Overview

Every earlier topic in this unit computed exact probabilities from a fully specified model. This closing topic asks a different question: what happens to a sample average, or a sum of many independent pieces, as the sample size grows without bound? The answer — the law of large numbers in two forms (weak and strong), and the Central Limit Theorem — is the theoretical justification for treating a sample mean as a reliable estimate of a population mean, and for approximating almost any sum of independent contributions as normal, regardless of the individual pieces' own distribution. The topic opens with two general-purpose tail-probability bounds (Markov, Chebyshev) that require nothing beyond a mean and a variance, and closes with a worked Monte Carlo application that ties every idea in the unit — expectation, variance, the CLT — into a single practical estimation procedure.