Let be a sequence of iid random variables with expectation and variance , then the distribution of

tends to be standard normal as .

Proof(I didn't do it in class):

Uses the notion of moment generating function(MGF).

A way of proof can be seen through the fact
(that I didn't prove) that if the generating functions
of a sequence of random variables converges to
the limiting generating function of a random
variable *Z* then the distribution functions
converge to the distribution function of *Z*.

Reminder: The generating function of the standard Normal is

We show through use of the fact that the moment generating function
of a sum of independent rv's is the product
of their mgf's and the fact that
they have the same distribution
then

We then prove that the log of this tended to

by using l'Hopital's rule.

The theorem is also true for independent variables who do not have the same distributions, along as they are bounded and the means and variances are finite.