AP®StatisticsCollege BoardStudy GuidesUnit 2: Probability, Random Variables, and Probability DistributionsDiscrete Random VariablesCumulative Probability Distributions for Discrete Random Variables

Cumulative Probability Distributions for Discrete Random Variables (College Board AP® Statistics): Study Guide

Written by: Dan Finlay

Reviewed by: Lucy Kirkham

Updated on 9 July 2026

Discrete cumulative probability distributions

What is a discrete cumulative probability distribution?

A discrete cumulative probability distribution shows the probability that a discrete random variable is less than or equal to each of its possible values
A discrete cumulative probability distribution can be given as either a table or a function
To find the cumulative probability $P (X \leq x)$
- identify the values of the random variable that are less than or equal to the $x$
- add together the probabilities of these values
The cumulative probability of the smallest value is always equal to the probability of that value
- e.g. if $X$ can only take the values 0, 2 or 4, then $P (X \leq 0) = P (X = 0)$
The cumulative probability of the largest value is always equal to 1
- e.g. if $X$ can only take the values 0, 2 or 4, then $P (X \leq 4) = 1$

Diagram showing probability and cumulative probability distributions. First probability is 0.5, second is 0.3, third is 0.2. Cumulative sums are 0.5, 0.8, and 1. — **Example of a cumulative probability distribution**

How can I find probabilities using a cumulative probability distribution?

To find $P (X \leq x)$
- if $X$ can take the value $x$
  - read $P (X \leq x)$ directly from the distribution
- if $X$ cannot take the value $x$
  - find the biggest value of $X$ that is less than $x$
  - find the cumulative probability of this value
    - e.g. if $X$ can take the values, 1, 2, 3 or 5 then $P (X \leq 4) = P (X \leq 3)$
To find $P (X < x)$
- find the biggest value of $X$ that is less than $x$
- find the cumulative probability of this value
  - e.g. if $X$ can take the values, 1, 2, 3 or 5 then $P (X < 5) = P (X \leq 3)$
To find $P (X > x)$ use the identity
- $P (X > x) = 1 - P (X \leq x)$
To find $P (X \geq x)$ use the identity
- $P (X \geq x) = 1 - P (X < x)$
To find $P (a \leq X \leq b)$ use the identity
- $P (X \leq b) - P (X < a)$
  - Note that $a$ is not included in the second inequality
To find $P (X = x)$
- find the biggest value of $X$ that is less than $x$
- find the cumulative probability of this value
- subtract this probability from $P (X \leq x)$
  - e.g. if $X$ can take the values, 1, 2, 3 or 5 then $P (X = 3) = P (X \leq 3) - P (X \leq 2)$

Worked Example

$X$ is a discrete random variable that can take any positive integer value. The cumulative probability distribution is given by the function $P (X \leq x) = \frac{1}{x + 1}$ , where $x$ is a positive integer.

(a) Find $P (X = 1)$ .

(b) Find $P (X \geq 5)$ .

(A) $0$

(B) $\frac{1}{4}$

(D) $\frac{1}{5}$

Answer:

(a)

1 is the first positive integer and so is the smallest value that $X$ can take

Therefore $P (X \leq 1) = P (X = 1)$

Substitute $x = 1$ into the function

$\begin{array}{rcl} P (X \leq 1) & = & \frac{1}{1 + 1} \\ = & \frac{1}{2} \end{array}$

$P (X = 1) = \frac{1}{2}$

(b)

Use the identity $P (X \geq x) = 1 - P (X < x)$

$P (X \geq 5) = 1 - P (X < 5)$

4 is the largest value that $X$ can take that is also less than 5

$P (X < 5) = P (X \leq 4)$

Substitute $x = 4$ into the function

$\begin{array}{rcl} P (X \leq 4) & = & \frac{1}{4 + 1} \\ = & \frac{1}{5} \end{array}$

This is also $P (X < 5)$ so substitute into $P (X \geq 5) = 1 - P (X < 5)$

$\begin{array}{rcl} P (X \geq 5) & = & 1 - \frac{1}{5} \\ = & \frac{4}{5} \end{array}$

$P (X \geq 5) = \frac{4}{5}$

(c)

$X$ cannot take the value of 3.5

3 is the largest value that $X$ can take that is also less than 3.5

Substitute $x = 3$ into the function

$\begin{array}{rcl} P (X \leq 3) & = & \frac{1}{3 + 1} \\ = & \frac{1}{4} \end{array}$

The correct answer is B

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