In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.
Let follow an ordinary normal distribution, . Then, follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero.
Using the 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 is near zero), obtained by setting , the probability density function is given by
where .
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 and is the inverse error function
The expectation is then given by
The variance is given by
Since this is proportional to the variance ÃÂ<sup>2</sup> of X, ÃÂ 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.
Given numbers 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