Half-normal distribution

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

Half-normal distribution
Probability density function
Probability density function of the half-normal distribution '"`UNIQ--postMath-00000001-QINU`"'
Cumulative distribution function
Cumulative distribution function of the half-normal distribution '"`UNIQ--postMath-00000003-QINU`"'
Parameters — (scale)
Ex. kurtosis

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 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.[1][2]

Parameter estimationEdit

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 distributionsEdit

See alsoEdit


  1. ^ Gelman, A. (2006), "Prior distributions for variance parameters in hierarchical models", Bayesian Analysis, 1 (3): 515–534, doi:10.1214/06-ba117a
  2. ^ Röver, C.; Bender, R.; Dias, S.; Schmid, C.H.; Schmidli, H.; Sturtz, S.; Weber, S.; Friede, T. (2020), On weakly informative prior distributions for the heterogeneity parameter in Bayesian random-effects meta-analysis, arXiv:2007.08352

Further readingEdit

External linksEdit

(note that MathWorld uses the parameter