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Discrete Random Variable — Definition, Formula & Examples

A discrete random variable is a variable whose value comes from a chance process and can only land on separate, listable values, such as the number of heads in 5 coin flips. You can list every possible value and give each one its own probability.

A random variable XX is discrete if its set of possible values is countable, meaning finite or countably infinite. Its distribution is given by a probability mass function p(x)=P(X=x)p(x) = P(X = x). This function must satisfy 0≤p(x)≤10 \le p(x) \le 1 for every value xx, and ∑p(x)=1\sum p(x) = 1, where the sum is taken over all possible values.

Key Formula

E(X)=μX=∑xi piE(X) = \mu_X = \sum x_i \, p_i
Where:
  • xix_i = each possible value of the random variable
  • pip_i = the probability that X equals x_i
  • E(X) or μXE(X) \text{ or } \mu_X = the expected value (mean) of X, its long-run average

Worked Example

Problem: Let X be the number of televisions in a randomly selected household. The possible values are 0, 1, 2, and 3, with probabilities 0.1, 0.3, 0.4, and 0.2. Confirm that this is a valid probability distribution, then find the expected value of X.
Step 1: Check that every probability is between 0 and 1. Then check that the probabilities add up to 1.
0.1+0.3+0.4+0.2=10.1 + 0.3 + 0.4 + 0.2 = 1
Step 2: Multiply each value by its probability, then add the products.
E(X)=0(0.1)+1(0.3)+2(0.4)+3(0.2)=0+0.3+0.8+0.6=1.7E(X) = 0(0.1) + 1(0.3) + 2(0.4) + 3(0.2) = 0 + 0.3 + 0.8 + 0.6 = 1.7
Answer: The distribution is valid. The expected value is E(X)=1.7E(X) = 1.7 televisions per household. No single household owns 1.7 TVs. The number is the long-run average across many randomly chosen households.

Why It Matters

Discrete random variables are the core of AP Statistics Unit 4. You use them to build binomial and geometric distributions and to compute means and standard deviations of counts. Actuaries use the same expected-value calculation to set insurance premiums from the probabilities of different claim amounts.

Common Mistakes

Mistake: Believing a discrete variable must have a finite list of whole-number values.
Correction: Discrete only means the values are countable and separated. The number of coin flips until the first head (1, 2, 3, ...) has infinitely many values and is still discrete. Shoe sizes such as 7, 7.5, and 8 are not whole numbers, yet they are also discrete. A variable is continuous only when it can take any value in an interval, such as height or time.

Related Terms