\begin{definition}[Definition:Probability Mass Function/Joint]
Let $X: \Omega \to \R$ and $Y: \Omega \to \R$ both be discrete random variables on $\struct {\Omega, \Sigma, \Pr}$.
Then the '''joint (probability) mass function''' of $X$ and $Y$ is the (real-valued) function $p_{X, Y}: \R^2 \to \closedint 0 1$ defined as:
:$\forall \tuple {x, y} \in \R^2: \map {p_{X, Y} } {x, y} = \begin {cases}
\map \Pr {\set {\omega \in \Omega: \map X \omega = x \land \map Y \omega = y} } & : x \in \Omega_X \text { and } y \in \Omega_Y \\
0 & : \text {otherwise}
\end {cases}$
That is, $\map {p_{X, Y} } {x, y}$ is the probability that the discrete random variable $X$ takes the value $x$ at the same time that the discrete random variable $Y$ takes the value $y$.
\end{definition}