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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 σ^{2}, 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.## Related distributions

## References

The cumulative distribution function (CDF) is given by

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

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

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

The expectation is then given by

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

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

The variance is given by

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

Both the mean, μ, and the variance, σ^{2}, 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.

- Leone FC, Nottingham RB, Nelson LS (1961). "The Folded Normal Distribution".
*Technometrics*3 (4): 543–550. - Johnson NL (1962). "The folded normal distribution: accuracy of the estimation by maximum likelihood".
*Technometrics*4 (2): 249–256. - Nelson LS (1980). "The Folded Normal Distribution".
*J Qual Technol*12 (4): 236–238. - Elandt RC (1961). "The folded normal distribution: two methods of estimating parameters from moments".
*Technometrics*3 (4): 551–562. - Lin PC (2005). "Application of the generalized folded-normal distribution to the process capability measures".
*Int J Adv Manuf Technol*26 825–830.

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Last updated on Monday June 02, 2008 at 04:42:19 PDT (GMT -0700)

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Last updated on Monday June 02, 2008 at 04:42:19 PDT (GMT -0700)

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