Lesson 8: Mathematical Expectation

Overview Section

In this lesson, we learn a general definition of mathematical expectation, as well as some specific mathematical expectations, such as the mean and variance.

Objectives

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

  • To get a general understanding of the mathematical expectation of a discrete random variable.
  • To learn a formal definition of \(E[u(X)]\), the expected value of a function of a discrete random variable.
  • To understand that the expected value of a discrete random variable may not exist.
  • To learn and be able to apply the properties of mathematical expectation.
  • To learn a formal definition of the mean of a discrete random variable.
  • To derive a formula for the mean of a hypergeometric random variable.
  • To learn a formal definition of the variance and standard deviation of a discrete random variable.
  • To learn and be able to apply a shortcut formula for the variance of a discrete random variable.
  • To be able to calculate the mean and variance of a linear function of a discrete random variable.
  • To learn a formal definition of the sample mean and sample variance.
  • To learn and be able to apply a shortcut formula for the sample variance.
  • To understand the steps involved in each of the proofs in the lesson.
  • To be able to apply the methods learned in the lesson to new problems.