Lesson 26: Random Functions Associated with Normal Distributions

Overview Section

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 \(X_1, X_2, \ldots, X_n\) is taken from a normal population (distribution). We'll also learn about a new probability distribution called the (Student's) t distribution.

Objectives

Upon completion of this lesson, you should be able to:

  • To learn the probability distribution of a linear combination of independent normal random variables \(X_1, X_2, \ldots, X_n\).
  • To learn how to find the probability that a linear combination of independent normal random variables \(X_1, X_2, \ldots, X_n\) takes on a certain interval of values.
  • To learn the sampling distribution of the sample mean when \(X_1, X_2, \ldots, X_n\) are a random sample from a normal population with mean \(\mu\) and variance \(\sigma^2\).
  • To use simulation to get a feel for the shape of a probability distribution.
  • To learn the sampling distribution of the sample variance when \(X_1, X_2, \ldots, X_n\) are a random sample from a normal population with mean \(\mu\) and variance \(\sigma^2\).
  • 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.