Probability Distributions and their Stories

Continuous Multivariate distributions

Story. The Dirichlet distribution is a generalization of the Beta distribution. It is a probability distribution describing probabilities of outcomes. Instead of describing probability of one of two outcomes of a Bernoulli trial, like the Beta distribution does, it describes probability of $K−1$ of $K$ outcomes. The Beta distribution is the special case of $K=2$ .
Parameters. The parameters are $α_1,α_2,…α_K$ , all strictly positive, defined analogously to $α$ and $β$ of the Beta distribution.
Support. The Dirichlet distribution has support on the interval [0, 1] such that $\sum_{i=1}^K y_i = 1$ .
Probability density function.

$\begin{align} f(\boldsymbol{\theta};\boldsymbol{\alpha}) = \frac{1}{B(\boldsymbol{\alpha})}\,\prod_{i=1}^K y_i^{\alpha_i-1} \end{align}$

where

$\begin{align}B(\boldsymbol{\alpha}) = \frac{\prod_{i=1}^K\Gamma(\alpha_i)}{\Gamma\left(\sum_{i=1}^K \alpha_i\right)}\end{align}$

is the multivariate Beta function.
Related distributions.
- The special case where $K=2$ is a Beta distribution with parameters $α=α_1$ and $β=α_2$ .