TOPICS
Search

Inverse Gaussian Distribution


The inverse Gaussian distribution, also known as the Wald distribution, is the distribution over [0,infty) with probability density function and distribution function given by

P(x)=sqrt(lambda/(2pix^3))e^(-lambda(x-mu)^2/(2xmu^2))
(1)
D(x)=1/2{1+erf[sqrt(lambda/(2x))(x/mu-1)]}+1/2e^(2lambda/mu){1-erf[sqrt(lambda/(2x))(x/mu+1)]},
(2)

where mu>0 is the mean and lambda>0 is a scaling parameter.

The inverse Gaussian distribution is implemented in the Wolfram Language as InverseGaussianDistribution[mu, lambda].

The nth raw moment is given by

 mu_n^'=e^(lambda/mu)sqrt((2lambda)/pi)mu^(n-1/2)K_(1/2-n)(lambda/mu),
(3)

where K_n(z) is a modified Bessel function of the second kind, giving the first few as

mu_1^'=mu
(4)
mu_2^'=(mu^2(lambda+mu))/lambda
(5)
mu_3^'=(mu^3(lambda^2+3lambdamu+3mu^2))/(lambda^2).
(6)

Using K_(-n-1)(z)=(2n/z)K_(-n)(z)+K_(-n)(z) gives a recursion relation for the raw moments as

 mu_(n+1)^'=((2n-1)mu^2)/lambdamu_n^'+mu^2mu_(n-1)^'.
(7)

The first few central moments are

mu_2=(mu^3)/lambda
(8)
mu_3=(3mu^5)/(lambda^2)
(9)
mu_4=(3mu^6(lambda+5mu))/(lambda^3).
(10)

The cumulants kappa_n are given by

 kappa_(n+1)=((2n)!)/(2^nn!mu^(2n+1)lambda^n).
(11)

The variance, skewness, and kurtosis excess are given by

sigma^2=(mu^3)/lambda
(12)
gamma_1=3sqrt(mu/lambda)
(13)
gamma_2=(15mu)/lambda.
(14)

See also

Normal Distribution

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Inverse Gaussian Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseGaussianDistribution.html

Subject classifications