Half-normal distribution

The half-normal distribution is the probability distribution of the absolute value of a random variable that is normally distributed with expected value 0 and variance σ². I.e. if X is normally distributed with mean 0, then Y = |X| is half-normally distributed.

The cumulative distribution function (CDF) is given by

<math>F_Y(y; \sigma) = \int_0^y \frac{1}{\sigma}\sqrt{\frac{2}{\pi}} \, \exp \left( -\frac{x^2}{2\sigma^2} \right)\, dx</math>

Using the change-of-variables z = x/σ, the CDF can be written as

<math>F_Y(y; \sigma) = \int_0^{y/\sigma} \sqrt{\frac{2}{\pi}} \, \exp \left(-\frac{z^2}{2}\right) dz.

</math>

The expectation is then given by

<math>E(y) = \sigma \sqrt{2/\pi},</math>

The variance is given by

<math>\operatorname{Var}(y) = \sigma^2\left(1 - \frac{2}{\pi}\right). </math>

Since this is proportional to the variance σ² of X, σ can be seen as a scale parameter of the new distribution.

Related distributions

• The distribution is a special case of the folded normal distribution with μ = 0.
• (Y/σ) has a chi distribution with 1 degree of freedom.

External links

• Half-Normal Distribution at MathWorld