Probability mass function of the categorical distribution

Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Multivariate discrete distributions ▷ Categorical distribution ▷ Probability mass function

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

\[\label{eq:cat} X \sim \mathrm{Cat}(\left[p_1, \ldots, p_k \right]) \; .\]

Then, the probability mass function of $X$ is

\[\label{eq:cat-pmf} f_X(x) = \left\{ \begin{array}{rl} p_1 \; , & \text{if} \; x = e_1 \\ \vdots \; \hphantom{,} & \quad \vdots \\ p_k \; , & \text{if} \; x = e_k \; . \\ \end{array} \right.\]

where $e_1, \ldots, e_k$ are the $1 \times k$ elementary row vectors.

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

∎

Sources:

original work

Metadata: ID: P98 | shortcut: cat-pmf | author: JoramSoch | date: 2020-05-11, 22:58.