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

Formal definition

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