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knowl/xref/def-normal-distribution.html
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<article class="definition definition-like"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.8</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical">
<div class="para">Consider the parametrized family of functions:</div>
<div class="displaymath process-math">
\begin{gather*}
f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{-(x-\mu)^2}{2\sigma^2}}.
\end{gather*}
</div>
<div class="para">The function <span class="process-math">\(f(x; \mu, \sigma^2)\)</span> is called a <dfn class="terminology">normal density function</dfn> with parameters <span class="process-math">\(\mu\)</span> and <span class="process-math">\(\sigma^2\text{.}\)</span> A continuous random variable <span class="process-math">\(X\)</span> with pdf <span class="process-math">\(f(x; \mu, \sigma^2)\)</span> is said to have a <dfn class="terminology">normal distribution</dfn>. We’ll write <span class="process-math">\(X \sim \Norm(\mu, \sigma^2)\text{.}\)</span>
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#def-normal-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para">In the specific case that <span class="process-math">\(\mu = 0\)</span> and <span class="process-math">\(\sigma^2 = 1\text{,}\)</span> we call the resulting distribution the <dfn class="terminology">standard normal distribution</dfn>. We’ll use the notation <span class="process-math">\(\phi(x)\)</span> for the standard normal density function, and <span class="process-math">\(\Phi(x)\)</span> for the standard normal cumulative distribution function.<div class="autopermalink" data-description="Paragraph"><a href="#def-normal-distribution-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Definition 2.2.8"><a href="#def-normal-distribution" title="Copy heading and permalink for Definition 2.2.8" aria-label="Copy heading and permalink for Definition 2.2.8">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Continuous-RVs.html#def-normal-distribution">in-context</a></span>
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knowl/xref/thm-CLT.html
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<article class="theorem theorem-like"><h3 class="heading">
<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>
</h3>
<div class="para logical">
<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>
<div class="displaymath process-math">
\begin{align*}
S_n \amp = X_1 + X_2 + \dotsb + X_n \\
A_n \amp = \frac{X_1 + X_2 + \dotsb + X_n}{n}
\end{align*}
</div>
<div class="para">Then, for sufficiently large <span class="process-math">\(n\text{:}\)</span>
</div>
<div class="displaymath process-math">
\begin{gather*}
S_n \approx \Norm(n \mu, n \sigma^2) \\
A_n \approx \Norm\left(\mu, \frac{\sigma^2}{n}\right)
\end{gather*}
</div>
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<div class="autopermalink" data-description="Theorem 4.2.1"><a 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></article><span class="incontext"><a class="internal" href="sec-CLT.html#thm-CLT">in-context</a></span>
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