41 lines
3.3 KiB
HTML
41 lines
3.3 KiB
HTML
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<article class="definition definition-like"><h2 class="heading">
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<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>
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</h2>
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<div class="para logical">
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<div class="para">Consider the parametrized family of functions:</div>
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<div class="displaymath process-math">
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\begin{gather*}
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f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{-(x-\mu)^2}{2\sigma^2}}.
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\end{gather*}
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<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>
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</div>
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<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#def-normal-distribution-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="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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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" aria-hidden="true" data-description="Definition 2.2.8"><a tabindex="-1" 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>
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</article><span class="incontext"><a class="internal" href="sec-Continuous-RVs.html#def-normal-distribution">In Context</a></span>
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