Half-normal distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Half-normal distribution

Probability density function ${\displaystyle \sigma =1}$
Cumulative distribution function ${\displaystyle \sigma =1}$
Parameters	${\displaystyle \sigma >0}$ — (scale)
Support	${\displaystyle x\in [0,\infty )}$
PDF	${\displaystyle f(x;\sigma )={\frac {\sqrt {2}}{\sigma {\sqrt {\pi }}}}\exp \left(-{\frac {x^{2}}{2\sigma ^{2}}}\right)\quad x>0}$
CDF	${\displaystyle F(x;\sigma )=\operatorname {erf} \left({\frac {x}{\sigma {\sqrt {2}}}}\right)}$
Quantile	${\displaystyle Q(F;\sigma )=\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(F)}$
Mean	${\displaystyle {\frac {\sigma {\sqrt {2}}}{\sqrt {\pi }}}}$
Median	${\displaystyle \sigma {\sqrt {2}}\operatorname {erf} ^{-1}(1/2)}$
Mode	${\displaystyle 0}$
Variance	${\displaystyle \sigma ^{2}\left(1-{\frac {2}{\pi }}\right)}$
Skewness	${\displaystyle {\frac {{\sqrt {2}}(4-\pi )}{(\pi -2)^{3/2}}}\approx 0.9952717}$
Ex. kurtosis	${\displaystyle {\frac {8(\pi -3)}{(\pi -2)^{2}}}\approx 0.869177}$
Entropy	${\displaystyle {\frac {1}{2}}\log _{2}\left(2\pi e\sigma ^{2}\right)-1}$

Let ${\displaystyle X}$ follow an ordinary normal distribution, ${\displaystyle N(0,\sigma ^{2})}$. Then, ${\displaystyle Y=|X|}$ follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero.