Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability mass function ▷ Joint probability mass function

Definition: Let $X$ be a discrete random vector with possible outcomes $\mathcal{X} = \mathcal{X}_1 \times \ldots \times \mathcal{X}_n$ where $\mathcal{X}_1, \ldots, \mathcal{X}_n$ are the sets of possible values for the entries $X_1, \ldots, X_n$. Then, a function $f_X(x): \mathcal{X} \to \mathbb{R}$ is the joint probability mass function of $X$, if

\[\label{eq:pmf-joint-def-s0} f_X(x) \in [0, 1]\]

for all $x \in \mathcal{X}$,

\[\label{eq:pmf-joint-def-s1} \mathrm{Pr}(X_1 = x_1, \ldots, X_n = x_n) = f_X(x_1, \ldots, x_n)\]

for all $(x_1, \ldots, x_n) \in \mathcal{X}$ and

\[\label{eq:pmf-joint-def-s2} \sum_{x_1 \in \mathcal{X}_1} \ldots \sum_{x_n \in \mathcal{X}_n} f_X(x_1, \ldots, x_n) = 1 \; .\]
 
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Metadata: ID: D228 | shortcut: pmf-joint | author: JoramSoch | date: 2026-03-26, 11:04.