Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate discrete distributions ▷ Bernoulli distribution ▷ Special case of categorical distribution

Theorem: The Bernoulli distribution with success probability $p$ is a special case of the categorical distribution with category probabilities $p_1 = p$ and $p_2 = 1-p$:

\[\label{eq:bern-cat} X \sim \mathrm{Cat}\left( [p, 1-p] \right) \quad \Rightarrow \quad X \sim \mathrm{Bern}(p) \; .\]

Proof: The probability mass function of the categorical distribution, where $x$ is a $1 \times k$ vector and $e_1, \ldots, e_k$ are the $1 \times k$ elementary row vectors, is as follows:

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

If we let $k = 2$, $p_1 = p$ and $p_2 = 1-p$, we obtain

\[\label{eq:cat-pmf-bern-s1} \mathrm{Cat}\left( x; [p, 1-p] \right) = \left\{ \begin{array}{rl} p \; , & \text{if} \; x = [1, 0] \\ 1-p \; , & \text{if} \; x = [0, 1] \; . \\ \end{array} \right.\]

Writing this in terms of the first entry $x_1 \in \left\lbrace 0, 1 \right\rbrace$, we get

\[\label{eq:cat-pmf-bern-s2} \mathrm{Cat}\left( x; [p, 1-p] \right) = \left\{ \begin{array}{rl} p \; , & \text{if} \; x_1 = 1 \\ 1-p \; , & \text{if} \; x_1 = 0 \\ \end{array} \right.\]

which is equivalent to the probability mass function of the Bernoulli distribution.

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Metadata: ID: P494 | shortcut: bern-cat | author: JoramSoch | date: 2025-04-04, 14:38.