Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Multivariate continuous distributions ▷ Dirichlet distribution ▷ Definition

Definition: Let $X$ be a $k \times 1$ random vector. Then, $X$ is said to follow a Dirichlet distribution with concentration parameters $\alpha = \left[ \alpha_1, \ldots, \alpha_k \right]$

\[\label{eq:Dir} X \sim \mathrm{Dir}(\alpha) \; ,\]

if and only if its probability density function is given by

\[\label{eq:beta-pdf} \mathrm{Dir}(x; \alpha) = \frac{\Gamma\left( \sum_{i=1}^k \alpha_i \right)}{\prod_{i=1}^k \Gamma(\alpha_i)} \, \prod_{i=1}^k {x_i}^{\alpha_i-1}\]

where $\alpha_i > 0$ for all $i = 1, \ldots, k$, and the density is zero, if $x_i \notin [0,1]$ for any $i = 1, \ldots, k$ or $\sum_{i=1}^k x_i \neq 1$.

 
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Metadata: ID: D54 | shortcut: dir | author: JoramSoch | date: 2020-05-10, 20:36.