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) = [画像:(2theta)/pie^(-x^2theta^2/pi)]
(1)
D(x) = [画像: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^' = [画像:(3pi^2)/(4theta^4).]
(7)
The first few central moments are
mu_2 = [画像:(pi-2)/(2theta^2)]
(8)
mu_3 = [画像:(4-pi)/(2theta^3)]
(9)
mu_4 = [画像:(3pi^2-4pi-12)/(4theta^4),]
(10)
giving the mean, variance, skewness, and kurtosis excess as
mu = 1/theta
(11)
sigma^2 = [画像:(pi-2)/(2theta^2)]
(12)
gamma_1 = [画像:(sqrt(2)(4-pi))/((pi-2)^(3/2))]
(13)
gamma_2 = [画像:(8(pi-3))/((pi-2)^2).]
(14)
See also
Normal DistributionExplore with Wolfram|Alpha
WolframAlpha
More things to try:
Cite this as:
Weisstein, Eric W. "Half-Normal Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Half-NormalDistribution.html