# Inverse Gaussian distribution

Format: InvGauss(m, l)

The Inverse Gaussian distribution is a right-skewed distribution bounded at zero. Sometimes given the notation IG(m,l). Examples of the Inverse Gaussian distribution are given below:

## Uses

The Inverse Gaussian is a distribution seldom used in risk analysis. Its primary uses are:

• As a population distribution where a Lognormal distribution has too heavy a right tail

• To model stock returns and interest rate processes (e.g. Madan (1998))

Most uses are rather obscure: it has been used, for example, in physics to model the time until a particle, moving with Brownian motion with a drift, will exceed a certain distance from its original position for the first time. However, this parallels a problem in stock price modeling, or any other stochastic variable exhibiting geometric Brownian motion, where one wants to know the time until a share price first exceeds a certain value above (or below) its current market value.

Seshadri (1999) offers a very complete guide to the Inverse Gaussian. Its name comes from the fact that its cumulant generating function is the inverse of that of the Gaussian distribution.

It is also sometimes called the First Passage Time Distribution of Brownian motion with drift.

It is a member of the exponential family of distributions.

It is often called the Wald distribution but it was, in fact, first written about by Schroedinger in 1915.

If X takes an Inverse Gaussian distribution, then 1/X takes a distribution known as the Random Walk Distribution.

## ModelRisk functions added to Microsoft Excel for the Inverse Gaussian (IG) distribution

VoseInvGauss generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseInvGaussObject constructs a distribution object for this distribution.

VoseInvGaussProb returns the probability density or cumulative distribution function for this distribution.

VoseInvGaussProb10 returns the log10 of the probability density or cumulative distribution function.

VoseInvGaussFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseInvGaussFitObject constructs a distribution object of this distribution fitted to data.

VoseInvGaussFitP returns the parameters of this distribution fitted to data.