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,∞)
PDF	$\frac{1}{\sigma\sqrt{2\pi}} \, e^{ -\frac{(x-\mu)^2}{2\sigma^2} } + \frac{1}{\sigma\sqrt{2\pi}} \, e^{ -\frac{(x+\mu)^2}{2\sigma^2} }$
CDF	$\frac{1}{2}\left[ \mbox{erf}\left(\frac{x+\mu}{\sigma\sqrt{2}}\right) + \mbox{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right]$
Mean	$\mu_Y = \sigma \sqrt{\tfrac{2}{\pi}} \, e^{(-\mu^2/2\sigma^2)} + \mu \left(1 - 2\,\Phi(\tfrac{-\mu}{\sigma}) \right)$
Variance	$\sigma_Y^2 = \mu^2 + \sigma^2 - \mu_Y^2$

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. In the physics of heat conduction, the folded normal distribution is a fundamental solution of the heat equation on the upper plane (i.e. a heat kernel).

The probability density function (PDF) is given by

$f_Y(x;\mu,\sigma)= \frac{1}{\sigma\sqrt{2\pi}} \, e^{ -\frac{(x-\mu)^2}{2\sigma^2} } + \frac{1}{\sigma\sqrt{2\pi}} \, e^{ -\frac{(x+\mu)^2}{2\sigma^2} }$

for x≥0, and 0 everywhere else. It follows that the cumulative distribution function (CDF) is given by:

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

for x≥0, where erf() is the error function. This expression reduces to the CDF of the half-normal distribution when μ = 0.

The mean of the folded distribution is then

$\mu_Y = \sigma \sqrt{\frac{2}{\pi}} \,\, \exp\left(\frac{-\mu^2}{2\sigma^2}\right) - \mu \, \mbox{erf}\left(\frac{-\mu}{\sqrt{2} \sigma}\right),$

The variance then is expressed easily in terms of the mean:

$\sigma_Y^2 = \mu^2 + \sigma^2 - \mu_Y^2.$

Both the mean (μ) and variance (σ²) of X in the original normal distribution can be interpreted as the location and scale parameters of Y in the folded distribution.

Differential equations

The PDF of the folded normal distribution can also be defined by the system of differential equations

$\begin{cases} \sigma^4 f''(x) + 2\sigma^2 x f'(x) + \left(-\mu ^2+\sigma^2+x^2\right) f(x) = 0 \\ f(0) = \sqrt{2/\pi} \, \frac{1}{\sigma} \, e^{-\frac{\mu^2}{2\sigma^2}} \\ f'(0) = 0 \end{cases}$

Related distributions

• When μ = 0, the distribution of Y is a half-normal distribution.
• The random variable (Y/σ)² has a noncentral chi-squared distribution with 1 degree of freedom and noncentrality equal to (μ/σ)².

See also

• Folded cumulative distribution

External links

• Virtual Laboratories: The Folded Normal Distribution