Error function

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

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