Error function

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In mathematics, the error function is a function associated with the cumulative distribution function of the normal distribution.

The definition is

\operatorname{erf}(x) = \frac{2}{\sqrt\pi} \int_{0}^{x} \exp(-t^2) dt .\,

The complementary error function is defined as

\operatorname{erfc}(x) = 1 - \operatorname{erf}(x) .\,

The probability that a normally distributed random variable X with mean μ and variance σ2 exceeds x is

F(x;\mu,\sigma)=\frac{1}{2} \left[ 1 + \operatorname{erf} \left( \frac{x-\mu}{\sigma\sqrt{2}} \right) \right].
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