# Inverse Gaussian

InvGauss(m,l)

Inverse Gaussian equations

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:

• 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.

### Generation

It is not easy to generate values from the InvGaussian Distribution using any direct algorithm. However, you can construct a General distribution from the equation for the probability density function, as shown in the model InvGaussian. This method has the advantage of retaining the benefits of Latin Hypercube sampling if used.

The links to the InvGaussian software specific models are provided here:

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.

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