44 lines
2.6 KiB
HTML
Executable File
44 lines
2.6 KiB
HTML
Executable File
<!DOCTYPE html>
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<html lang="en-US" dir="ltr">
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<!--******************************************-->
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<!--* Authored with PreTeXt *-->
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<!--* pretextbook.org *-->
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<!--* Theme: boulder *-->
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<!--* Generated from PreTeXt source *-->
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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">3.1.2</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">
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<div class="para">If <span class="process-math">\(X\)</span> takes on the values <span class="process-math">\(x_1, x_2, \dotsc, x_n\)</span> uniformly (i.e., each value having probability <span class="process-math">\(\frac{1}{n}\)</span>), then the expected value is the usual average:</div>
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<div class="displaymath process-math">
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\begin{align*}
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\E(X) \amp = x_1 \cdot \frac{1}{n} + x_2\cdot \frac{1}{n} + \dotsb + x_n \cdot \frac{1}{n} \\
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\amp = \frac{x_1 + x_2 + \dotsb + x_n}{n}
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\end{align*}
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</div>
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<div class="para">For example, the expected value of a fair 6-sided die roll <span class="process-math">\(R\)</span> is:</div>
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<div class="displaymath process-math">
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\begin{align*}
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\E(R) \amp = 1 \left(\frac{1}{6}\right) + 2 \left(\frac{1}{6}\right) + 3 \left(\frac{1}{6}\right) + 4 \left(\frac{1}{6}\right) + 5 \left(\frac{1}{6}\right) + 6 \left(\frac{1}{6}\right) \\
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\amp = \frac{1 + 2 + \dotsb + 6}{6} = \frac{21}{6} = \frac{7}{2}.
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\end{align*}
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</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#example-EV-die-1-1" 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 3.1.2"><a href="#example-EV-die" title="Copy heading and permalink for Example 3.1.2" aria-label="Copy heading and permalink for Example 3.1.2">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Expected-Value.html#example-EV-die">in-context</a></span>
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</body>
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</html>
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