Half-normal distribution

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.

Properties

Using the parametrization of the normal distribution, the probability density function of the half-normal is given by
where.
Alternatively using a scaled precision parametrization, obtained by setting, the probability density function is given by
where.
The cumulative distribution function 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 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 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. 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 or negative is no longer necessary. Thus,

Applications

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

Parameter estimation

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

Related distributions

The distribution is a special case of the folded normal distribution with μ = 0.
It also coincides with a zero-mean normal distribution truncated from below at zero
If Y has a half-normal distribution, then ² has a chi square distribution with 1 degree of freedom, i.e. Y/''σ has a chi distribution with 1 degree of freedom.
The half-normal distribution is a special case of the generalized gamma distribution with d'' = 1, p = 2, a = .
If Y has a half-normal distribution, Y^-2 has a Lévy distribution
The Rayleigh distribution is a moment-tilted and scaled generalization of the half-normal distribution.
Modified half-normal distribution with the pdf on is given as, where denotes the Fox–Wright Psi function.