# Half-normal distribution

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

Probability density function | |||

Cumulative distribution function | |||

Parameters | — (scale) | ||
---|---|---|---|

Support | |||

CDF | |||

Quantile | |||

Mean | |||

Median | |||

Mode | |||

Variance | |||

Skewness | |||

Ex. kurtosis | |||

Entropy |

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 (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 σ^{2} 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 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,

## 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 (see truncated normal distribution)
- If
*Y*has a half-normal distribution, then (*Y*/*σ*)^{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*= .

## See also

## References

## Further reading

- Leone, F. C.; Nelson, L. S.; Nottingham, R. B. (1961), "The folded normal distribution",
*Technometrics*,**3**(4): 543–550, doi:10.2307/1266560, JSTOR 1266560