Law of Large Numbers: Weak and Strong Forms for Actuaries
Learn both forms of the Law of Large Numbers and their actuarial significance.
Weak Law of Large Numbers
The Weak Law of Large Numbers (WLLN) states that for iid random variables X1, X2, ..., Xn with mean mu, the sample mean X_bar converges to mu in probability: for every epsilon > 0, P(|X_bar - mu| > epsilon) approaches 0 as n approaches infinity. This can be proved using Chebyshev's inequality: P(|X_bar - mu| > epsilon) <= Var(X_bar) / epsilon^2 = sigma^2 / (n * epsilon^2).
In actuarial terms, the WLLN guarantees that the average claim cost per policy converges to the true expected claim cost as the portfolio grows. This is the mathematical foundation of insurance pricing.
Strong Law of Large Numbers
The Strong Law (SLLN) provides a stronger guarantee: X_bar converges to mu almost surely, meaning P(lim X_bar = mu) = 1. The SLLN requires only that E[|X|] < infinity (the Kolmogorov version). While the distinction between weak and strong convergence is important theoretically, Exam P primarily tests the WLLN and its applications.
Actuarial Significance
The LLN justifies the insurance business model. By pooling a large number of independent, identically distributed risks, an insurer can predict its average loss per policy with high accuracy. The risk per policy decreases as the portfolio grows. However, the total risk (absolute variability of total losses) still increases with n, growing proportionally to sqrt(n). This distinction between risk per policy and total risk is conceptually important for actuaries and occasionally appears in exam problems.