Discrete random variables
P(X = x) for x in a finite set. E(X) = Σ xP(x); Var(X) = E(X²) − (E(X))².
Y12 Stats. A discrete RV takes specific values. Probability mass function P(X = x) sums to 1. Expected value E(X) is the long-run mean.
Worked examples
X = roll on a fair die: E(X) = (1+2+...+6)/6 = 3.5.
Var(X) for fair die = E(X²) − 3.5² = 91/6 − 12.25 ≈ 2.917.
P(X ≤ 3) for fair die = 1/2.
Frequently asked questions
E(X+Y)?
Always = E(X) + E(Y), regardless of independence.
Var(X+Y)?
If independent: Var(X) + Var(Y). Otherwise depends on covariance.