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Half-normal distribution
Probability density function ${\displaystyle \sigma =1}$
Cumulative distribution function ${\displaystyle \sigma =1}$
Parameters	${\displaystyle \sigma >0}$ — (scale)
Support	${\displaystyle x\in [0,\mathrm{\infty })}$
PDF	${\displaystyle f(x;\sigma )=\frac{\sqrt{2}}{\sigma \sqrt{\pi }}\exp \left(-\frac{x^2}{2{\sigma }^2}\right)x>0}$
CDF	${\displaystyle F(x;\sigma )=\mathrm{erf}\left(\frac{x}{\sigma \sqrt{2}}\right)}$
Quantile	${\displaystyle Q(F;\sigma )=\sigma \sqrt{2}{\mathrm{erf}}^{-1}(F)}$
Mean	${\displaystyle \frac{\sigma \sqrt{2}}{\sqrt{\pi }}\approx 0.797885\sigma }$
Median	${\displaystyle \sigma \sqrt{2}{\mathrm{erf}}^{-1}(1/2)\approx 0.674490\sigma }$
Mode	${\displaystyle 0}$
Variance	${\displaystyle {\sigma }^2\left(1-\frac{2}{\pi }\right)}$
Skewness	${\displaystyle \frac{\sqrt{2}(4-\pi )}{(\pi -2{)}^{3/2}}\approx 0.9952717}$
Excess kurtosis	${\displaystyle \frac{8(\pi -3)}{(\pi -2{)}^2}\approx 0.869177}$
Entropy	${\displaystyle \frac{1}{2}{\log }_2\left(2\pi e{\sigma }^2\right)-1}$
MGF	${\displaystyle \exp \left(\frac{{\sigma }^2{t}^2}{2}\right)\mathrm{erfc}\left(-\frac{\sigma t}{\sqrt{2}}\right)}$
CF	${\displaystyle w\left(\frac{\sigma t}{\sqrt{2}}\right)}$ where${\displaystyle w(x)}$ is theFaddeeva function

In probability theory and statistics, thehalf-normal distribution is a special case of thefolded normal distribution.

Let${\displaystyle X}$ follow an ordinarynormal distribution,${\displaystyle N(0,{\sigma }^2)}$. Then,${\displaystyle Y=|X|}$ follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero.

Using the${\displaystyle \sigma }$ parametrization of the normal distribution, theprobability density function (PDF) of the half-normal is given by

${\displaystyle f_Y(y;\sigma )=\frac{\sqrt{2}}{\sigma \sqrt{\pi }}\exp \left(-\frac{y^2}{2{\sigma }^2}\right)y\ge 0,}$

where${\displaystyle E[Y]=\mu =\frac{\sigma \sqrt{2}}{\sqrt{\pi }}}$.

Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if${\displaystyle \sigma }$ is near zero), obtained by setting${\displaystyle \theta =\frac{\sqrt{\pi }}{\sigma \sqrt{2}}}$, theprobability density function is given by

${\displaystyle f_Y(y;\theta )=\frac{2\theta }{\pi }\exp \left(-\frac{y^2{\theta }^2}{\pi }\right)y\ge 0,}$

where${\displaystyle E[Y]=\mu =\frac{1}{\theta }}$.

Thecumulative distribution function (CDF) is given by

${\displaystyle F_Y(y;\sigma )={\int }_0^y\frac{1}{\sigma }\sqrt{\frac{2}{\pi }}\exp \left(-\frac{x^2}{2{\sigma }^2}\right)dx}$

Using the change-of-variables${\displaystyle z=x/(\sqrt{2}\sigma )}$, the CDF can be written as

${\displaystyle F_Y(y;\sigma )=\frac{2}{\sqrt{\pi }}{\int }_0^{y/(\sqrt{2}\sigma )}\exp \left(-z^2\right)dz=\mathrm{erf}\left(\frac{y}{\sqrt{2}\sigma }\right),}$

where erf is theerror function, a standard function in many mathematical software packages.

The quantile function (or inverse CDF) is written:

${\displaystyle Q(F;\sigma )=\sigma \sqrt{2}{\mathrm{erf}}^{-1}(F)}$

where${\displaystyle 0\le F\le 1}$ and${\displaystyle {\mathrm{erf}}^{-1}}$ is theinverse error function

The expectation is then given by

${\displaystyle E[Y]=\sigma \sqrt{2/\pi },}$

The variance is given by

${\displaystyle \mathrm{var}(Y)={\sigma }^2\left(1-\frac{2}{\pi }\right).}$

Since this is proportional to the variance σ² ofX,σ can be seen as ascale parameter of the new distribution.

The differential entropy of the half-normal distribution is exactly one bit less the differential entropy of a zero-mean normal distribution with the same second moment about 0. This can be understood intuitively since the magnitude operator reduces information by one bit (if the probability distribution at its input is even). Alternatively, since a half-normal distribution is always positive, the one bit it would take to record whether a standard normal random variable were positive (say, a 1) or negative (say, a 0) is no longer necessary. Thus,

${\displaystyle h(Y)=\frac{1}{2}{\log }_2\left(\frac{\pi e{\sigma }^2}{2}\right)=\frac{1}{2}{\log }_2\left(2\pi e{\sigma }^2\right)-1.}$

The half-normal distribution is commonly utilized as aprior probability distribution forvariance parameters inBayesian inference applications.^[1][2]

Given numbers${\displaystyle \{x_i{\}}_{i=1}^n}$ drawn from a half-normal distribution, the unknown parameter${\displaystyle \sigma }$ of that distribution can be estimated by the method ofmaximum likelihood, giving

${\displaystyle \hat{\sigma }=\sqrt{\frac{1}{n}\sum _{i=1}^n{x}_i^2}}$

The bias is equal to

${\displaystyle b\equiv \mathrm{E}[({\hat{\sigma }}_{\mathrm{m}\mathrm{l}\mathrm{e}}-\sigma )]=-\frac{\sigma }{4n}}$

which yields thebias-corrected maximum likelihood estimator

${\displaystyle {\hat{\sigma }}_{\text{mle}}^{\ast }={\hat{\sigma }}_{\text{mle}}-\hat{b}.}$

• The distribution is a special case of thefolded normal distribution withμ = 0.
• It also coincides with a zero-mean normal distribution truncated from below at zero (seetruncated normal distribution)
• IfY has a half-normal distribution, then (Y/σ)² has achi square distribution with 1 degree of freedom, i.e.Y/σ has achi distribution with 1 degree of freedom.
• The half-normal distribution is a special case of thegeneralized gamma distribution withd = 1,p = 2,a = ${\displaystyle \sqrt{2}\sigma }$.
• IfY has a half-normal distribution,Y ^-2 has aLévy distribution
• TheRayleigh distribution is a moment-tilted and scaled generalization of the half-normal distribution.
• Modified half-normal distribution^[3] with the pdf on${\displaystyle (0,\mathrm{\infty })}$ is given as${\displaystyle f(x)=\frac{2{\beta }^{\frac{\alpha }{2}}{x}^{\alpha -1}\exp (-\beta x^2+\gamma x)}{\mathrm{\Psi }\left(\frac{\alpha }{2},\frac{\gamma }{\sqrt{\beta }}\right)}}$, where${\displaystyle \mathrm{\Psi }(\alpha ,z)={}_1{\mathrm{\Psi }}_1\left(\begin{array}{c}\left(\alpha ,\frac{1}{2}\right)\\ (1,0)\end{array};z\right)}$ denotes theFox–Wright Psi function.

• Half-t distribution
• Truncated normal distribution
• Folded normal distribution
• Rectified Gaussian distribution

• Leone, F. C.; Nelson, L. S.; Nottingham, R. B. (1961), "The folded normal distribution",Technometrics,3 (4):543–550,doi:10.2307/1266560,hdl:2027/mdp.39015095248541,JSTOR 1266560

• Half-Normal Distribution atMathWorld

(note that MathWorld uses the parameter${\displaystyle \theta =\frac{1}{\sigma }\sqrt{\pi /2}}$

• Continuous distributions
• Normal distribution

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