\begin{definition}[Definition:Joint Distribution]
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 $X$.
\end{definition}