Central Limit Theorem: Statement, Proof Sketch, and Applications
Understand the Central Limit Theorem and how to apply it on Exam P.
Statement of the CLT
Let X1, X2, ..., Xn be independent and identically distributed random variables with mean mu and finite variance sigma^2. Then the standardized sample mean (X_bar - mu) / (sigma / sqrt(n)) converges in distribution to a standard normal N(0, 1) as n approaches infinity. Equivalently, for large n, X_bar is approximately Normal(mu, sigma^2/n).
The CLT also applies to sums: S_n = X1 + ... + Xn is approximately Normal(n*mu, n*sigma^2). Either form may appear on Exam P.
Applying the CLT on Exam P
The typical Exam P CLT problem specifies a distribution (not necessarily normal), gives n (usually n >= 30), and asks for an approximate probability involving the sample mean or sum. Steps: (1) Find mu and sigma^2 of the individual Xi. (2) Compute the mean and variance of X_bar or S_n. (3) Standardize to Z and use the normal table.
Example: Claim sizes have mean 500 and standard deviation 200. For 100 independent claims, find P(total > 52,000). S ~ Normal(50000, 100*40000) = Normal(50000, 4000000). Z = (52000 - 50000)/2000 = 1.0. P(S > 52000) = 1 - Phi(1.0) = 0.1587.
Limitations and Refinements
The CLT requires finite variance. For heavy-tailed distributions without finite variance, the CLT does not apply. The rate of convergence depends on the skewness of the underlying distribution: symmetric distributions converge faster. For discrete distributions, applying a continuity correction of 0.5 improves accuracy. The Berry-Esseen theorem quantifies the approximation error, though this level of detail is not tested on Exam P.