Half-normal distribution

Half-normal distribution
Parameters	$\sigma ^{2}>0$ — (scale)
Support	$x\in[0,\infty)$
PDF	$f(x;\sigma )={\frac {{\sqrt {2}}}{\sigma {\sqrt {\pi }}}}\exp \left(-{\frac {x^{2}}{2\sigma ^{2}}}\right)\quad x>0$
CDF	$F(x;\sigma )=\operatorname {erf} \left({\frac {x}{\sigma {\sqrt {2}}}}\right)$
Quantile	$Q(F;\sigma )=\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(F)$
Mean	${\frac {\sigma {\sqrt {2}}}{{\sqrt {\pi }}}}$
Median	$\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(1/2)$
Variance	$\sigma ^{2}\left(1-{\frac {2}{\pi }}\right)$
Entropy	${\frac {1}{2}}\log \left({\frac {\pi \sigma ^{2}}{2}}\right)+{\frac {1}{2}}$

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

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

Properties

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

f_{Y}(y;\sigma )={\frac {\sqrt {2}}{\sigma {\sqrt {\pi }}}}\exp \left(-{\frac {y^{2}}{2\sigma ^{2}}}\right)\quad y\geq 0,

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

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

f_{Y}(y;\theta )={\frac {2\theta }{\pi }}\exp \left(-{\frac {y^{2}\theta ^{2}}{\pi }}\right)\quad y\geq 0,

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

The cumulative distribution function (CDF) is given by

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 $z=x/({\sqrt {2}}\sigma )$ , the CDF can be written as

F_{Y}(y;\sigma )={\frac {2}{\sqrt {\pi }}}\,\int _{0}^{y/({\sqrt {2}}\sigma )}\exp \left(-z^{2}\right)dz=\operatorname {erf} \left({\frac {y}{{\sqrt {2}}\sigma }}\right),

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

The quantile function (or inverse CDF) is written:

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

where $0\leq F\leq 1$ and $\operatorname {erf}^{{-1}}$ is the inverse error function

The expectation is then given by

E(Y)=\sigma {\sqrt {2/\pi }},

The variance is given by

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

Since this is proportional to the variance σ² of X, σ can be seen as a scale parameter of the new distribution.

The entropy of the half-normal distribution is exactly one bit less the 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,

H(Y)={\frac {1}{2}}\log \left({\frac {\pi \sigma ^{2}}{2}}\right)+{\frac {1}{2}},

The density functions satisfy the differential equations

\sigma ^{2}f'(x)+xf(x)=0,\qquad f(1)={\frac {{\sqrt {\frac {2}{\pi }}}e^{-1/(2\sigma ^{2})}}{\sigma }}.

and

\pi f'(x)+2\theta ^{2}xf(x)=0,\qquad f(1)={\frac {2e^{-\theta ^{2}/\pi }\theta }{\pi }}

Parameter estimation

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

{\hat \sigma }={\sqrt {{\frac 1n}\sum _{{i=1}}^{n}x_{i}^{2}}}

Related distributions

The distribution is a special case of the folded normal distribution with μ = 0.
It also coincides with a zero-mean normal distribution truncated from below at zero (see truncated normal distribution)
If Y has a half-normal distribution, then (Y/σ)² has a chi square distribution with 1 degree of freedom, i.e. Y/σ has a chi distribution with 1 degree of freedom.
If Y has a half-normal distribution, then Y/σ has a chi distribution with 1 degree of freedom.

External links

Half-Normal Distribution at MathWorld

(note that MathWorld uses the parameter

\theta ={\frac {1}{\sigma }}{\sqrt {\pi /2}}

)

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Half-normal distribution

Properties

Parameter estimation

Related distributions

See also

External links