Folded normal distribution

The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable X with mean μ and variance σ2, the random variable Y = |X| has a folded normal distribution. Such a case may be encountered if only the magnitude of some variable is recorded, but not its sign. The distribution is called Folded because probability mass to the left of the x = 0 is "folded" over by taking the absolute value.

The cumulative distribution function (CDF) is given by


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

+ \int_0^{y} \frac{1}{\sigma\sqrt{2\pi}} \, \exp \left( -\frac{(x-\mu)^2}{2\sigma^2} \right)\, dx.$$

Using the change-of-variables z = (x &minus; μ)/σ, the CDF can be written as


 * $$F_Y(y; \mu, \sigma) = \int_{-\mu/\sigma}^{(y-\mu)/\sigma} \frac{1}{\sqrt{2\pi}} \, \exp \left(-\frac{1}{2}\left(z + \frac{2\mu}{\sigma}\right)^2\right) dz

+ \int_{-\mu/\sigma}^{(y-\mu)/\sigma} \frac{1}{\sqrt{2\pi}} \, \exp \left( -\frac{z^2}{2} \right) dz. $$

The expectation is then given by


 * $$E(y) = \sigma \sqrt{2/\pi} \exp(-\mu^2/2\sigma^2) + \mu\left[1-2\Phi(-\mu/\sigma)\right],$$

where Φ(•) denotes the cumulative distribution function of a standard normal distribution.

The variance is given by


 * $$\operatorname{Var}(y) = \mu^2 + \sigma^2 - \left\{ \sigma \sqrt{2/\pi} \exp(-\mu^2/2\sigma^2) + \mu\left[1-2\Phi(-\mu/\sigma)\right] \right\}^2. $$

Both the mean, μ, and the variance, σ2, of X can be seen to location and scale parameters of the new distribution.

Related distributions

 * When μ = 0, the distribution of Y is a half-normal distribution.
 * (Y/σ) has a noncentral chi distribution with 1 degree of freedom and noncentrality equal to μ/σ.