The Central Limit Theorem

The Central Limit Theorem:
Let $X_1,X_2,X_3,\ldots$ be a sequence of iid random variables with expectation $\mu$ and variance $\sigma^2$, then the distribution of

$$\frac{X_1+X_2+\cdots X_n -n\mu}{\sigma \sqrt{n}}$$

tends to be standard normal as $n\longrightarrow \infty$.

Proof(I didn't do it in class):
Uses the notion of moment generating function(MGF).

M_X(t)=E(e^tX)

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

$$M_Z(t)=e^{\frac{t^2}{2}}$$

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

$$MGF(\frac{\sum X_i}{\sqrt{n}})=\left[M(\frac{t}{\sqrt{n}})\right]^n$$

We then prove that the log of this tended to

$$\frac{t^2}{2}$$

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.