# Definition:Joint Distribution

## Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\struct {S, \Sigma'}$ be a measurable space.

Let $X_1, X_2, \ldots, X_d$ be random variables on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.

Define a random vector $X : \Omega \to S^d$ by:

$\map X \omega = \tuple {\map {X_1} \omega, \map {X_2} \omega, \ldots, \map {X_d} \omega}$

for each $\omega \in \Omega$.

Then the joint distribution of $X_1, X_2, \ldots, X_d$ is defined as the probability distribution of $X$.