Famous Theorems of Mathematics/Law of large numbers

From Wikibooks, open books for an open world
Jump to: navigation, search

Given X1, X2, ... an infinite sequence of i.i.d. random variables with finite expected value E(X1) = E(X2) = ... = µ < ∞, we are interested in the convergence of the sample average

The weak law[edit]

Theorem:

Proof:


This proof uses the assumption of finite variance (for all ). The independence of the random variables implies no correlation between them, and we have that

The common mean μ of the sequence is the mean of the sample average:

Using Chebyshev's inequality on results in

This may be used to obtain the following:

As n approaches infinity, the expression approaches 1. And by definition of convergence in probability (see Convergence of random variables), we have obtained