Mixture Distributions and Their Actuarial Applications
Understand mixture distributions and how they model heterogeneous insurance portfolios.
Finite Mixtures
A finite mixture distribution has PDF f(x) = w1*f1(x) + w2*f2(x) + ... + wk*fk(x), where the weights wi are non-negative and sum to 1. Each fi represents a component distribution. In insurance, mixtures model heterogeneous risk classes: a portfolio might contain 60% low-risk policies with claim distribution f1 and 40% high-risk policies with distribution f2.
The mean of a mixture is E[X] = w1*mu1 + w2*mu2 + ... + wk*muk. For Exam P, the variance formula is critical: Var(X) = sum(wi * (sigma_i^2 + mu_i^2)) - (sum(wi * mu_i))^2, which is equivalent to E[Var(X|Class)] + Var(E[X|Class]).
Continuous Mixtures
In a continuous mixture, a parameter Theta itself has a distribution. Given Theta = theta, X has conditional density f(x|theta). The unconditional (marginal) density of X is f(x) = integral of f(x|theta) * g(theta) d_theta. A classic example: if X|Lambda ~ Poisson(Lambda) and Lambda ~ Gamma(alpha, beta), then X marginally follows a Negative Binomial distribution.
Exam P Mixture Problems
A typical Exam P problem describes two or three risk groups with different loss distributions and asks for the overall expected loss, the probability that a randomly selected policy produces a claim above a threshold, or the conditional probability of belonging to a risk group given a claim has occurred (Bayes' theorem). The double expectation and total variance formulas are the primary tools: E[X] = E[E[X|Theta]] and Var(X) = E[Var(X|Theta)] + Var(E[X|Theta]).