# Lesson 26: Random Functions Associated with Normal Distributions

Printer-friendly version

### 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 S2. 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 X1, X2, ..., Xn 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 X1X2, ... , Xn
• To learn how to find the probability that a linear combination of independent normal random variables X1X2, ... , Xn takes on a certain interval of values.
• To learn the sampling distribution of the sample mean when X1X2, ... , Xn are a random sample from a normal population with mean μ and variance σ2
• To use simulation to get a feel for the shape of a probability distribution.
• To learn the sampling distribution of the sample variance when X1X2, ... , Xn are a random sample from a normal population with mean μ and variance σ2
• 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.