Folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable X with mean μ and variance σ², the random variable Y = |X| has a folded normal distribution. Such a case may be encountered if only the magnitude of some variable is recorded, but not its sign. The distribution is called "folded" because probability mass to the left of x = 0 is folded over by taking the absolute value. In the physics of heat conduction, the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds to having a perfect insulator on a hyperplane through the origin.

Probability density function μ=1, σ=1
Cumulative distribution function μ=1, σ=1
Parameters	μ ∈ R   (location) σ2 > 0   (scale)
Support	x ∈ [0,∞)
PDF	${\displaystyle {\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}+{\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(x+\mu )^{2}}{2\sigma ^{2}}}}}$
CDF	${\displaystyle {\frac {1}{2}}\left[{\mbox{erf}}\left({\frac {x+\mu }{\sigma {\sqrt {2}}}}\right)+{\mbox{erf}}\left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}$
Mean	${\displaystyle \mu _{Y}=\sigma {\sqrt {\tfrac {2}{\pi }}}\,e^{(-\mu ^{2}/2\sigma ^{2})}+\mu \left(1-2\,\Phi (-{\tfrac {\mu }{\sigma }})\right)}$
Variance	${\displaystyle \sigma _{Y}^{2}=\mu ^{2}+\sigma ^{2}-\mu _{Y}^{2}}$