# Definition:Probability Mass Function/Joint

## Definition

Let $X: \Omega \to \R$ and $Y: \Omega \to \R$ both be discrete random variables on $\left({\Omega, \Sigma, \Pr}\right)$.

Then the **joint (probability) mass function** of $X$ and $Y$ is the (real-valued) function $p_{X, Y}: \R^2 \to \left[{0 \,.\,.\, 1}\right]$ defined as:

- $\forall \left({x, y}\right) \in \R^2: p_{X, Y} \left({x, y}\right) = \begin{cases} \Pr \left({\left\{{\omega \in \Omega: X \left({\omega}\right) = x \land Y \left({\omega}\right) = y}\right\}}\right) & : x \in \Omega_X \text { and } y \in \Omega_Y \\ 0 & : \text {otherwise} \end{cases}$

That is, $p_{X, Y} \left({x, y}\right)$ 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 $y$.

$p_{X, Y} \left({x, y}\right)$ can also be written:

- $\Pr \left({X = x, Y = y}\right)$

Similarly to the individual probability mass functions of $X$ and $Y$, we have:

\(\displaystyle \sum_{x \mathop \in \Omega_X \atop y \mathop \in \Omega_Y} p_{X, Y} \left({x, y}\right)\) | \(=\) | \(\displaystyle \Pr \left({\bigcup_{x \mathop \in \Omega_X \atop y \mathop \in \Omega_Y} \left\{{\omega \in \Omega: X \left({\omega}\right) = x, Y \left({\omega}\right) = y}\right\}}\right)\) | $\quad$ by the definition of probability measure | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \Pr \left({\Omega}\right)\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1\) | $\quad$ | $\quad$ |

The latter is usually written:

- $\displaystyle \sum_{x \mathop \in \R} p_{X, Y} \left({x, y}\right) = 1$

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 3.1$: Bivariate discrete distributions: $(1)$