Inverse Gaussian distribution

Family of continuous probability distributions
the distribution of 1/x when x is Gaussian

In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two parameter family of continuous probability distributions with support on (0,∞). Its probability density function is given by

for x > 0, where is the mean and is the shape parameter. The inverse Gaussian distribution has several properties analogous to a Gaussian distribution. The name can be misleading: it is an "inverse" only in that, while the Gaussian describes a Brownian motion's level at a fixed time, the inverse Gaussian describes the distribution of the time a Brownian motion with positive drift takes to reach a fixed positive level. Its cumulant generating function (logarithm of the characteristic function) is the inverse of the cumulant generating function of a Gaussian random variable. To indicate that a random variable X is inverse Gaussian distributed with mean μ and shape parameter λ we write .

A common special case of the above arises when the Brownian motion has no drift. In that case, parameter μ tends to infinity, and the first passage time for fixed level α has probability density function

(see also Bachelier). This is a Lévy distribution with parameters and .

This distribution appears to have been first derived in 1900 by Louis Bachelier as the time a stock reaches a certain price for the first time. In 1915 it was used independently by Erwin Schrödinger and Marian v. Smoluchowski as the time to first passage of a Brownian motion. In the field of reproduction modeling it is known as the Hadwiger function, after Hugo Hadwiger who described it in 1940. Abraham Wald re derived this distribution in 1944 as the limiting form of a sample in a sequential probability ratio test. The name inverse Gaussian was proposed by Maurice Tweedie in 1945. Tweedie investigated this distribution in 1956 and 1957 and established some of its statistical properties. The distribution was extensively reviewed by Folks and Chhikara in 1978.

Despite the simple formula for the probability density function, numerical probability calculations for the inverse Gaussian distribution nevertheless require special care to achieve full machine accuracy in floating point arithmetic for all parameter values. Functions for the inverse Gaussian distribution are provided for the R programming language by several packages including rmutil, SuppDists, STAR, invGauss, LaplacesDemon, and statmod.