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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

\begin{displaymath}\frac{X_1+X_2+\cdots X_n -n\mu}{\sigma \sqrt{n}}
\end{displaymath}

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).

MX(t)=E(etX)

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

\begin{displaymath}M_Z(t)=e^{\frac{t^2}{2}}\end{displaymath}

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

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

We then prove that the log of this tended to

\begin{displaymath}\frac{t^2}{2}\end{displaymath}

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.



 
next up previous index
Next: Polar Method for Generating Up: Limit Theorems Previous: Weak Law of Large
Susan Holmes
1998-12-07