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.

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,

In the single parameter form, the MGF simplifies to An inverse Gaussian distribution in double parameter form

This approach is better in the sense that it clearly shows dimensionless nature of the single parameter form (note that

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.

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.

Let the stochastic process Xt be given by where Wt is a standard Brownian motion.

defined by: And suppose that we wish to find the probability density function for the time when the process first hits some barrier

The Fokker-Planck equation describing the evolution of the probability distribution

Based on the initial condition, the fundamental solution to the Fokker-Planck equation, denoted by

This will allow the original and mirror solutions to cancel out exactly at the barrier at each instant in time.

Due to the linearity of the BVP, the solution to the Fokker-Planck equation with this initial condition is: Now we must determine the value of

The fully absorbing boundary condition implies that: At

Substituting this back into the above equation, we find that: Therefore, the full solution to the BVP is: Now that we have the full probability density function, we are ready to find the first passage time distribution

The simplest route is to first compute the survival function

The survival function gives us the probability that the Brownian motion process has not crossed the barrier

, the first passage time follows an inverse Gaussian distribution: 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[7]: 74 [8]: 39 ).

The model where with all wi known, (μ, λ) unknown and all Xi independent has the following likelihood function Solving the likelihood equation yields the following maximum likelihood estimates

[9] Generate a random variate from a normal distribution with mean 0 and standard deviation equal 1 Square the value and use the relation Generate another random variate, this time sampled from a uniform distribution between 0 and 1 If

In 1915 it was used independently by Erwin Schrödinger[5] and Marian v. Smoluchowski[6] as the time to first passage of a Brownian motion.

[13] Abraham Wald re-derived this distribution in 1944[14] as the limiting form of a sample in a sequential probability ratio test.

The name inverse Gaussian was proposed by Maurice Tweedie in 1945.

[15] Tweedie investigated this distribution in 1956[16] and 1957[17][18] and established some of its statistical properties.

The distribution was extensively reviewed by Folks and Chhikara in 1978.

[2] Assuming that the time intervals between occurrences of a random phenomenon follow an inverse Gaussian distribution, the probability distribution for the number of occurrences of this event within a specified time window is referred to as rated inverse Gaussian.

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.

[20] Functions for the inverse Gaussian distribution are provided for the R programming language by several packages including rmutil,[21][22] SuppDists,[23] STAR,[24] invGauss,[25] LaplacesDemon,[26] and statmod.