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

: $F\_Y(y;\; mu,\; sigma)\; =\; int\_0^y\; frac\{1\}\{sigmasqrt\{2pi\; ,\; exp\; left(\; -frac\{(-x-mu)^2\}\{2sigma^2\}\; ight),\; dx+\; int\_0^\{y\}\; frac\{1\}\{sigmasqrt\{2pi\; ,\; exp\; left(\; -frac\{(x-mu)^2\}\{2sigma^2\}\; ight),\; dx.$

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

: $F\_Y(y;\; mu,\; sigma)\; =\; int\_\{-mu/sigma\}^\{(y-mu)/sigma\}\; frac\{1\}\{sqrt\{2pi\; ,\; exp\; left(-frac\{1\}\{2\}left(z\; +\; frac\{2mu\}\{sigma\}\; ight)^2\; ight)\; dz+\; int\_\{-mu/sigma\}^\{(y-mu)/sigma\}\; frac\{1\}\{sqrt\{2pi\; ,\; exp\; left(\; -frac\{z^2\}\{2\}\; ight)\; dz.$

The expectation is then given by

: $E(y)\; =\; sigma\; sqrt\{2/pi\}\; exp(-mu^2/2sigma^2)\; +\; muleft\; [1-2Phi(-mu/sigma)\; ight]\; ,$

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

The variance is given by

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

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.

References

* cite journal | author=Leone FC, Nottingham RB, Nelson LS | year = 1961
title = The Folded Normal Distribution
journal = Technometrics | volume = 3 | issue = 4 | pages = 543–550
doi = 10.2307/1266560
* cite journal | author=Johnson NL | year = 1962
title = The folded normal distribution: accuracy of the estimation by maximum likelihood
journal = Technometrics | volume = 4 | issue = 2 | pages = 249–256
doi = 10.2307/1266622
* cite journal | author=Nelson LS | year = 1980
title = The Folded Normal Distribution
journal = J Qual Technol | volume = 12 | issue = 4 | pages = 236–238
doi =
* cite journal | author=Elandt RC | year = 1961
title = The folded normal distribution: two methods of estimating parameters from moments
journal = Technometrics | volume = 3 | issue = 4 | pages = 551–562
doi = 10.2307/1266561
* cite journal | author=Lin PC | year = 2005
title = Application of the generalized folded-normal distribution to the process capability measures
journal = Int J Adv Manuf Technol | volume = 26 | pages = 825–830
doi = 10.1007/s00170-003-2043-x