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

${\displaystyle \mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}{\displaystyle \underset{0}{\overset{x}{\int }}}\exp (-t^2)dt.}$

The complementary error function is defined as

${\displaystyle \mathrm{erfc}(x)=1-\mathrm{erf}(x).}$

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

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