46 lines
2.5 KiB
HTML
46 lines
2.5 KiB
HTML
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<article class="theorem theorem-like"><h2 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">4.2.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h2>
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<div class="para logical">
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<div class="para">Suppose <span class="process-math">\(X_1, X_2, \dotsc, X_n\)</span> are <dfn class="terminology">independent and identically distributed</dfn> (or <dfn class="terminology">iid</dfn>) random variables with finite expected value <span class="process-math">\(\mu\)</span> and finite variance <span class="process-math">\(\sigma^2\text{.}\)</span> Let:</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}
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\end{align*}
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</div>
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<div class="para">Then, for sufficiently large <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{gather*}
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S_n \approx \Norm(n \mu, n \sigma^2) \\
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A_n \approx \Norm\left(\mu, \frac{\sigma^2}{n}\right)
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\end{gather*}
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</div>
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<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Theorem 4.2.1"><a tabindex="-1" href="#thm-CLT" title="Copy heading and permalink for Theorem 4.2.1" aria-label="Copy heading and permalink for Theorem 4.2.1">🔗</a></div>
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</article><span class="incontext"><a class="internal" href="sec-CLT.html#thm-CLT">In Context</a></span>
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