Definitions

# 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

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

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

The expectation is then given by

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

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

The variance is given by

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

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

## References

• 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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