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 and variance σ², then Y = |X| is half-normally distributed.

The cumulative distribution function (CDF) is given by

$F_{Y}(y;\sigma )=\int _{0}^{y}{\frac {1}{\sigma }}{\sqrt {\frac {2}{\pi }}}\,\exp \left(-{\frac {x^{2}}{2\sigma ^{2}}}\right)\,dx$

Using the change-of-variables $z=x/({\sqrt {2}}\sigma )$, the CDF can be written as

$F_{Y}(y;\sigma )={\frac {2}{\sqrt {\pi }}}\,\int _{0}^{y/({\sqrt {2}}\sigma )}\exp \left(-z^{2}\right)dz={\mbox{erf}}\left({\frac {y}{{\sqrt {2}}\sigma }}\right),$

where erf(x) is the error function, a standard function in many mathematical software packages.

The expectation is then given by

$E(Y)=\sigma {\sqrt {2/\pi }},$

The variance is given by

$\operatorname {Var} (Y)=\sigma ^{2}\left(1-{\frac {2}{\pi }}\right).$

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

The entropy of the half-normal distribution is exactly one bit less the entropy of a zero-mean normal distribution with the same second moment. This can be understood intuitively since the magnitude operator reduces information by one bit (if the probability distribution at its input is even). Thus,

$H(Y)={\frac {1}{2}}\log \left({\frac {\pi \sigma ^{2}}{2}}\right)+{\frac {1}{2}}$

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

• Half-Normal Distribution at MathWorld