Statistics · Unit 4: Discrete Random Variables · 14 min read · Updated 2026-05-11
Discrete Random Variables — AP Statistics
AP Statistics · Unit 4: Discrete Random Variables · 14 min read
1. What is a Discrete Random Variable?★☆☆☆☆⏱ 3 min
A discrete random variable (DRV) is a variable that takes on a countable number of distinct values, where each value has an associated probability of occurring. Unlike continuous random variables (which take uncountably infinite values over an interval), discrete random variables can be listed one by one.
Discrete random variable questions make up 10-15% of Unit 4's exam weight, appearing on both multiple-choice and free-response sections. DRVs are the foundation for all common discrete probability distributions tested on the AP exam, including binomial and geometric distributions.
2. Probability Distributions (PMFs and CDFs)★★☆☆☆⏱ 3 min
A probability distribution for a discrete random variable is called a probability mass function (PMF). It describes all possible values of the DRV and the probability that the variable takes each value. For a DRV $X$, the PMF is written $p(x) = P(X = x)$.
A valid PMF must satisfy two core requirements:
$0 \leq p(x) \leq 1$ for all possible $x$
The sum of all $p(x)$ over all possible $x$ equals 1
\sum_{\text{all } x} p(x) = 1
A cumulative distribution function (CDF) gives the probability that $X$ is less than or equal to a specific value: $F(x) = P(X \leq x)$. This simplifies calculating probabilities for ranges of outcomes: $P(a < X \leq b) = F(b) - F(a)$.
3. Expected Value and Variance★★☆☆☆⏱ 3 min
The expected value (or mean) of a discrete random variable $X$, written $\mu_X = E(X)$, is the long-run average value of $X$ we would expect to observe if we repeated the random process infinitely many times. The formula for expected value is:
E(X) = \mu_X = \sum_{\text{all } x} x \cdot P(X=x)
Variance, written $\sigma_X^2 = Var(X)$, measures the spread of the distribution, or the average squared deviation of $X$ from its mean. A calculation-friendly formula used for most AP exam problems is:
The standard deviation $\sigma_X$ is the square root of the variance, and it measures spread in the original units of $X$, unlike variance which is in squared units.
4. Linear Transformations of Discrete Random Variables★★★☆☆⏱ 2 min
A linear transformation changes a random variable $X$ into a new random variable $Y = aX + b$, where $a$ and $b$ are fixed constants. Common examples include unit conversion or adding a fixed fee to a variable payout.
Variance: $Var(aX + b) = a^2 Var(X)$ (always holds)
The constant $b$ does not affect variance because adding $b$ shifts all values of $X$ by the same amount, so the spread of the distribution does not change. Only scaling by $a$ changes spread, and since variance is in squared units, we square $a$.
5. Combining Independent Discrete Random Variables★★★☆☆⏱ 3 min
Two random variables $X$ and $Y$ are independent if knowing the value of one does not change the probability distribution of the other. We can combine independent DRVs to find the expected value and variance of their sum or difference.
For any constants $a, b$, $E(aX + bY) = aE(X) + bE(Y)$. This always holds, even for dependent variables.
For independent $X$ and $Y$, $Var(aX + bY) = a^2 Var(X) + b^2 Var(Y)$. Variances always add, even when taking the difference of two variables: $Var(X-Y) = Var(X) + Var(Y)$.
Common Pitfalls
Why: Students incorrectly extend the expected value rule (difference of expectations is expectation of differences) to variance.
Why: Expected value uses $a$ to the first power, so students carry this over to variance by mistake.
Why: Students rush and do not read inequality signs carefully.
Why: Students confuse the long-run average (expected value) with the mode (most common outcome).
Why: Students assume independence automatically if it is not explicitly stated.
Why: Students forget the fundamental requirement that all probabilities sum to 1.