The strong law of large numbers

1. The strong law of large numbers • Theorem. Let X1, X2, ... be a sequence of independent and identically distributed random variables, each having the same finite mean µ. Then, with probability 1, Proof. Left for graduate school. • Example. Suppose a sequence of independent trials of an experiment is performed. Let E be a fixed event of the experiment and let P(E) be the probability that E occurs on any particular trial. Let Xi = 1 if E occurs on the ith trial and let Xi = 0 otherwise. Note that E[Xi] = P(E). By the above theorem, except for a set of probability zero,

2. The central limit theorem • Theorem. Let X1, X2, ... be a sequence of independent and identically distributed random variables, each having the same finite mean µ and the same finite variance 2. Then the distribution of tends to the standard normal as n . That is , for all real numbers a, where  is the c.d.f. for the standard normal distribution.

3. An application of the central limit theorem • Problem. Let Xi, i = 1, 2, ... , 10 be independent r.v.’s, each uniformly distributed over (0, 1). Calculate an approximation to • Solution. Since E[Xi] = 1/2 and Var(Xi) = 1/12, we have by the central limit theorem,