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Generate your own distribution via a relationship to another distribution available in your software

Many distributions are connected together via mathematical relationships. If we know the relationship to a distribution or distributions available in your software, we can use that relationship to generate the required distribution. This is useful if for example the distribution you would like to use is not available, or if showing the relationship in the model makes it easier for you or the user to understand.

Example 1: The Student t distribution

A Student(0,1,n) distribution is a  Normal distribution with mean 0 and a variance that is a random variable equal to (n/Chisquared(n)), where Chisquared(n) is a Chi-Squared distribution. You can simply use the Student distribution.

Example 2: The Beta-Binomial distribution

A Beta-Binomial(n,a,b) distribution is a Binomial distribution with n trials and a probability that is also a random variable, = Beta(a,b,1). To create a Beta-Binomial(100,17,23), for example, we would use the following nested construction:


The generation of the Beta-binomial distribution is as described above is shown in the model BetaBinomial.

Example 3: The Chi Squared distribution

A ChiSquared(n) distribution is a Gamma(0,2,n/2) distribution.

Other examples in ModelAssist

The Dirichlet, Multinomial and Multivariate Hypergeometric distributions can be created with nested Beta, Gamma,  Binomial and Hypergeometric distributions respectively, all of which are available in most software.

The Johnson, Kemp, and Kotz (2005), and Johnson, Kotz, and Balakrishnan Vol 1 and 2 (1994) books are an excellent resource for learning about rare distributions and ways of generating them through their relationships with other distributions.