# Definition:Probability Mass Function/Joint

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## Contents

## Definition

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 $y$.

## Also denoted as

$\map {p_{X, Y} } {x, y}$ can also be written:

- $\map \Pr {X = x, Y = y}$

## Also see

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

\(\displaystyle \sum_{x \mathop \in \Omega_X \atop y \mathop \in \Omega_Y} \map {p_{X, Y} } {x, y}\) | \(=\) | \(\displaystyle \map \Pr {\bigcup_{x \mathop \in \Omega_X \atop y \mathop \in \Omega_Y} \set {\omega \in \Omega: \map X \omega = x, \map Y \omega = y} }\) | Definition of Probability Measure | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \Pr \Omega\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1\) |

The latter is usually written:

- $\displaystyle \sum_{x \mathop \in \R} \map {p_{X, Y} } {x, y} = 1$

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 3.1$: Bivariate discrete distributions: $(1)$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**joint probability mass function**