# Joint probability distribution

A probability distribution over the collection of joint configurations of all the variables you care about.

[summary: $$\newcommand{\bR}{\mathbb{R}} \newcommand{\bP}{\mathbb{P}} \newcommand{\cS}{\mathcal{S}} \newcommand{\cF}{\mathcal{F}} \newcommand{\gO}{\Omega} \newcommand{\go}{\omega} \newcommand{\ts}{\times}$$

A joint probability distribution of real-valued random variables $X_1, X_2, \cdots, X_n$ is a probability distribution $\bP$ over $\bR^n$. The probability of the event $(X_1 \in A_1, X_2 \in A_2, \cdots, X_n \in A_n)$ is $\bP(A_1,A_2, \cdots, A_n)$. ]

$$\newcommand{\bR}{\mathbb{R}} \newcommand{\bP}{\mathbb{P}} \newcommand{\cS}{\mathcal{S}} \newcommand{\cF}{\mathcal{F}} \newcommand{\gO}{\Omega} \newcommand{\go}{\omega} \newcommand{\ts}{\times}$$

A joint probability distribution of real-valued random variables $X_1, X_2, \cdots, X_n$ is a probability distribution $\bP$ over $\bR^n$. The probability of the event $(X_1 \in A_1, X_2 \in A_2, \cdots, X_n \in A_n)$ is $\bP(A_1,A_2, \cdots, A_n)$.

# Formal definition

Let $\{X_i \}_{i \in I}$ be a collection of random variables taking values in the spaces $(S_i, \cS_i)$. 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 $(\gO, \cF, \bP)$, then $\bP$ induces a joint distribution of the $X_i$. The induced function is a distribution because an event $(X_1 \in A_1, X_2 \in A_2, \cdots, X_n \in A_n)$ with $A_k \in \cS_k$ can be viewed as the event $\{ \go \in \gO : X_1(\go) \in A_1, \cdots, X_n(\go) \in A_n\}$, which is in $\cF$ because $\go \mapsto (X_1(\go), \cdots, X_n(\go))$ is a measurable map $\gO \to S_1 \ts \cdots \ts S_k$.