In AP Statistics, a discrete random variable is a variable that can take only a countable number of numerical values, each with an associated probability, where all the probabilities sum to 1 (CED 4.7.A). Examples include the number of heads in three coin flips or the sum of two dice.
A discrete random variable assigns a number to each outcome of a random process, and that number can only come from a countable list of values. Think "things you count, not things you measure." The number of heads in three coin flips, the number of puppies in a litter, the sum when you roll two dice. You can list every possible value (0, 1, 2, 3, ...), and each value gets its own probability.
The full picture of a discrete random variable is its probability distribution, which you can show as a table, graph, or function pairing each value with its probability. Two rules always hold. Every probability is between 0 and 1, and the probabilities across all possible values sum to exactly 1. Once you have the distribution, you can compute the mean (expected value) μ_X = Σ x_i · P(x_i) and the standard deviation σ_X = √(Σ(x_i − μ_X)² · P(x_i)). Those two numbers are parameters, fixed values describing the whole distribution, not statistics from a sample.
Discrete random variables live in Unit 4, specifically Topics 4.7 and 4.8. Learning objective 4.7.A asks you to represent the probability distribution of a discrete random variable, 4.7.B asks you to interpret it (shape, center, spread, in context), and 4.8.A and 4.8.B ask you to calculate and interpret its parameters with correct units. This is also the on-ramp to the rest of the course. The binomial and geometric distributions in later Unit 4 topics are just famous discrete random variables, and Unit 5's sampling distributions (Topic 5.1) treat a sample statistic itself as a random variable. If you can read a probability table and pull out a mean and standard deviation, the back half of the course gets dramatically easier.
Keep studying AP Statistics Unit 4
Probability Distribution (Unit 4)
A discrete random variable and its probability distribution are a package deal. The variable tells you what gets counted, and the distribution (table, graph, or function) tells you how likely each count is. AP questions almost always hand you one and ask about the other.
Expected Value (Unit 4)
Expected value is just the mean of a discrete random variable. It is a weighted average where each value is weighted by its probability, μ_X = Σ x_i · P(x_i). It tells you the long-run average outcome if the random process ran forever, which is exactly how you should interpret it on an FRQ.
Variance (Unit 4)
Variance and standard deviation measure how spread out a discrete random variable's values are around the expected value. The formula weights each squared deviation by its probability. Interpreting σ_X in context, with units, is the second half of LO 4.8.B.
Sampling Distributions (Unit 5)
Here's the big payoff. A sample statistic like x̄ or p̂ is itself a random variable, because random sampling produces random numerical outcomes. Topic 5.1 asks why your sample differs from someone else's, and the answer is that statistics vary randomly, exactly the behavior random variables describe.
Multiple-choice questions typically give you a probability distribution table and ask you to find a missing probability (using the sum-to-1 rule), compute or interpret the expected value, or describe what happens to the distribution under a transformation like W = 2Z + 1 (the mean transforms the same way, the spread stretches by the multiplier). Another common stem gives you a list of possible values like {0, 1, 2, 3, 4} and asks what they represent, which tests whether you know the values are numerical outcomes of a random process. On FRQs, you may need to construct a distribution from scratch (like the number of heads in three coin flips), calculate μ_X and σ_X, and interpret them in context with units. The interpretation is where points get lost, so always say what the expected value means as a long-run average in the problem's context, not just the number.
A discrete random variable takes countable values, so P(X = some exact value) can be a real, nonzero number you read off a table. A continuous random variable (like height or time) takes any value in an interval, so probabilities come from areas under a density curve and P(X = one exact value) is 0. Quick gut check: if you'd count it, it's discrete; if you'd measure it, it's continuous. Note that discrete doesn't mean finite. The number of emails you get in an hour is discrete even though it could be 0, 1, 2, ... with no upper limit, because the values are still countable.
A discrete random variable takes a countable number of numerical values, and each value has a probability attached to it.
All probabilities in a discrete distribution must be between 0 and 1 and must sum to exactly 1, which is how you find missing probabilities in a table.
The mean (expected value) is μ_X = Σ x_i · P(x_i), a probability-weighted average that describes the long-run average outcome of the random process.
The standard deviation σ_X = √(Σ(x_i − μ_X)² · P(x_i)) measures typical distance from the expected value, and both parameters must be interpreted in context with units.
Mean and standard deviation of a random variable are parameters, meaning fixed values for the distribution, not statistics calculated from a sample.
Discrete random variables set up the binomial and geometric distributions later in Unit 4 and the idea of a sampling distribution in Unit 5.
It's a variable that assigns numbers to outcomes of a random process and can take only a countable number of values, like the number of heads in three coin flips (0, 1, 2, or 3). Each value has a probability, and all probabilities sum to 1.
Discrete variables are counted (number of puppies in a litter, sum of two dice), while continuous variables are measured (height, time) and can take any value in an interval. For continuous variables, the probability of any single exact value is 0, but discrete variables can have nonzero probability at exact values.
No. It has to take a countable number of values, which can be infinite. The number of emails you receive in an hour is discrete with possible values 0, 1, 2, 3, ... going on forever, because you can still list them.
Multiply each value by its probability and add everything up: μ_X = Σ x_i · P(x_i). For example, if P(Z = -1) = 0.3, P(Z = 0) = 0.4, and P(Z = 2) = 0.3, the mean is (-1)(0.3) + (0)(0.4) + (2)(0.3) = 0.3.
No. A random variable's values are always numbers (the numerical outcomes of random behavior), so a discrete random variable is quantitative. Categorical variables like eye color or political party have non-numeric labels and don't have means or standard deviations at all.