# Generalized Gaussian Probability Density Function

## Contents |

## Density Function

The Generalized Gaussian density has the following form:

where (rho) is the "shape parameter". The density is plotted in the following figure:

Matlab code used to generate this figure is available here: ggplot.m.

Adding an arbitrary location parameter, , and inverse scale parameter, , the density has the form,

Matlab code used to generate this figure is available here: ggplot2.m.

## Generating Random Samples

Samples from the Generalized Gaussian can be generated by a transformation of Gamma random samples, using the fact that if is a distributed random variable, and is an independent random variable taking the value -1 or +1 with equal probability, then,

is distributed . That is,

where the density of is written in a non-standard but suggestive form.

## Matlab Code

Matlab code to generate random variates from the Generalized Gaussian density with parameters as described here is here:

As an example, we generate random samples from the example Generalized Gaussian densities shown above.

Matlab code used to generate this figure is available here: ggplot3.m.

## Mixture Densities

A more general family of densities can be constructed from mixtures of Generalized Gaussians. A mixture density, , is made up of constituent densities together with probabilities associated with each constituent density.

The densities have different forms, or parameter values. A random variable with a mixture density can be thought of as being generated by a two-part process: first a decision is made as to which constituent density to draw from, where the density is chosen with probability , then the value of the random variable is drawn from the chosen density. Independent repetitions of this process result in a sample having the mixture density .

As an example consider the density,

Matlab code used to generate these figures is available here: ggplot4.m.