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Half-Normal Distribution


HalfNormalDistribution

The half-normal distribution is a normal distribution with mean 0 and parameter theta limited to the domain x in [0,infty). It has probability and distribution functions given by

P(x)=(2theta)/pie^(-x^2theta^2/pi)
(1)
D(x)=erf((thetax)/(sqrt(pi))).
(2)

It is implemented in the Wolfram Language as HalfNormalDistribution[theta].

The nth raw moment is given by

 mu_n^'=pi^((n-1)/2)theta^(-n)Gamma(1/2(n+1)),
(3)

where Gamma(z) is the gamma function, giving the first few raw moments as

mu_1^'=1/theta
(4)
mu_2^'=pi/(2theta^2)
(5)
mu_3^'=pi/(theta^3)
(6)
mu_4^'=(3pi^2)/(4theta^4).
(7)

The first few central moments are

mu_2=(pi-2)/(2theta^2)
(8)
mu_3=(4-pi)/(2theta^3)
(9)
mu_4=(3pi^2-4pi-12)/(4theta^4),
(10)

giving the mean, variance, skewness, and kurtosis excess as

mu=1/theta
(11)
sigma^2=(pi-2)/(2theta^2)
(12)
gamma_1=(sqrt(2)(4-pi))/((pi-2)^(3/2))
(13)
gamma_2=(8(pi-3))/((pi-2)^2).
(14)

See also

Normal Distribution

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Cite this as:

Weisstein, Eric W. "Half-Normal Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Half-NormalDistribution.html

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