Probability density function of the Dirichlet distribution

Index: The Book of Statistical Proofs ▷ Probability 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:

original work

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