Loss Models: From Data to Decisions, 4th Edition

3.3 Generating functions and sums of random variables

Consider a portfolio of insurance risks covered by insurance policies issued by an insurance company. The total claims paid by the insurance company on all policies is the sum of all payments made by the insurer. Thus, it is useful to be able to determine properties of S_k = X₁ + ··· + X_k. The first result is a version of the central limit theorem.

Theorem 3.7 For a random variable S_k as previously defined, E(S_k) = E(X₁) + ··· + E(X_k). Also, if X₁, …, X_k are independent, Var(S_k) =Var(X₁) + ··· + Var(X_k). If the random variables X₁, X₂, … are independent and their first two moments meet certain conditions, lim_k→∞ [S_k − E(S_k)]/ has a normal distribution with mean 0 and variance 1.

When working with a sequence of random variables there are many types of limit. The limit used in the theorem is called convergence in distribution. It means that for a given argument the distribution function converges to its limiting case. Here, if we define Z_k = [S_k − E(S_k)]/, then, for any value of z, lim_k→∞ F_Zk (z) = F_Z(z), where Z has a standard normal distribution. Thus, probabilities of sums of random variables can often be approximated by those from the normal distribution.

Obtaining the exact distribution or density function of S_k is usually very ...