Chapter 10: Expected Values of Discrete RVs

Meike Niederhausen and Nicky Wakim

2023-10-18

Learning Objectives

  1. Calculate the mean (expected value) of discrete random variables

Our good and fair friend, the 6-sided die

Example 1

Suppose you roll a fair 6-sided die. What value do you expect to get?

What is an expected value?

Definition: Expected value

The expected value of a discrete r.v. \(X\) that takes on values \(x_1, x_2, \ldots, x_n\) is \[\mathbb{E}[X] = \sum_{i=1}^n x_ip_X(x_i).\]

  • Expected values are not necessarily an actual outcome
    • In previous example, we cannot roll a 3.5
    • It could be that our expected value is not in the sample space (\(E(X) \notin S\))
  • Definition holds when \(X\) takes on countably infinitely many values (think \(n=\infty\))

Our good and not-so-fair friend, the 6-sided die

Example 2

Suppose the die is 6-sided, but not fair. And the probabilities of each side is distributed as:

\(x\) \(p_X(x)\)
1 0.10
2 0.05
3 0.02
4 0.30
5 0.50
6 0.03

What value do you expect to get on a roll?

Expected value of a Bernoulli distribution

Example 3

Suppose \[X = \left\{ \begin{array}{ll} 1 & \quad \mathrm{with\ probability}\ p \quad\mathrm{(success)}\\ 0 & \quad \mathrm{with\ probability}\ 1-p \quad\mathrm{(failure)} \end{array} \right.\] Find the expected value of \(X\).

Let’s slightly change our random variable

Example 5

Suppose \[X = \left\{ \begin{array}{ll} 1 & \quad \mathrm{with\ probability}\ p \\ -1 & \quad \mathrm{with\ probability}\ 1-p \end{array} \right.\] Find the expected value of \(X\).

Ghost! đŸ‘»

Example 6

A ghost is trick-or-treating. It comes to a house where it is known that there are 30 candies in the bag and only one is a watermelon Jolly Rancher, which is the ghost’s favorite. The ghost takes pieces of candy without replacement until it gets the watermelon Jolly Rancher. How many pieces of candy do we expect the ghost to take?