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<h1>
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Normal Distribution
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</h1>
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<time class="published" datetime="2018-11-12T10:21:00+01:00">
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12 novembre 2018
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</time>
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</li>
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<li>Mathematics</li>
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<li>Statistics</li>
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<h2>Normal Distribution</h2>
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<p>The probability density of normal distribution is:
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</p>
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<div class="math">$$\mathcal{N}(\mu,\sigma^2)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$</div>
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<p>where,
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* <span class="math">\(\mu\)</span> is the mean (or expectation) of the distribution. It is also equal to median and mode of the distribution.
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* <span class="math">\(\sigma^2\)</span> is the variance.
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* <span class="math">\(\sigma\)</span> is the standard deviation. </p>
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<h2>Standard Normal Distribution</h2>
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<p>If <span class="math">\(\mu=0\)</span> and <span class="math">\(\sigma=1\)</span>, then the normal distribution is known as standard normal distribution:
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</p>
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<div class="math">$$\phi(x)=\frac{e^{-\frac{x^2}{2}}}{\sigma\sqrt{2\pi}}$$</div>
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<p>Every normal distribution can be represented as standard normal distribution:</p>
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<div class="math">$$\mathcal{N}(\mu,\sigma^2)=\frac{1}{\sigma}\phi(\frac{x-\mu}{\sigma})$$</div>
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<h2>Cumulative Probability</h2>
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<p>Consider a real-valued random variable, <span class="math">\(X\)</span>. The cumulative distribution function of <span class="math">\(X\)</span> (or just the distribution function of <span class="math">\(X\)</span>) evaluated at <span class="math">\(x\)</span> is the probability that <span class="math">\(X\)</span> will take a value less than or equal to <span class="math">\(x\)</span>:</p>
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<div class="math">$$F_X(x)=P(X\leq x)$$</div>
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<p>
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also,
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</p>
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<div class="math">$$P(a\leq X\leq b)=P(a\lt X\lt b)=F_X(b)-F_X(a)$$</div>
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<p>the cumulative distribution function for a function with normal distribution is:
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</p>
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<div class="math">$$\Phi(x)=\frac{1}{2}\left(1+erf\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right)$$</div>
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<p>
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where <span class="math">\(erf\)</span> is the error function:
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</p>
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<div class="math">$$erf(z)=\frac{2}{\sqrt{\pi}}\int_0^ze^{-x^2}dx$$</div>
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