Index: The Book of Statistical ProofsProbability Distributions ▷ Multivariate continuous distributions ▷ Dirichlet distribution ▷ Probability density function

Theorem: Let $X$ be a random vector following a Dirichlet distribution:

$\label{eq:Dir} X \sim \mathrm{Dir}(\alpha) \; .$

Then, the probability density function of $X$ is

$\label{eq:Dir-pdf} f_X(x) = \frac{\Gamma\left( \sum_{i=1}^k \alpha_i \right)}{\prod_{i=1}^k \Gamma(\alpha_i)} \, \prod_{i=1}^k {x_i}^{\alpha_i-1} \; .$

Proof: This follows directly from the definition of the Dirichlet distribution.

Sources:

Metadata: ID: P95 | shortcut: dir-pdf | author: JoramSoch | date: 2020-05-05, 21:22.