Definition:Probability Mass Function

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Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

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


Then the probability mass function of $X$ is the (real-valued) function $p_X: \R \to \closedint 0 1$ defined as:

$\quad \forall x \in \R: \map {p_X} x = \begin {cases} \map \Pr {\set {\omega \in \Omega: \map X \omega = x} } & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end {cases}$

where $\Omega_X$ is defined as $\Img X$, the image of $X$.

That is, $\map {p_X} x$ is the probability that the discrete random variable $X$ takes the value $x$.


$\map {p_X} x$ can also be written:

$\map \Pr {X = x}$


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

\(\ds \sum_{x \mathop \in \Omega_X} \map {p_X} x\) \(=\) \(\ds \map \Pr {\bigcup_{x \mathop \in \Omega_X} \set {\omega \in \Omega: \map X \omega = x} }\) Definition of Probability Measure
\(\ds \) \(=\) \(\ds \map \Pr \Omega\)
\(\ds \) \(=\) \(\ds 1\)

The latter is usually written:

$\ds \sum_{x \mathop \in \R} \map {p_X} x = 1$


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 $\map \Delta Z$.


Joint Probability Mass Function

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:

$\quad \forall \tuple {x, y} \in \R^2: \map {p_{X, Y} } {x, y} = \begin {cases} \map \Pr {\set {\omega \in \Omega: \map X \omega = x \land \map Y \omega = y} } & : x \in \Omega_X \text { and } y \in \Omega_Y \\ 0 & : \text {otherwise} \end {cases}$

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 $y$.


General Definition

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

Then the joint (probability) mass function of $X$ is (real-valued) function $p_X: \R^n \to \closedint 0 1$ defined as:

$\forall x = \tuple {x_1, x_2, \ldots, x_n} \in \R^n: \map {p_X} x = \map \Pr {X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n}$

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


Examples

Arbitrary Example

Consider a population consisting of children the state of whose teeth is being monitored.

The following table consists of a count of the number of teeth with dental caries in a group of $100$ schoolchildren:

$\quad \begin {array} {|l|l|} \hline \text {Number of Teeth} & \text {Number of Children} \\ \hline 0 & 53 \\ 1 & 29 \\ 2 & 14 \\ 3 & 1 \\ 4 & 3 \\ \hline \end {array}$

The values of the probability mass function, in this case better referred to as a relative frequency function, are:

\(\ds \map f 0\) \(=\) \(\ds 0 \cdotp 53\)
\(\ds \map f 1\) \(=\) \(\ds 0 \cdotp 29\)
\(\ds \map f 2\) \(=\) \(\ds 0 \cdotp 14\)
\(\ds \map f 3\) \(=\) \(\ds 0 \cdotp 01\)
\(\ds \map f 4\) \(=\) \(\ds 0 \cdotp 03\)


Also known as

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

Some sources refer to it just as a mass function, or a probability function.

It is also known as a frequency function, which is also used for probability density function.

When used in the context of a set of raw data, it can also be called a relative frequency function.


Also see

  • Results about probability mass functions can be found here.


Sources

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