Define expected value and variance. What are their key properties?

How to think about it

These two quantities underpin every loss function, every risk metric, and every A/B test power calculation. Know the definitions cold and the algebra by heart.

Expected value

For a discrete random variable X:

E[X] = Σ x · P(X = x)

For a continuous random variable:

E[X] = ∫ x · f(x) dx

Key properties:

• Linearity: E[aX + bY] = a·E[X] + b·E[Y] for any constants a, b — even when X and Y are dependent.
• E[c] = c for any constant c.

Variance

Var(X) = E[(X - E[X])²] = E[X²] - (E[X])²

The second form — “mean of squares minus square of mean” — is usually easier to compute.

Key properties:

• Var(aX + b) = a²·Var(X) — shifting by b has no effect; scaling multiplies by a².
• If X and Y are independent: Var(X + Y) = Var(X) + Var(Y).
• If dependent, add the covariance term: Var(X + Y) = Var(X) + Var(Y) + 2·Cov(X,Y).

Worked numeric example

A game pays £5 with probability 0.4 and -£2 with probability 0.6.

E[X] = 5×0.4 + (-2)×0.6 = 2.0 - 1.2 = 0.80
E[X²] = 25×0.4 + 4×0.6 = 10 + 2.4 = 12.4
Var(X) = 12.4 - (0.80)² = 12.4 - 0.64 = 11.76
SD(X) = √11.76 ≈ 3.43

Expected profit is £0.80 per game, but the standard deviation is £3.43 — high risk relative to the reward.