Half-normal distribution

The half-normal distribution is the probability distribution of the absolute value of a random variable that is normally distributed with expected value 0 and variance σ². I.e. if "X" is normally distributed with mean 0, then "Y" = |"X"| is half-normally distributed.

The cumulative distribution function (CDF) is given by

: $F\_Y(y;\; sigma)\; =\; int\_0^y\; frac\{1\}\{sigma\}sqrt\{frac\{2\}\{pi\; ,\; exp\; left(\; -frac\{x^2\}\{2sigma^2\}\; ight),\; dx$

Using the change-of-variables z = "x"/σ, the CDF can be written as

: $F\_Y(y;\; sigma)\; =\; int\_0^\{y/sigma\}\; sqrt\{frac\{2\}\{pi\; ,\; exp\; left(-frac\{z^2\}\{2\}\; ight)\; dz.$

The expectation is then given by

: $E(y)\; =\; sigma\; sqrt\{2/pi\},$

The variance is given by

: $operatorname\{Var\}(y)\; =\; sigma^2left(1\; -\; frac\{2\}\{pi\}\; ight).$

Since this is proportional to the variance σ² of "X", σ can be seen as a scale parameter of the new distribution.

Related distributions

* The distribution is a special case of the folded normal distribution with μ = 0.
* ("Y"/σ) has a chi distribution with 1 degree of freedom.

External links

* [http://mathworld.wolfram.com/Half-NormalDistribution.html Half-Normal Distribution] at MathWorld

References