Double Expectation and Total Variance Formulas
Master the law of total expectation and total variance for Exam P.
Law of Total Expectation
The law of total expectation (also called the tower property or iterated expectation) states E[X] = E[E[X|Y]]. Here, E[X|Y] is a random variable (a function of Y), and taking its expectation over Y gives E[X]. This is the most frequently used formula on Exam P for problems involving mixed or hierarchical models.
Example: Suppose the number of claims N ~ Poisson(Lambda) where Lambda ~ Gamma(2, 1). Then E[N] = E[E[N|Lambda]] = E[Lambda] = 2. Without the tower property, you would need to find the marginal distribution of N first.
Law of Total Variance
Var(X) = E[Var(X|Y)] + Var(E[X|Y]). The first term captures the average variability within groups, and the second term captures the variability between group means. This decomposition is essential for Exam P and is the foundation of credibility theory.
Continuing the example: Var(N) = E[Var(N|Lambda)] + Var(E[N|Lambda]) = E[Lambda] + Var(Lambda) = 2 + 2 = 4. Since N|Lambda ~ Poisson(Lambda), Var(N|Lambda) = Lambda and E[N|Lambda] = Lambda.
Application Strategy
On Exam P, whenever you see a hierarchical model (X depends on Theta, and Theta has its own distribution), immediately apply these formulas. Step 1: compute E[X|Theta] and Var(X|Theta) using the conditional distribution. Step 2: treat these as functions of Theta and compute their expectations and variances using Theta's distribution. Common pairings include Poisson-Gamma (yielding negative binomial), Normal-Normal, and Binomial-Beta. Recognizing the structure quickly saves time on the exam.