Expected Value and Variance: Computational Techniques for Exam P

Shortcut Variance Formula

The most important computational formula is Var(X) = E[X^2] - (E[X])^2. On Exam P, always compute E[X] and E[X^2] separately, then combine. This avoids computing E[(X - mu)^2] directly, which requires knowing mu first and typically involves harder integrals.

For linear transformations: E[aX + b] = a*E[X] + b and Var(aX + b) = a^2 * Var(X). The constant b shifts the mean but does not affect variance.

LOTUS and Indicator Methods

The Law of the Unconscious Statistician (LOTUS) states E[g(X)] = sum of g(x)*p(x) for discrete X or integral of g(x)*f(x) dx for continuous X. You do not need to find the distribution of g(X) first. This saves significant time on Exam P.

Indicator random variables simplify counting problems. If I_A = 1 when event A occurs and 0 otherwise, then E[I_A] = P(A) and Var(I_A) = P(A)(1 - P(A)). The total number of events among A1, ..., An is N = I_1 + ... + I_n, so E[N] = sum of P(Ai). For Var(N), include covariance terms: Var(N) = sum of Var(I_i) + 2 * sum_{i

Conditional Expectation Approach

When a problem has a natural conditioning variable, the tower property E[X] = E[E[X|Y]] often simplifies calculations dramatically. Similarly, Var(X) = E[Var(X|Y)] + Var(E[X|Y]). Choose the conditioning variable to make inner expectations easy. On Exam P, the conditioning variable is often a discrete type indicator or a mixing parameter. Practice recognizing when conditioning simplifies a seemingly difficult calculation.