Discrete random variables

A discrete random variable takes separate values, each with a probability, like the score on a dice. Its expected value is the long-run average.

Key facts

  • The probabilities add to 1.
  • Expectation: E(X) = Σ x P(X = x).
  • Variance: Var(X) = E(X²) − (E(X))².
  • E(aX + b) = aE(X) + b; Var(aX + b) = a² Var(X).

Worked examples

Example 1

X takes 1, 2, 3 with probabilities 0.2, 0.5, 0.3. Find E(X) and Var(X).

  1. E(X) = 0.2 + 1.0 + 0.9 = 2.1
  2. E(X²) = 0.2 + 2.0 + 2.7 = 4.9
  3. Var(X) = 4.9 − 2.1² = 0.49

Answer: E(X) = 2.1, Var(X) = 0.49