a cell, measuring the time until a toxin molecule leaves the cell. Then for some , with pdf If we see a toxin molecule leave at 0.3 min, what's the MLE for ? We want to maximize over , an open interval. So we'll use the \"Open Interval Method\". Then when . Checking values to the left and the right: So (and therefore is increasing) on , and (and therefore is decreasing) on . Now we can conclude that is the location of a global (and not just local) maximum value. More generally, if the observed time is , then the MLE will be . What if we had more data points? For example, suppose two toxin molecules leave the cell at min and min? Waiting Times Molecule Time Rate Estimate How do we combine these data points? We could take the average of the rate estimates: Alternatively, we could average the times first, then create a new rate estimate from the average time: Both of these make some sense, but let's do a careful computation to be certain which way is correct (if either of them is!). Using the Open Interval Method: So when . Testing points to the left and right: Now ( is increasing) on , and ( is decreasing) on . Therefore, the MLE is . Tracing the values and throughout the calculation, we can see that the value will generally match the number of waiting times collected, and the value will be the sum of the waiting times. So, generally, with collected waiting times of , the MLE will be: "
|
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|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
+14
-2
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
+14
-2
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
+14
-2
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
+14
-2
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
+14
-2
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
+220
-5
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -135,7 +135,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
<textarea class="embed-code-textbox" id="embed-code-textbox" readonly aria-label="textbox"><iframe src="https://example.com/embed" width="100%" height="1000"></iframe></textarea><button class="copy-embed-button button" id="copy-embed-button" title="Copy embed code"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Copy</span></button>
|
||||
</div>
|
||||
</div>
|
||||
<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="notes-01-29.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="ch-notes.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><a class="next-button button" href="quizzes.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
|
||||
<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="notes-01-29.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="ch-notes.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><a class="next-button button" href="notes-02-05.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
|
||||
</div></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N}
|
||||
\newcommand{\Z}{\mathbb Z}
|
||||
\newcommand{\Q}{\mathbb Q}
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list active"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
@@ -274,9 +286,212 @@ eBookConfig.enable_chatcodes = false;
|
||||
<span class="process-math">\(\sigma = \sqrt{\Var(X)}\)</span> is called the <dfn class="terminology">standard deviation</dfn>.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-variance-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 90"><a href="#def-variance" title="Copy heading and permalink for Definition 90" aria-label="Copy heading and permalink for Definition 90">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Variance"><a href="#notes-02-03-3" title="Copy heading and permalink for Subsection: Variance" aria-label="Copy heading and permalink for Subsection: Variance">🔗</a></div></section><div class="autopermalink" data-description="Section: Tuesday, Feb 3"><a href="#notes-02-03" title="Copy heading and permalink for Section: Tuesday, Feb 3" aria-label="Copy heading and permalink for Section: Tuesday, Feb 3">🔗</a></div></section></div>
|
||||
<div class="autopermalink" data-description="Definition 90"><a href="#def-variance" title="Copy heading and permalink for Definition 90" aria-label="Copy heading and permalink for Definition 90">🔗</a></div></article><div class="para logical" id="notes-02-03-3-4">
|
||||
<div class="para">We can write an alternative formula here:</div>
|
||||
<div class="displaymath process-math" id="notes-02-03-3-4-1">
|
||||
\begin{align*}
|
||||
\Var(X) \amp = \E\left[ (X - \mu)^2 \right] \\
|
||||
\amp = \E\left[ X^2 - 2\mu X + \mu^2 \right] \\
|
||||
\amp = \E(X^2) - 2\mu \E(X) + \E(\mu^2) \\
|
||||
\amp = \E(X^2) - 2\mu^2 + \mu^2 \\
|
||||
\amp = \E(X^2) - \mu^2.
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">This formula is generally more useful for performing computations.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="notes-02-03-3-5"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">91</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="notes-02-03-3-5-1-1">Let <span class="process-math">\(X\)</span> indicate event <span class="process-math">\(A\)</span> with <span class="process-math">\(\Pr(A) = p\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="sidebyside"><div class="sbsrow" style="margin-left:10%;margin-right:10%;grid-template-columns:37.5% 37.5%;column-gap:20%;">
|
||||
<div class="sbspanel fixed-width sbspanel--top top" style=""><figure class="table table-like" id="notes-02-03-3-5-1-2-1"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = k)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1 - p\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
|
||||
</tr>
|
||||
</table>
|
||||
<figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">92<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for <span class="process-math">\(X\)</span><div class="autopermalink" data-description="Table 92: Distribution for X"><a href="#notes-02-03-3-5-1-2-1" title="Copy heading and permalink for Table 92: Distribution for X" aria-label="Copy heading and permalink for Table 92: Distribution for X">🔗</a></div></figcaption></figure></div>
|
||||
<div class="sbspanel fixed-width sbspanel--top top" style=""><figure class="table table-like" id="notes-02-03-3-5-1-2-2"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X^2 = k)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0^2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1 - p\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1^2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
|
||||
</tr>
|
||||
</table>
|
||||
<figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">93<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for <span class="process-math">\(X^2\)</span><div class="autopermalink" data-description="Table 93: Distribution for X^2"><a href="#notes-02-03-3-5-1-2-2" title="Copy heading and permalink for Table 93: Distribution for X^2" aria-label="Copy heading and permalink for Table 93: Distribution for X^2">🔗</a></div></figcaption></figure></div>
|
||||
</div></div> <div class="para logical" id="notes-02-03-3-5-1-3">
|
||||
<div class="para">Then <span class="process-math">\(\E(X) = p\)</span> and <span class="process-math">\(\E(X^2) = p\text{,}\)</span> so:</div>
|
||||
<div class="displaymath process-math" id="notes-02-03-3-5-1-3-3">
|
||||
\begin{gather*}
|
||||
\Var(X) = \E(X^2) - \left(\E(X)\right)^2 = p - p^2 = p(1 - p).
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-5-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 91"><a href="#notes-02-03-3-5" title="Copy heading and permalink for Example 91" aria-label="Copy heading and permalink for Example 91">🔗</a></div></article><article class="example example-like" id="notes-02-03-3-6"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">94</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="notes-02-03-3-6-1-1">Let <span class="process-math">\(R\)</span> be the roll of a fair D6.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="sidebyside"><div class="sbsrow" style="margin-left:10%;margin-right:10%;grid-template-columns:37.5% 37.5%;column-gap:20%;">
|
||||
<div class="sbspanel fixed-width sbspanel--top top" style=""><figure class="table table-like" id="notes-02-03-3-6-1-2-1"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(R = k)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
</table>
|
||||
<figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">95<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for <span class="process-math">\(R\)</span><div class="autopermalink" data-description="Table 95: Distribution for R"><a href="#notes-02-03-3-6-1-2-1" title="Copy heading and permalink for Table 95: Distribution for R" aria-label="Copy heading and permalink for Table 95: Distribution for R">🔗</a></div></figcaption></figure></div>
|
||||
<div class="sbspanel fixed-width sbspanel--top top" style=""><figure class="table table-like" id="notes-02-03-3-6-1-2-2"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(R^2 = k)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1^2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2^2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3^2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4^2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5^2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6^2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
|
||||
</tr>
|
||||
</table>
|
||||
<figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">96<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for <span class="process-math">\(R^2\)</span><div class="autopermalink" data-description="Table 96: Distribution for R^2"><a href="#notes-02-03-3-6-1-2-2" title="Copy heading and permalink for Table 96: Distribution for R^2" aria-label="Copy heading and permalink for Table 96: Distribution for R^2">🔗</a></div></figcaption></figure></div>
|
||||
</div></div> <div class="para logical" id="notes-02-03-3-6-1-3">
|
||||
<div class="para">Then <span class="process-math">\(\E(R) = \frac{7}{2}\)</span> and <span class="process-math">\(\E(R^2) = \frac{91}{6}\text{,}\)</span> so:</div>
|
||||
<div class="displaymath process-math" id="notes-02-03-3-6-1-3-3">
|
||||
\begin{gather*}
|
||||
\Var(R) = \E(R^2) - \left(\E(R)\right)^2 = \frac{91}{6} - \frac{49}{4} = \frac{35}{12}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-6-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 94"><a href="#notes-02-03-3-6" title="Copy heading and permalink for Example 94" aria-label="Copy heading and permalink for Example 94">🔗</a></div></article><article class="example example-like" id="notes-02-03-3-7"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">97</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-02-03-3-7-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X \in [0, 1]\)</span> with pdf <span class="process-math">\(f(x) = 2x\text{.}\)</span>
|
||||
</div>
|
||||
<div class="displaymath process-math" id="notes-02-03-3-7-1-1-3">
|
||||
\begin{align*}
|
||||
\E(X) \amp = \int_0^1 x \cdot 2x\ dx \\
|
||||
\amp = \int_0^1 2x^2\ dx \\
|
||||
\amp = \frac{2x^3}{3}\bigg|_0^1 \\
|
||||
\amp = \frac{2}{3} - 0 \\
|
||||
\amp = \frac{2}{3} \\
|
||||
\E(X^2) \amp = \int_0^1 x^2\cdot 2x\ dx \\
|
||||
\amp = \int_0^1 2x^3\ dx \\
|
||||
\amp = \frac{2x^4}{4}\bigg|_0^1 \\
|
||||
\amp = \frac{1}{2} - 0 \\
|
||||
\amp = \frac{1}{2} \\
|
||||
\Rightarrow \quad \Var(X) \amp = \E(X^2) - \left(\E(X)\right)^2 \\
|
||||
\amp = \frac{1}{2} - \frac{4}{9} \\
|
||||
\amp = \frac{1}{18}.
|
||||
\end{align*}
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</div>
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|
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<div class="para">WARNING!!! In general, variance is <em class="emphasis">not</em> linear! That is,</div>
|
||||
<div class="displaymath process-math" id="notes-02-03-3-8-2">
|
||||
\begin{align*}
|
||||
\Var(X + Y) \amp \neq \Var(X) + \Var(Y) \\
|
||||
\Var(kX) \amp \neq k \Var(X)
|
||||
\end{align*}
|
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</div>
|
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<div class="para">But there are still relevant properties we can state here:</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-03-3-8" 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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<article class="theorem theorem-like" id="notes-02-03-3-9"><h4 class="heading">
|
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<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">98</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-02-03-3-9-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X\)</span> be a random variable and <span class="process-math">\(k\in \R\text{.}\)</span> Then:</div>
|
||||
<div class="displaymath process-math" id="notes-02-03-3-9-1-1-3">
|
||||
\begin{align*}
|
||||
\Var(X + k) \amp = \Var(X) \amp \Var(kX) \amp = k^2\Var(X).
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Moreover, if <span class="process-math">\(Y\)</span> is another random variable and <span class="process-math">\(X, Y\)</span> are independent, then:</div>
|
||||
<div class="displaymath process-math" id="notes-02-03-3-9-1-1-6">
|
||||
\begin{gather*}
|
||||
\Var(X + Y) = \Var(X) + \Var(Y).
|
||||
\end{gather*}
|
||||
</div>
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|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">99</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
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</h4>
|
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<div class="para logical" id="notes-02-03-3-10-1-1">
|
||||
<div class="para">Let <span class="process-math">\(H_1, H_2, \dotsc, H_n\)</span> be indicators with parameter <span class="process-math">\(p\text{.}\)</span> Let <span class="process-math">\(S = H_1 + \dotsb + H_n\text{,}\)</span> so <span class="process-math">\(S \sim \Bin(n, p)\text{.}\)</span> We know <span class="process-math">\(\Var(H_i) = p(1-p)\text{,}\)</span> and <span class="process-math">\(H_1, \dotsc, H_n\)</span> are independent. So:</div>
|
||||
<div class="displaymath process-math" id="notes-02-03-3-10-1-1-7">
|
||||
\begin{align*}
|
||||
\Var(S) \amp = \Var(H_1 + \dotsb + H_n) \\
|
||||
\amp = \Var(H_1) + \dotsb + \Var(H_n) \\
|
||||
\amp = p(1-p) + \dotsb + p(1-p) \\
|
||||
\amp = np(1-p)
|
||||
\end{align*}
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</div>
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<ul class="structural toc-item-list contains-active">
|
||||
<li class="toc-item toc-section">
|
||||
<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>
|
||||
<ul class="structural toc-item-list">
|
||||
<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">
|
||||
<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Tuesday, 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>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section contains-active">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list active">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="quizzes.html" class="internal"><span class="title">Quizzes</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-01.html" class="internal"><span class="title">Quiz 1</span></a></div></li>
|
||||
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="review.html" class="internal"><span class="title">Exam Review</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="Exam-1-Review.html" class="internal"><span class="title">Exam 1 Review</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="Exam-1-Review.html#Exam-1-Review-3" class="internal"><span class="title">Allowed Materials</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="Exam-1-Review.html#Exam-1-Review-4" class="internal"><span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li></ul>
|
||||
</li>
|
||||
</ul></nav></div>
|
||||
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-02-05"><h2 class="heading hide-type">
|
||||
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Thursday, Feb 5</span>
|
||||
</h2>
|
||||
<section class="introduction" id="notes-02-05-2"><div class="para" id="notes-02-05-2-1">This is an outline of the topics we covered in class. These notes are <em class="emphasis">not</em> a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></section><section class="subsection" id="notes-02-05-3"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Summary</span>
|
||||
</h3>
|
||||
<div class="para" id="notes-02-05-3-2">Here are the expected value and variance formulas for common distributions. Some of these, we’ve shown justification for. Others requires techniques beyond the scope of the class to justify.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-3-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<figure class="table table-like" id="notes-02-05-3-3"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">100<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Expected Value and Variance Formulas<div class="autopermalink" data-description="Table 100: Expected Value and Variance Formulas"><a href="#notes-02-05-3-3" title="Copy heading and permalink for Table 100: Expected Value and Variance Formulas" aria-label="Copy heading and permalink for Table 100: Expected Value and Variance Formulas">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines">Distribution</td>
|
||||
<td class="c m b1 r0 l0 t0 lines">Parameters</td>
|
||||
<td class="c m b1 r0 l0 t0 lines">Expected Value</td>
|
||||
<td class="c m b1 r0 l0 t0 lines">Variance</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">Indicator</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(p(1-p)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">Binomial</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(n, p\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(np\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(np(1-p)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">Geometric</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\frac{1}{p}\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\frac{1-p}{p^2}\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">Poisson</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\lambda\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\lambda\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\lambda\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines">Exponential</td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\lambda\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\frac{1}{\lambda}\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\frac{1}{\lambda^2}\)</span></td>
|
||||
</tr>
|
||||
</table></div></figure><div class="autopermalink" data-description="Subsection: Summary"><a href="#notes-02-05-3" title="Copy heading and permalink for Subsection: Summary" aria-label="Copy heading and permalink for Subsection: Summary">🔗</a></div></section><section class="subsection" id="notes-02-05-4"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Covariance</span>
|
||||
</h3>
|
||||
<article class="definition definition-like" id="notes-02-05-4-2"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">101</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-02-05-4-2-1-1">
|
||||
<div class="para">If <span class="process-math">\(X, Y\)</span> are random variables with expected values of <span class="process-math">\(\mu_X, \mu_Y\text{,}\)</span> then the <dfn class="terminology">covariance</dfn> of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> is:</div>
|
||||
<div class="displaymath process-math" id="notes-02-05-4-2-1-1-6">
|
||||
\begin{align*}
|
||||
\Cov(X, Y) \amp = \E\left[ (X - \mu_X)(Y - \mu_Y)\right].
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 101"><a href="#notes-02-05-4-2" title="Copy heading and permalink for Definition 101" aria-label="Copy heading and permalink for Definition 101">🔗</a></div></article><div class="para logical" id="notes-02-05-4-3">
|
||||
<div class="para">Observe that:</div>
|
||||
<div class="displaymath process-math" id="notes-02-05-4-3-1">
|
||||
\begin{align*}
|
||||
\Cov(X, X) \amp = \E\left[(X - \mu_X)(X - \mu_X)\right] \\
|
||||
\amp = \E\left[(X - \mu_X)^2\right] \\
|
||||
\amp = \Var(X),
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">so covariance generalizes the variance formula to two variables. As with variance, there’s an alternative formula more suited to doing computations:</div>
|
||||
<div class="displaymath process-math" id="notes-02-05-4-3-2">
|
||||
\begin{align*}
|
||||
\Var(X) \amp = \E\left[(X - \mu_X)(X - \mu_X)\right] = \E(X^2) - \mu_X^2 \\
|
||||
\Cov(X, Y) \amp = \E\left[(X - \mu_X)(Y - \mu_Y)\right] = \E(XY) - \mu_X\mu_Y
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="notes-02-05-4-4"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">102</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="notes-02-05-4-4-1-1">Consider the joint distribution table:<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="notes-02-05-4-4-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">103<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution<div class="autopermalink" data-description="Table 103: Joint Distribution"><a href="#notes-02-05-4-4-1-2" title="Copy heading and permalink for Table 103: Joint Distribution" aria-label="Copy heading and permalink for Table 103: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r1 l0 t0 lines"></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.2</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.05</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">0.65</td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para logical" id="notes-02-05-4-4-1-3">
|
||||
<div class="para">From the table, we can calculate the marginal distributions:</div>
|
||||
<div class="displaymath process-math" id="notes-02-05-4-4-1-3-1">
|
||||
\begin{align*}
|
||||
\Pr(X = 0) \amp = 0.25 \amp \Pr(Y = 0) \amp = 0.3 \\
|
||||
\Pr(X = 1) \amp = 0.75 \amp \Pr(Y = 1) \amp = 0.7
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">So <span class="process-math">\(\E(X) = \mu_X = 0.75\)</span> and <span class="process-math">\(\E(Y) = \mu_Y = 0.7\text{.}\)</span> Then:</div>
|
||||
<div class="displaymath process-math" id="notes-02-05-4-4-1-3-4">
|
||||
\begin{align*}
|
||||
\E(XY) \amp = (0)(0)(0.2) + (1)(0)(0.1) + (0)(1)(0.05) + (1)(1)(0.65) \\
|
||||
\amp = 0.65 \\
|
||||
\Cov(X, Y) \amp = \E(XY) - \mu_X \mu_Y \\
|
||||
\amp = 0.65 - (0.75)(0.7) \\
|
||||
\amp = 0.125.
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-4-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 102"><a href="#notes-02-05-4-4" title="Copy heading and permalink for Example 102" aria-label="Copy heading and permalink for Example 102">🔗</a></div></article><div class="para" id="notes-02-05-4-5">Question: the formula <span class="process-math">\(\Var(X) = \E\left[(X - \mu_X)^2\right]\)</span> makes it clear that variance cannot be negative, since squares are nonnegative. What about <span class="process-math">\(\Cov(X, Y)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="example example-like" id="notes-02-05-4-6"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">104</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="notes-02-05-4-6-1-1">In the previous example, since <span class="process-math">\(X, Y\)</span> were both indicator random variables, the variances for each were simply equal to the sum of the second row/column. Similarly, <span class="process-math">\(\E(XY)\)</span> was equal to the <span class="process-math">\(X = 1, Y = 1\)</span> entry in the table. Using two indicator random variables, significantly simplifies the covariance calculation, so we can vary the table and recalculate covariance quickly. Consider the following joint distribution:<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="notes-02-05-4-6-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">105<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution<div class="autopermalink" data-description="Table 105: Joint Distribution"><a href="#notes-02-05-4-6-1-2" title="Copy heading and permalink for Table 105: Joint Distribution" aria-label="Copy heading and permalink for Table 105: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
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\begin{gather*}
|
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\Cov(X, Y) = 0.2 - (0.6)(0.5) = -0.1 \lt 0.
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<div class="autopermalink" data-description="Example 104"><a href="#notes-02-05-4-6" title="Copy heading and permalink for Example 104" aria-label="Copy heading and permalink for Example 104">🔗</a></div></article><div class="para" id="notes-02-05-4-7">Question: how do we interpret <span class="process-math">\(\Cov(X, Y)\text{?}\)</span> <span class="process-math">\(X - \mu_X\)</span> is positive when <span class="process-math">\(X \gt \mu_X\)</span> and negative when <span class="process-math">\(X \lt \mu_X\text{.}\)</span> <span class="process-math">\(Y - \mu_Y\)</span> is positive when <span class="process-math">\(Y \gt \mu_Y\)</span> and negative when <span class="process-math">\(Y \lt \mu_Y\text{.}\)</span> So the product <span class="process-math">\((X - \mu_X)(Y - \mu_Y)\)</span> is positive when <span class="process-math">\(X, Y\)</span> are both larger or both smaller than their expected values, and negative when one is larger and one is smaller. That is, covariance tries to quantify the tendency of <span class="process-math">\(X, Y\)</span> to get big/small at the same time.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-05-4-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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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">
|
||||
<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Tuesday, 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>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-01-29.html" class="internal"><span class="title">Thursday, Jan 29</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-29.html#notes-01-29-3" class="internal"><span class="title">Expected Value</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section contains-active">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list active"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="quizzes.html" class="internal"><span class="title">Quizzes</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-01.html" class="internal"><span class="title">Quiz 1</span></a></div></li>
|
||||
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="recitations.html" class="internal"><span class="title">Recitations</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="review.html" class="internal"><span class="title">Exam Review</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="Exam-1-Review.html" class="internal"><span class="title">Exam 1 Review</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="Exam-1-Review.html#Exam-1-Review-3" class="internal"><span class="title">Allowed Materials</span></a></div></li>
|
||||
<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="Exam-1-Review.html#Exam-1-Review-4" class="internal"><span class="title">Exercises</span></a></div></li>
|
||||
</ul>
|
||||
</li></ul>
|
||||
</li>
|
||||
</ul></nav></div>
|
||||
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-02-10"><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">Tuesday, Feb 10</span>
|
||||
</h2>
|
||||
<section class="introduction" id="notes-02-10-2"><div class="para" id="notes-02-10-2-1">This is an outline of the topics we covered in class. These notes are <em class="emphasis">not</em> a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div></section><section class="subsection" id="notes-02-10-3"><h3 class="heading hide-type">
|
||||
<span class="type">Subsection</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Likelihood</span>
|
||||
</h3>
|
||||
<div class="para" id="notes-02-10-3-2">In probability theory, we start with some probability model and its parameter values, and we try to find the probabilities of seeing certain types of data. In statistics, we start with the collected data, and we try to find the most likely values of parameters for some underlying probability model.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="notes-02-10-3-3"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">106</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="notes-02-10-3-3-1-1">An <dfn class="terminology">estimator</dfn> is a way of estimating a parameter value based on data collected.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 106"><a href="#notes-02-10-3-3" title="Copy heading and permalink for Definition 106" aria-label="Copy heading and permalink for Definition 106">🔗</a></div></article><article class="example example-like" id="notes-02-10-3-4"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">107</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-02-10-3-4-1-1">
|
||||
<div class="para">Suppose we flip a coin 100 times and see 52 heads. Let <span class="process-math">\(p\)</span> be the bias of the coin. Then we might estimate:</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-4-1-1-2">
|
||||
\begin{gather*}
|
||||
\widehat{p} = \frac{52}{100}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">(The notation <span class="process-math">\(\widehat{p}\)</span> is sometimes used to indicate an estimator for <span class="process-math">\(p\text{.}\)</span>)</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 107"><a href="#notes-02-10-3-4" title="Copy heading and permalink for Example 107" aria-label="Copy heading and permalink for Example 107">🔗</a></div></article><article class="definition definition-like" id="notes-02-10-3-5"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">108</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="notes-02-10-3-5-1-1">If we see <span class="process-math">\(k\)</span> heads in <span class="process-math">\(n\)</span> flips, then the estimator <span class="process-math">\(\widehat{p} = \frac{k}{n}\)</span> is called the <dfn class="terminology">common sense estimator</dfn> for the binomial distribution parameter <span class="process-math">\(p\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 108"><a href="#notes-02-10-3-5" title="Copy heading and permalink for Definition 108" aria-label="Copy heading and permalink for Definition 108">🔗</a></div></article><article class="definition definition-like" id="notes-02-10-3-6"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">109</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para" id="notes-02-10-3-6-1-1">An estimator <span class="process-math">\(\widehat{p}\)</span> is <dfn class="terminology">unbiased</dfn> if <span class="process-math">\(\E(\widehat{p}) = p\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 109"><a href="#notes-02-10-3-6" title="Copy heading and permalink for Definition 109" aria-label="Copy heading and permalink for Definition 109">🔗</a></div></article><article class="example example-like" id="notes-02-10-3-7"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">110</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-02-10-3-7-1-1">
|
||||
<div class="para">Flip a coin 100 times, and let <span class="process-math">\(S\)</span> be the number of heads. Then <span class="process-math">\(S \sim \Bin(100, p)\text{.}\)</span> Let <span class="process-math">\(\widehat{p} = \frac{S}{100} = \frac{S}{n}\text{.}\)</span> Then:</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-7-1-1-4">
|
||||
\begin{align*}
|
||||
\E(\widehat{p}) \amp = \E\left(\frac{S}{n}\right) \\
|
||||
\amp = \frac{1}{n} \E\left(S\right) \\
|
||||
\amp = \frac{1}{n} (np) \\
|
||||
\amp = p.
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">So <span class="process-math">\(\E(\widehat{p} = p\text{,}\)</span> i.e., the common sense estimator is unbiased.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 110"><a href="#notes-02-10-3-7" title="Copy heading and permalink for Example 110" aria-label="Copy heading and permalink for Example 110">🔗</a></div></article><article class="definition definition-like" id="notes-02-10-3-8"><h4 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">111</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-02-10-3-8-1-1">
|
||||
<div class="para">We perform an experiment and collect data. Let <span class="process-math">\(p\)</span> be an unknown parameter value. The <dfn class="terminology">likelihood function</dfn> is:</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-8-1-1-3">
|
||||
\begin{gather*}
|
||||
\L(p) = \Pr(\text{data} \mid \text{paramater value is } p).
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Definition 111"><a href="#notes-02-10-3-8" title="Copy heading and permalink for Definition 111" aria-label="Copy heading and permalink for Definition 111">🔗</a></div></article><article class="example example-like" id="notes-02-10-3-9"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">112</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-02-10-3-9-1-1">
|
||||
<div class="para">Let <span class="process-math">\(S \sim \Bin(100, p)\text{.}\)</span> Suppose we see 52 heads. Then:</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-9-1-1-2">
|
||||
\begin{align*}
|
||||
\Pr(S = k) \amp = b(k; 100, p) = {100 \choose k}p^k (1 - p)^{100 - k} \\
|
||||
\L(p) \amp = {100 \choose 52}p^{52} (1 - p)^{48}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">In the first line, the variable <span class="process-math">\(k\)</span> represents the data. In the second line, the variable <span class="process-math">\(p\)</span> represents the parameter value. Our goal, given the collected data, is to find the <dfn class="terminology">maximum likelihood estimation</dfn> (MLE) for the parameter value.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="notes-02-10-3-9-1-2">
|
||||
<div class="para">
|
||||
<span class="process-math">\(\L(p)\)</span> is a continuous function over a closed interval <span class="process-math">\(p \in [0, 1]\text{,}\)</span> so we use the Closed Interval Method.</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-9-1-2-3">
|
||||
\begin{align*}
|
||||
\L'(p) \amp = {100 \choose 52}\left[ 52 p^{51} (1 - p)^{48} + p^{52}48 (1-p)^{47}(-1)\right] \\
|
||||
\amp = {100 \choose 52} p^{51} (1-p)^{47}\left[ 52 (1 - p) - 48p \right] \\
|
||||
\amp = {100 \choose 52} p^{51} (1-p)^{47}\left[ 52 - 100p \right]
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Now, we look for critical numbers in the interior of the interval:</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-9-1-2-4">
|
||||
\begin{gather*}
|
||||
\L'(p) = 0 \text{ when } p = \frac{52}{100}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Finally, we test the critical numbers and the endpoints of the interval to find the max:</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-9-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="notes-02-10-3-9-1-3"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">113<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Check Candidate Locations for Max<div class="autopermalink" data-description="Table 113: Check Candidate Locations for Max"><a href="#notes-02-10-3-9-1-3" title="Copy heading and permalink for Table 113: Check Candidate Locations for Max" aria-label="Copy heading and permalink for Table 113: Check Candidate Locations for Max">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\L(p)\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(52/100\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\gt 0\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0\)</span></td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para" id="notes-02-10-3-9-1-4">So <span class="process-math">\(\widehat{p} = \frac{52}{100}\)</span> is the MLE.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-9-1-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para" id="notes-02-10-3-9-1-5">More generally, a similar calculation will show that, with <span class="process-math">\(k\)</span> heads in <span class="process-math">\(n\)</span> flips, the MLE will be <span class="process-math">\(\widehat{p} = \frac{k}{n}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-9-1-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 112"><a href="#notes-02-10-3-9" title="Copy heading and permalink for Example 112" aria-label="Copy heading and permalink for Example 112">🔗</a></div></article><article class="example example-like" id="notes-02-10-3-10"><h4 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">114</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h4>
|
||||
<div class="para logical" id="notes-02-10-3-10-1-1">
|
||||
<div class="para">Suppose we observe a cell, measuring the time <span class="process-math">\(T\)</span> until a toxin molecule leaves the cell. Then <span class="process-math">\(T \sim \Exp(\lambda)\)</span> for some <span class="process-math">\(\lambda\text{,}\)</span> with pdf</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-10-1-1-4">
|
||||
\begin{gather*}
|
||||
f(t) =\lambda e^{-\lambda t}, \lambda \gt 0
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">If we see a toxin molecule leave at 0.3 min, what’s the MLE for <span class="process-math">\(\lambda\text{?}\)</span>
|
||||
</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-10-1-1-6">
|
||||
\begin{gather*}
|
||||
\L(\lambda) = \lambda e^{-0.3 \lambda} \quad \text{(density, not probability)}
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">We want to maximize <span class="process-math">\(\L(\lambda)\)</span> over <span class="process-math">\(\lambda \in (0, \infty)\text{,}\)</span> an open interval. So we’ll use the "Open Interval Method".</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-10-1-1-9">
|
||||
\begin{align*}
|
||||
\L'(\lambda) \amp = e^{-0.3\lambda} + \lambda e^{-0.3\lambda}(-0.3) \\
|
||||
\amp = e^{-0.3\lambda}\left( 1 - 0.3 e^{-0.3\lambda}\right)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Then <span class="process-math">\(\L'(\lambda) = 0\)</span> when <span class="process-math">\(\lambda = \frac{1}{0.3}\approx 3.33\text{.}\)</span> Checking <span class="process-math">\(\lambda\)</span> values to the left and the right:</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-10-1-1-13">
|
||||
\begin{align*}
|
||||
\L'(1) \amp = (+)(+) = (+) \\
|
||||
\L'(10) \amp = (+)(-) = (-)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">So <span class="process-math">\(\L'(\lambda) \gt 0\)</span> (and therefore <span class="process-math">\(\L(\lambda)\)</span> is increasing) on <span class="process-math">\((0, 1/0.3)\text{,}\)</span> and <span class="process-math">\(\L'(\lambda) \lt 0\)</span> (and therefore <span class="process-math">\(\L(\lambda)\)</span> is decreasing) on <span class="process-math">\((1/0.3, \infty)\text{.}\)</span> Now we can conclude that <span class="process-math">\(\widehat{\lambda} = 1/0.3 \approx 3.33\)</span> is the location of a global (and not just local) maximum value.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-10-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para" id="notes-02-10-3-10-1-2">More generally, if the observed time is <span class="process-math">\(t\text{,}\)</span> then the MLE will be <span class="process-math">\(\widehat{\lambda} = \frac{1}{t}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-10-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para" id="notes-02-10-3-10-1-3">What if we had more data points? For example, suppose two toxin molecules leave the cell at <span class="process-math">\(t_1 = 0.3\)</span> min and <span class="process-math">\(t_2 = 0.5\)</span> min?<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-10-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="notes-02-10-3-10-1-4"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">115<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Waiting Times<div class="autopermalink" data-description="Table 115: Waiting Times"><a href="#notes-02-10-3-10-1-4" title="Copy heading and permalink for Table 115: Waiting Times" aria-label="Copy heading and permalink for Table 115: Waiting Times">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines">Molecule</td>
|
||||
<td class="c m b1 r0 l0 t0 lines">Time</td>
|
||||
<td class="c m b1 r0 l0 t0 lines">Rate Estimate</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.3\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/0.3 \approx 3.33\)</span></td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.5\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/0.5 = 2\)</span></td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para logical" id="notes-02-10-3-10-1-5">
|
||||
<div class="para">How do we combine these data points? We could take the average of the rate estimates:</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-10-1-5-1">
|
||||
\begin{gather*}
|
||||
\frac{3.33 + 2}{2} \approx 2.67
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Alternatively, we could average the times first, then create a new rate estimate from the average time:</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-10-1-5-2">
|
||||
\begin{align*}
|
||||
\frac{0.3 + 0.5}{2} \amp = 0.4 \\
|
||||
\frac{1}{0.4} \amp = 2.5
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Both of these make some sense, but let’s do a careful computation to be certain which way is correct (if either of them is!).</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-10-1-5-3">
|
||||
\begin{align*}
|
||||
\L(\lambda) \amp = \left( \lambda e^{-0.3 \lambda} \right)\left( \lambda e^{-0.5 \lambda} \right) \\
|
||||
\amp = \lambda^2 e^{-0.3 \lambda - 0.5 \lambda} \\
|
||||
\amp = \lambda^2 e^{- 0.8 \lambda}
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Using the Open Interval Method:</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-10-1-5-4">
|
||||
\begin{align*}
|
||||
\L'(\lambda) \amp = 2\lambda e^{-0.8 \lambda} + \lambda^2 e^{-0.8 \lambda} (-0.8) \\
|
||||
\amp = \lambda e^{-0.8 \lambda} \left[ 2 - 0.8\lambda \right]
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">So <span class="process-math">\(\L'(\lambda) = 0\)</span> when <span class="process-math">\(\lambda = \frac{2}{0.8} = 2.5\text{.}\)</span> Testing points to the left and right:</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-10-1-5-7">
|
||||
\begin{align*}
|
||||
\L'(1) \amp = (+)(+)(+) = (+) \\
|
||||
\L'(5) \amp = (+)(+)(-) = (-)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">Now <span class="process-math">\(\L'(\lambda) \gt 0\)</span> (<span class="process-math">\(\L(\lambda)\)</span> is increasing) on <span class="process-math">\((0, 2.5)\text{,}\)</span> and <span class="process-math">\(\L'(\lambda) \lt 0\)</span> (<span class="process-math">\(\L(\lambda)\)</span> is decreasing) on <span class="process-math">\((2.5, \infty)\text{.}\)</span> Therefore, the MLE is <span class="process-math">\(\widehat{\lambda} = \frac{2}{0.8} = 2.5\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-10-1-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <div class="para logical" id="notes-02-10-3-10-1-6">
|
||||
<div class="para">Tracing the values <span class="process-math">\(2\)</span> and <span class="process-math">\(0.8\)</span> throughout the calculation, we can see that the value <span class="process-math">\(2\)</span> will generally match the number of waiting times collected, and the value <span class="process-math">\(0.8\)</span> will be the sum of the waiting times. So, generally, with collected waiting times of <span class="process-math">\(t_1, \dotsc, t_n\text{,}\)</span> the MLE will be:</div>
|
||||
<div class="displaymath process-math" id="notes-02-10-3-10-1-6-6">
|
||||
\begin{gather*}
|
||||
\widehat{\lambda} = \frac{n}{t_1 + \dotsb + t_n} = \frac{1}{\text{avg time}}.
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-10-3-10-1-6" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 114"><a href="#notes-02-10-3-10" title="Copy heading and permalink for Example 114" aria-label="Copy heading and permalink for Example 114">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Likelihood"><a href="#notes-02-10-3" title="Copy heading and permalink for Subsection: Likelihood" aria-label="Copy heading and permalink for Subsection: Likelihood">🔗</a></div></section><div class="autopermalink" data-description="Section: Tuesday, Feb 10"><a href="#notes-02-10" title="Copy heading and permalink for Section: Tuesday, Feb 10" aria-label="Copy heading and permalink for Section: Tuesday, Feb 10">🔗</a></div></section></div>
|
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</div>
|
||||
</body>
|
||||
</html>
|
||||
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
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||||
eBookConfig.build_info = "";
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||||
eBookConfig.python3 = null;
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eBookConfig.runestone_version = '7.11.13';
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||||
eBookConfig.runestone_version = '7.11.14';
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eBookConfig.jobehost = '';
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eBookConfig.proxyuri_runs = '';
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eBookConfig.proxyuri_files = '';
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eBookConfig.enable_chatcodes = false;
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<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
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||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
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<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
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||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
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||||
</script>
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||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
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||||
\DeclareMathOperator{\E}{E}
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||||
\DeclareMathOperator{\Var}{Var}
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||||
\DeclareMathOperator{\Cov}{Cov}
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||||
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||||
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||||
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||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
|
||||
+14
-2
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
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||||
eBookConfig.build_info = "";
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||||
eBookConfig.python3 = null;
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||||
eBookConfig.runestone_version = '7.11.13';
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||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
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eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
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|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
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<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
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<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
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||||
</script>
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||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
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||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
|
||||
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
@@ -295,7 +307,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="introduction" id="quiz-02-3-2-1">
|
||||
<div class="para" id="quiz-02-3-2-1-1">Consider the joint distribution for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> below.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<figure class="table table-like" id="quiz-02-3-2-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">91<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution<div class="autopermalink" data-description="Table 91: Joint Distribution"><a href="#quiz-02-3-2-1-2" title="Copy heading and permalink for Table 91: Joint Distribution" aria-label="Copy heading and permalink for Table 91: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<figure class="table table-like" id="quiz-02-3-2-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">116<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution<div class="autopermalink" data-description="Table 116: Joint Distribution"><a href="#quiz-02-3-2-1-2" title="Copy heading and permalink for Table 116: Joint Distribution" aria-label="Copy heading and permalink for Table 116: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r1 l0 t0 lines"></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
|
||||
|
||||
+15
-3
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
@@ -280,7 +292,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="introduction" id="quiz-02-3-2-1">
|
||||
<div class="para" id="quiz-02-3-2-1-1">Consider the joint distribution for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> below.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-02-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<figure class="table table-like" id="quiz-02-3-2-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">91<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution<div class="autopermalink" data-description="Table 91: Joint Distribution"><a href="#quiz-02-3-2-1-2" title="Copy heading and permalink for Table 91: Joint Distribution" aria-label="Copy heading and permalink for Table 91: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<figure class="table table-like" id="quiz-02-3-2-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">116<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution<div class="autopermalink" data-description="Table 116: Joint Distribution"><a href="#quiz-02-3-2-1-2" title="Copy heading and permalink for Table 116: Joint Distribution" aria-label="Copy heading and permalink for Table 116: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r1 l0 t0 lines"></td>
|
||||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
|
||||
|
||||
+16
-4
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -135,7 +135,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
<textarea class="embed-code-textbox" id="embed-code-textbox" readonly aria-label="textbox"><iframe src="https://example.com/embed" width="100%" height="1000"></iframe></textarea><button class="copy-embed-button button" id="copy-embed-button" title="Copy embed code"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Copy</span></button>
|
||||
</div>
|
||||
</div>
|
||||
<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="notes-02-03.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="root-1-2.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><a class="next-button button" href="quiz-01.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
|
||||
<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="notes-02-10.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="up-button button" href="root-1-2.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Up</span></a><a class="next-button button" href="quiz-01.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
|
||||
</div></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N}
|
||||
\newcommand{\Z}{\mathbb Z}
|
||||
\newcommand{\Q}{\mathbb Q}
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter contains-active">
|
||||
@@ -251,7 +263,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
<li><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></li>
|
||||
</ul></nav><div class="autopermalink" data-description="Chapter: Quizzes"><a href="#quizzes" title="Copy heading and permalink for Chapter: Quizzes" aria-label="Copy heading and permalink for Chapter: Quizzes">🔗</a></div></section></div>
|
||||
<div id="ptx-content-footer" class="ptx-content-footer">
|
||||
<a class="previous-button button" href="notes-02-03.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="quiz-01.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
|
||||
<a class="previous-button button" href="notes-02-10.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Prev</span></a><a class="top-button button" href="#" title="Top"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Top</span></a><a class="next-button button" href="quiz-01.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a>
|
||||
</div></main>
|
||||
</div>
|
||||
<div id="ptx-page-footer" class="ptx-page-footer">
|
||||
|
||||
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
@@ -309,7 +321,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</div>
|
||||
<div class="workspace" data-space="2cm"></div>
|
||||
<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">92</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">117</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>
|
||||
@@ -320,7 +332,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-FTC-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Theorem 92: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 92: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 92: 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 117: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 117: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 117: 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>
|
||||
<article class="exercise exercise-like" id="recitation-calculus-review-4-3"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<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>
|
||||
|
||||
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
@@ -294,7 +306,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</div>
|
||||
<div class="workspace" data-space="2cm"></div>
|
||||
<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">92</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">117</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>
|
||||
@@ -305,7 +317,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-FTC-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Theorem 92: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 92: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 92: 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 117: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 117: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 117: 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>
|
||||
<article class="exercise exercise-like" id="recitation-calculus-review-4-3"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<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>
|
||||
|
||||
+14
-2
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
+14
-2
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
+14
-2
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
+14
-2
@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
|
||||
eBookConfig.enableScratchAC = false;
|
||||
eBookConfig.build_info = "";
|
||||
eBookConfig.python3 = null;
|
||||
eBookConfig.runestone_version = '7.11.13';
|
||||
eBookConfig.runestone_version = '7.11.14';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
|
||||
eBookConfig.proxyuri_files = '';
|
||||
eBookConfig.enable_chatcodes = false;
|
||||
</script>
|
||||
<!--*** Runestone Services ***-->
|
||||
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<script src="_static/prefix-runtime.d892923478694492.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1d6e4aba17c2c7f2.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
|
||||
<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.6ec5960970d8b364.css">
|
||||
<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
|
||||
</script>
|
||||
@@ -151,6 +151,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
\DeclareMathOperator{\E}{E}
|
||||
\DeclareMathOperator{\Var}{Var}
|
||||
\DeclareMathOperator{\Cov}{Cov}
|
||||
\newcommand{\L}{\mathcal{L}}
|
||||
\newcommand{\lt}{<}
|
||||
\newcommand{\gt}{>}
|
||||
\newcommand{\amp}{&}
|
||||
@@ -219,6 +220,17 @@ eBookConfig.enable_chatcodes = false;
|
||||
<div class="toc-title-box"><a href="notes-02-03.html" class="internal"><span class="title">Tuesday, Feb 3</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-03.html#notes-02-03-3" class="internal"><span class="title">Variance</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-05.html" class="internal"><span class="title">Thursday, Feb 5</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-3" class="internal"><span class="title">Summary</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-05.html#notes-02-05-4" class="internal"><span class="title">Covariance</span></a></div></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<div class="toc-title-box"><a href="notes-02-10.html" class="internal"><span class="title">Tuesday, Feb 10</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-02-10.html#notes-02-10-3" class="internal"><span class="title">Likelihood</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
|
||||
Reference in New Issue
Block a user