91 lines
5.7 KiB
HTML
Executable File
91 lines
5.7 KiB
HTML
Executable File
<!DOCTYPE html>
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<html lang="en-US" dir="ltr">
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<meta name="robots" content="noindex, nofollow">
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</head>
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<body class="ignore-math">
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<article class="example example-like"><h4 class="heading">
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<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h4>
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<div class="para">Suppose we roll a fair 6-sided die two times, and let <span class="process-math">\(S\)</span> be the sum of the rolls. It’s straightforward to check the distribution for <span class="process-math">\(S\)</span> shown below. We’ve also included a column containing the products <span class="process-math">\(k \Pr(S = k)\text{,}\)</span> which must be summed up to find <span class="process-math">\(\E(S)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#example-EV-sum-2-dice-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"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3.1.4<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 3.1.4: "><a href="#example-EV-sum-2-dice-1-2" title="Copy heading and permalink for Table 3.1.4: " aria-label="Copy heading and permalink for Table 3.1.4: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
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<tr class="header-horizontal">
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<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k\)</span></th>
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<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(S = k)\)</span></th>
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<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(k \cdot \Pr(S = k)\)</span></th>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">2</td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/36\)</span></td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">3</td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6/36\)</span></td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">4</td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3/36\)</span></td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(12/36\)</span></td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">5</td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4/36\)</span></td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(20/36\)</span></td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">6</td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5/36\)</span></td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(30/36\)</span></td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">7</td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6/36\)</span></td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(42/36\)</span></td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">8</td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5/36\)</span></td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(40/36\)</span></td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">9</td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4/36\)</span></td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(36/36\)</span></td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">10</td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3/36\)</span></td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(30/36\)</span></td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">11</td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/36\)</span></td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(22/36\)</span></td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">12</td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/36\)</span></td>
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<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(12/36\)</span></td>
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</tr>
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</table></div></figure> <div class="para">The sum of the third column gives <span class="process-math">\(\E(S) = \frac{252}{36} = 7\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#example-EV-sum-2-dice-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Example 3.1.3"><a href="#example-EV-sum-2-dice" title="Copy heading and permalink for Example 3.1.3" aria-label="Copy heading and permalink for Example 3.1.3">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-EV-sum-2-dice">in-context</a></span>
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</body>
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