# Definition:Probability Mass Function

## Contents

## Definition

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space.

Let $X: \Omega \to \R$ be a discrete random variable on $\left({\Omega, \Sigma, \Pr}\right)$.

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

- $\forall x \in \R: p_X \left({x}\right) = \begin{cases} \Pr \left({\left\{{\omega \in \Omega: X \left({\omega}\right) = x}\right\}}\right) & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end{cases}$

where $\Omega_X$ is defined as $\operatorname{Im} \left({X}\right)$, the image of $X$.

That is, $p_X \left({x}\right)$ is the probability that the discrete random variable $X$ takes the value $x$.

$p_X \left({x}\right)$ can also be written:

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

Note that for any discrete random variable $X$, the following applies:

\(\displaystyle \sum_{x \mathop \in \Omega_X} p_X \left({x}\right)\) | \(=\) | \(\displaystyle \Pr \left({\bigcup_{x \mathop \in \Omega_X} \left\{{\omega \in \Omega : X \left({\omega}\right) = x}\right\}}\right)\) | Definition of Probability Measure | ||||||||||

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

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

The latter is usually written:

- $\displaystyle \sum_{x \mathop \in \R} p_X \left({x}\right) = 1$

Thus it can be seen by definition that a **probability mass function** is an example of a normalized weight function.

The set of **probability mass functions** on a finite set $Z$ can be seen denoted $\Delta \left({Z}\right)$.

### Joint Probability Mass Function

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:

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

### General Definition

Let $X = \left\{{X_1, X_2, \ldots, X_n}\right\}$ be a set of discrete random variables on $\left({\Omega, \Sigma, \Pr}\right)$.

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

- $\forall x = \left({x_1, x_2, \ldots, x_n}\right) \in \R^n: p_X \left({x}\right) = \Pr \left({X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n}\right)$

The properties of the two-element case can be appropriately applied.

## Also known as

A **(probability) mass function** is often seen abbreviated **p.m.f.**, **pmf** or **PMF**.

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 2.1$: Probability mass functions - 1991: Roger B. Myerson:
*Game Theory*... (previous) ... (next): $1.2$ Basic Concepts of Decision Theory