A joint probability distribution of real-valued random variables is a probability distribution $P$ over $R^{n}$ . The probability of the event $(X_{1} \in A_{1}, X_{2} \in A_{2}, \dots, X_{n} \in A_{n})$ is $P (A_{1}, A_{2}, \dots, A_{n})$ .

Formal definition

Let ${X_{i}}_{i \in I}$ be a collection of random variables taking values in the spaces $(S_{i}, S_{i})$ . Then a joint distribution of the ${X_{i}}_{i \in I}$ is a probability distribution over $\prod_{i \in I} S_{i}$ .

If the ${X_{i}}_{i \in I}$ are defined on a probability space $(Ω, F, P)$ , then $P$ induces a joint distribution of the $X_{i}$ . The induced function is a distribution because an event $(X_{1} \in A_{1}, X_{2} \in A_{2}, \dots, X_{n} \in A_{n})$ with $A_{k} \in S_{k}$ can be viewed as the event ${ω \in Ω : X_{1} (ω) \in A_{1}, \dots, X_{n} (ω) \in A_{n}}$ , which is in $F$ because $ω \mapsto (X_{1} (ω), \dots, X_{n} (ω))$ is a measurable map $Ω \to S_{1} \times \dots \times S_{k}$ .

Joint probability distribution