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

<math>F_Y(y; \mu, \sigma) = \int_0^y \frac{1}{\sigma\sqrt{2\pi}} \, \exp \left( -\frac{(-x-\mu)^2}{2\sigma^2} \right)\, dx

+ \int_0^{y} \frac{1}{\sigma\sqrt{2\pi}} \, \exp \left( -\frac{(x-\mu)^2}{2\sigma^2} \right)\, dx.</math>

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

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

+ \int_{-\mu/\sigma}^{(y-\mu)/\sigma} \frac{1}{\sqrt{2\pi}} \, \exp \left( -\frac{z^2}{2} \right) dz. </math>

The expectation is then given by

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

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

The variance is given by

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

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

• Leone FC, Nottingham RB, Nelson LS (1961). "The Folded Normal Distribution". Technometrics 3 (4): 543-550. doi:10.2307/1266560.
• Johnson NL (1962). "The folded normal distribution: accuracy of the estimation by maximum likelihood". Technometrics 4 (2): 249–256. doi:10.2307/1266622.
• 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. doi:10.2307/1266561.
• Lin PC (2005). "Application of the generalized folded-normal distribution to the process capability measures". Int J Adv Manuf Technol 26: 825-830. doi:10.1007/s00170-003-2043-x.