# Lesson 27: The Central Limit Theorem

### Introduction

In the previous lesson, we investigated the probability distribution ("sampling distribution") of the sample mean when the random sample *X*_{1}, *X*_{2}, ..., *X*_{n} comes from a normal population with mean *μ *and variance *σ*^{2}, that is, when *X _{i} *~

*N*(

*μ*,

*σ*

^{2}),

*i*= 1, 2, ...,

*n*. Specifically, we learned that if

*X*1, 2,... ,

_{i}, i =*n,*is a random sample of size

*n*from a

*N*(

*μ*,

*σ*

^{2}) population, then:

\(\bar{X}\sim N\left(\mu,\dfrac{\sigma^2}{n}\right)\)

But what happens if the *X _{i}* follow some other non-normal distribution? For example, what distribution does the sample mean follow if the

*X*come from the

_{i}*Uniform*(0, 1) distribution? Or, what distribution does the sample mean follow if the

*X*come from a chi-square distribution with three degrees of freedom? Those are the kinds of questions we'll investigate in this lesson. As the title of this lesson suggests, it is the

_{i }**Central Limit Theorem**that will give us the answer.

### Objectives

- To learn the Central Limit Theorem.
- To get an intuitive feeling for the Central Limit Theorem.
- To use the Central Limit Theorem to find probabilities concerning the sample mean.
- To be able to apply the methods learned in this lesson to new problems.