Half-normal distribution

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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

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

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

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

The expectation is then given by

$ E(y) = \sigma \sqrt{2/\pi}, $

The variance is given by

$ \operatorname{Var}(y) = \sigma^2\left(1 - \frac{2}{\pi}\right). $

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

Related distributions

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