# Lesson 26: Random Functions Associated with Normal Distributions

### Introduction

In the previous lessons, we've been working our way up towards fully defining the probability distribution of the sample mean \(\bar{X}\) and the sample variance *S*^{2}. We have determined the expected value and variance of the sample mean. Now, in this lesson, we (finally) determine the probability distribution of the sample mean and sample variance when a random sample *X*_{1}, *X*_{2}, ..., *X _{n}* is taken from a normal population (distribution). We'll also learn about a new probability distribution called the

**(Student's) t distribution**.

**Objectives**

- To learn the probability distribution of a linear combination of independent normal random variables
*X*_{1},*X*_{2}, ... ,*X*_{n}. - To learn how to find the probability that a linear combination of independent normal random variables
*X*_{1},*X*_{2}, ... ,*X*_{n}takes on a certain interval of values. - To learn the sampling distribution of the sample mean when
*X*_{1},*X*_{2}, ... ,*X*_{n}_{ }are a random sample from a normal population with mean*μ*and variance*σ*^{2}. - To use simulation to get a feel for the shape of a probability distribution.
- To learn the sampling distribution of the sample variance when
*X*_{1},*X*_{2}, ... ,*X*_{n}_{ }are a random sample from a normal population with mean*μ*and variance*σ*^{2}. - To learn the formal definition of a
*T*random variable. - To learn the characteristics of Student's
*t*distribution. - To learn how to read a
*t*-table to find*t*-values and probabilities associated with*t*-values. - To understand each of the steps in the proofs in the lesson.
- To be able to apply the methods learned in this lesson to new problems.