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

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 mean or expected value of $X$ is

\[\label{eq:cat-mean} \mathrm{E}(X) = \left[p_1, \ldots, p_k \right] \; .\]

Proof: If we conceive the outcome of a categorical distribution to be a $1 \times k$ vector, then the elementary row vectors $e_1 = \left[1, 0, \ldots, 0 \right]$, …, $e_k = \left[0, \ldots, 0, 1 \right]$ are all the possible outcomes and they occur with probabilities $\mathrm{Pr}(X = e_1) = p_1$, …, $\mathrm{Pr}(X = e_k) = p_k$. Consequently, the expected value is

\[\label{eq:cat-mean-qed} \begin{split} \mathrm{E}(X) &= \sum_{x \in \mathcal{X}} x \cdot \mathrm{Pr}(X = x) \\ &= \sum_{i=1}^k e_i \cdot \mathrm{Pr}(X = e_i) \\ &= \sum_{i=1}^k e_i \cdot p_i \\ &= \left[p_1, \ldots, p_k \right] \; . \end{split}\]
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Metadata: ID: P24 | shortcut: cat-mean | author: JoramSoch | date: 2020-01-16, 11:17.