Inverse Gaussian distribution

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 . The probability density function (pdf) of the inverse Gaussian distribution has a single parameter form given by In this form, the mean and variance of the distribution are equal, Also, the cumulative distribution function (cdf) of the single parameter inverse Gaussian distribution is related to the standard normal distribution by where , and the is the cdf of standard normal distribution. The variables and are related to each other by the identity In the single parameter form, the MGF simplifies to An inverse Gaussian distribution in double parameter form can be transformed into a single parameter form by appropriate scaling where The standard form of inverse Gaussian distribution is If Xi has an distribution for i = 1, 2, ..., n and all Xi are independent, then Note that is constant for all i. This is a necessary condition for the summation. Otherwise S would not be Inverse Gaussian distributed. For any t > 0 it holds that The inverse Gaussian distribution is a two-parameter exponential family with natural parameters −λ/(2μ2) and −λ/2, and natural statistics X and 1/X.