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
Using the change-of-variables z = (x − μ)/σ, the CDF can be written as
The expectation is then given by
where Φ(•) denotes the cumulative distribution function of a standard normal distribution.
The variance is given by
Both the mean, μ, and the variance, σ2, of X can be seen to location and scale parameters of the new distribution.
- 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.
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