45 lines
2.4 KiB
HTML
Executable File
45 lines
2.4 KiB
HTML
Executable File
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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="theorem theorem-like"><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">117</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">Suppose <span class="process-math">\(X_1, \dotsc, X_n\)</span> are <dfn class="terminology">independent and identically distributed</dfn> (or <dfn class="terminology">i.i.d.</dfn>) with finite expected value <span class="process-math">\(\mu\)</span> and finite variance <span class="process-math">\(\sigma^2\text{.}\)</span> Define:</div>
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<div class="displaymath process-math">
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\begin{align*}
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S_n \amp = X_1 + X_2 + \dotsb + X_n \\
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A_n \amp = \frac{X_1 + X_2 + \dotsb + X_n}{n} = \frac{S_n}{n}
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\end{align*}
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</div>
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<div class="para">Then, for large enough <span class="process-math">\(n\text{:}\)</span>
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</div>
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<div class="displaymath process-math">
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\begin{align*}
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S_n \amp \approx \Norm(n\mu, n\sigma^2) \\
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A_n \amp \approx \Norm\left(\mu, \frac{\sigma^2}{n}\right)
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\end{align*}
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</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#thm-CLT-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="Theorem 117"><a href="#thm-CLT" title="Copy heading and permalink for Theorem 117" aria-label="Copy heading and permalink for Theorem 117">🔗</a></div></article><span class="incontext"><a class="internal" href="notes-02-19.html#thm-CLT">in-context</a></span>
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</body>
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</html>
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