AP Statistics · Topic 2.8 · Unit 2
AP Stats 2.8: Random Variables & Distributions
By Jude Wallis · Published
Topic 2.8 in the Fall 2026 AP Statistics course introduces random variables. They take numerical values from a random phenomenon. A discrete probability distribution lists the probability of every value, and those sum to 1. A cumulative distribution gives the probability at or below each value.
AP Statistics: Unit 2 (topics 2.8). In the Fall 2026 AP Statistics course, introduction to random variables and probability distributions is Unit 2 Topic 2.8, aligned to skill 3.A.
What topic 2.8 covers
Topic 2.8 shifts from events to numbers by introducing random variables. The objective is to construct a probability distribution for a discrete random variable.
A random variable is a variable whose values are numerical outcomes of a random phenomenon, usually written with a capital letter such as . A discrete random variable can take only countable values, such as the number of pets a household owns.
Building a probability distribution
A probability distribution for a discrete random variable shows the probability associated with every possible value of the variable. You can determine it from the rules of probability or estimate it with a simulation, as in topic 2.3, earlier in Unit 2.
One condition must always hold: the sum of the probabilities over all possible values equals 1. That is what lets you find a missing probability by subtracting the known ones from 1. A distribution can be represented three ways, as a graph, a table, or a function, and each shows the same probabilities attached to the values of the random variable.
Cumulative probability distributions
Alongside the ordinary distribution, a cumulative probability distribution can be written as a table or a function. It shows the probability of being less than or equal to each value of the discrete random variable, written .
You build it by adding probabilities from the smallest value upward, so each cumulative entry is a running total. Cumulative values make range questions quick, because 'at most 2' is a single cumulative lookup rather than a sum you rebuild each time. The parameters that summarize a distribution, its mean and standard deviation, come next in topic 2.9, later in Unit 2.
Representing and checking a distribution
A discrete probability distribution can be shown as a table, a graph, or a function, and each form carries the same information: every possible value paired with its probability. Two checks confirm that a distribution is valid, and both come straight from the rules of probability.
First, each probability must lie between 0 and 1. Second, the probabilities must sum to exactly 1, which is what lets you recover a single missing probability by subtraction. The cumulative form is a running total of these probabilities, so it climbs from 0 up to 1 as you move through the values in order. That running total is what makes an 'at most' question a single lookup rather than a fresh sum.
Completing a distribution and a cumulative value
Let X be the number of pets a household owns, with P(0) = 0.30, P(1) = 0.35, P(2) unknown, and P(3) = 0.10. Find the missing probability, then find the cumulative probability P(X at most 2).
All probabilities must sum to 1, so add the known ones: .
Solve for the missing value: .
Cumulative through 2 adds the first three probabilities: .
Compute the total: , then .
The missing probability is , and the cumulative probability .
Frequently asked questions
What is a discrete random variable?
A random variable is a variable whose values are numerical outcomes of a random phenomenon. It is discrete when it can take only countable values, such as 0, 1, 2, or 3 pets. Its probability distribution lists the probability of each possible value, and those probabilities add to 1.
What is a cumulative probability distribution?
A cumulative probability distribution shows the probability of being less than or equal to each value of the random variable, written P(X at most x). You build it by adding probabilities from the smallest value upward. It makes 'at most' range questions a single lookup instead of a fresh sum each time.