Probability Mass Function¶

Definition¶

Let $X$ be a discrete random variable. The probability mass function of $X$ is the function $p_X$ defined by

p_X(x)=\mathbb{P}(X=x).

It is zero except on the countable set of values that $X$ can take.

Interpretation¶

A PMF lists, for each possible value of a discrete random variable, the probability that the variable takes exactly that value. It describes the whole distribution of such a variable as a table or a bar chart.

Probability bars for the number of heads in three tosses, 1/8, 3/8, 3/8 and 1/8, with the bars for 2 and 3 heads highlighted.

The diagram shows $X\sim\operatorname{Bin}(3,1/2)$ $Bin (3, 1/2)$ , the number of heads in three fair tosses, with masses $1/8$ , $3/8$ , $3/8$ and $1/8$ , which sum to $1$ . The two red bars give $\mathbb{P}(X\ge2)=3/8+1/8=1/2$ $1/2$ .

Each bar is a value of $p_X(x)=\mathbb{P}(X=x)$ $x)$ ; together they add up to $1$ . Set $c$ and $d$ to add the masses of a range of values.

Each mass is the probability of a generated event $\{X=x\}$ $x}$ . These events are disjoint for different values and together make up the whole sample space, so the masses are nonnegative, add up to $1$ , and the probability of any set $B$ of values is the sum of the masses in $B$ ; summing up to a threshold $t$ gives the CDF, as in the Key Results.

The argument of $p_X$ is a value, not an outcome: in the figure, $p_X(1)=3/8$ collects the three outcomes $HTT$ , $THT$ and $TTH$ . The PMF is $0$ away from the possible values, for example $p_X(1.5)=0$ $0$ .

Key Results¶

$p_X(x)\ge0$ and $\displaystyle \sum_xp_X(x)=1$ $1$ .
$\displaystyle F_X(t)=\sum_{x\le t}p_X(x)$ $x \leq t \sum p_{X} (x)$ .

Examples¶

For a fair die, $p_X(3)=1/6$ and $\mathbb{P}(X\le3)=3/6$ $3/6$ .
If $p_X(0)=0.7$ and $p_X(1)=0.3$ $0.3$ , then $\mathbb{P}(X>0)=0.3$ $0.3$ .

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 3
Nicky van Foreest · Probability Distributions · Refresher

Aliases: PMF, frequency function, mass function, p.m.f. Symbols: sum

Last updated: 20 October 2026 03:02