Folded normal distribution

Probability density function μ = 1,σ = 1
Cumulative distribution function μ = 1,σ = 1
Parameters	μ ∈ R — (location) σ2 > 0 — (scale)
Support	x ∈ [0,$\infty$)
PDF	(see article)
CDF	(see article)
Mean	(see article)
Variance	(see article)

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 probability density function (PDF) is given by

$f(x;\mu,\sigma)= \frac{1}{\sigma\sqrt{2\pi}} \, \exp \left( -\frac{(-x-\mu)^2}{2\sigma^2} \right) + \frac{1}{\sigma\sqrt{2\pi}} \, \exp \left( -\frac{(x-\mu)^2}{2\sigma^2} \right)\qquad(x \ge 0\,)$

The cumulative distribution function (CDF) is given by

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

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

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

Alternatively, using the change of variables $z = -(x+\mu)/\sqrt{2}\sigma$ in the first integral and $z = (x-\mu)/\sqrt{2}\sigma$ in the second integral, one can show that

$F_Y(y; \mu, \sigma) = \frac{1}{2}\left[ \mbox{erf}\left(\frac{y+\mu}{\sqrt{2}\sigma}\right) + \mbox{erf}\left(\frac{y-\mu}{\sqrt{2}\sigma}\right)\right],$

where erf(x) is the error function, which is a standard function in many mathematical software packages. This expression reduces to the CDF of the half-normal distribution when μ = 0.

The expectation is then given by

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

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

The variance is given by

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

Both the mean, μ, and the variance, σ², of X can be seen as the 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 (Technometrics, Vol. 3, No. 4) 3 (4): 543–550. doi:10.2307/1266560. JSTOR 1266560. 
• Johnson NL (1962). "The folded normal distribution: accuracy of the estimation by maximum likelihood". Technometrics (Technometrics, Vol. 4, No. 2) 4 (2): 249–256. doi:10.2307/1266622. JSTOR 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 (Technometrics, Vol. 3, No. 4) 3 (4): 551–562. doi:10.2307/1266561. JSTOR 1266561. 
• Lin PC (2005). "Application of the generalized folded-normal distribution to the process capability measures". Int J Adv Manuf Technol 26 (7–8): 825–830. doi:10.1007/s00170-003-2043-x.