[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$~$.