Expectation of a discrete random variable
Multiply each possible value by its probability and add the products.
For a finite distribution, E(X) = ΣxP(X = x). Check that all probabilities are nonnegative and sum to one before calculating. The expected value need not be a possible outcome: it is a probability-weighted average.
For a function g, E(g(X)) = Σg(x)P(X = x). In general E(X²) is not [E(X)]²; their difference is the variance. Keep units meaningful, particularly when a variable represents money or a measured quantity.
Worked example
X is 0, 1 or 2 with probabilities 0.2, 0.5 and 0.3. Find E(X).
- The probabilities sum to 1.
- Calculate 0 × 0.2 + 1 × 0.5 + 2 × 0.3.
Answer: 1.1
Practise expectation of a discrete random variable
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 9FM0 · FS1 · Discrete random variables: Expected value of a discrete variable
- 7367 · statistics · Discrete random variables: Expected value of a discrete variable
- H245 · statistics · Discrete random variables: Expected value of a discrete variable
- H645 · Y422 · Discrete random variables: Expected value of a discrete variable
- H645 · Y432 · Discrete random variables: Expected value of a discrete variable
- 9FM0 · FS2 · 3.1 Geometric distributions: A geometric waiting-time probability
- H245 · statistics · 5.02f–g Geometric distributions: A geometric waiting-time probability
- H645 · Y422 · SR16–R17 Geometric distributions: A geometric waiting-time probability
- H645 · Y432 · SR16–R17 Geometric distributions: A geometric waiting-time probability
- H635 · Y412 · SR16–R17 Geometric distributions: A geometric waiting-time probability
Next practice: Binomial probabilities.