folded over

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} 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(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
+ 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(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, σ2, of X can be seen to location and scale parameters of the new distribution.

Related distributions


  • 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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