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

follow an ordinary normal distribution,

follows a half-normal distribution.

parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by where

Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if

The cumulative distribution function (CDF) is given by Using the change-of-variables

, the CDF can be written as where erf is the error function, a standard function in many mathematical software packages.

The quantile function (or inverse CDF) is written: where

is the inverse error function The expectation is then given by The variance is given by Since this is proportional to the variance σ2 of X, σ can be seen as a scale parameter of the new distribution.

The differential entropy of the half-normal distribution is exactly one bit less the differential entropy of a zero-mean normal distribution with the same second moment about 0.

This can be understood intuitively since the magnitude operator reduces information by one bit (if the probability distribution at its input is even).

Alternatively, since a half-normal distribution is always positive, the one bit it would take to record whether a standard normal random variable were positive (say, a 1) or negative (say, a 0) is no longer necessary.

Thus, The half-normal distribution is commonly utilized as a prior probability distribution for variance parameters in Bayesian inference applications.

drawn from a half-normal distribution, the unknown parameter

of that distribution can be estimated by the method of maximum likelihood, giving The bias is equal to which yields the bias-corrected maximum likelihood estimator