Chapter 11: Expected Values of Sums of Discrete RVs

Meike Niederhausen and Nicky Wakim

2023-10-18

Table of contents

• Learning Objectives
• Revisiting our two card draw
• What if we draw A LOT of cards?
• Sum of discrete RVs
• Corollaries from Thm 11.1
• Revisiting our ghost! 👻
• Cost of hotel rooms

Learning Objectives

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

Revisiting our two card draw

Example 1

Suppose you draw 2 cards from a standard deck of cards with replacement. Let \(X\) be the number of hearts you draw. Find \(\mathbb{E}[X]\).

Recall Binomial RV with \(n=2\):

\[p_X(x) = {2 \choose x}p^x(1-p)^{2-x} \text{  for } x = 0, 1, 2\]

What if we draw A LOT of cards?

Example 2

What is the expected number of hearts in Example 1 if you draw 200 cards?

Recall Binomial RV with \(n=200\):

\[p_X(x) = {200 \choose x}p^x(1-p)^{200-x}\] \[\text{  for } x = 0, 1, 2, ..., 200\]

Sum of discrete RVs

Theorem 11.1: Sum of discrete RVs

For discrete r.v.’s \(X_i\) and constants \(a_i\), \(i=1,2,\dots, n\), \[\mathbb{E}\Bigg[\sum_{i=1}^n a_iX_i\Bigg] = \sum_{i=1}^n a_i\mathbb{E}[X_i] .\] Remark: The theorem holds for infinitely r.v.’s \(X_i\) as well.

• For two RVs, \(X\) and \(Y\):
  • We can say \(E[X+Y] = E[X] + E[Y]\)
  • … and constant numbers \(a\) and \(b\), we can also say \(E[aX+bY] = aE[X] + bE[Y]\)
  • We can also also say \(E[X-Y] = E[X] - E[Y]\), since \(b=-1\)

Corollaries from Thm 11.1

Corollary 11.1.1

For a discrete r.v. \(X\), and constants \(a\) and \(b\), \[\mathbb{E}[aX+b] = a\mathbb{E}[X] + b.\]

Corollary 11.1.2

If \(X_i\), \(i=1,2,\dots, n\), are identically distributed r.v.’s, then \[\mathbb{E}\bigg[\sum_{i=1}^n X_i\bigg] = n\mathbb{E}[X_1] .\]

Revisiting our ghost! 👻

Example 3

The ghost is trick-or-treating at a different house now. In this case it is known that the bag of candy has 10 chocolates, 20 lollipops, and 30 laffy taffies. The ghost grabs a handful of five pieces of candy. How many pieces of chocolate do we expect the ghost to take?

Cost of hotel rooms

Example 4

A tour group is planning a visit to the city of Minneapolis and needs to book 30 hotel rooms. The average price of a room is $200. In addition, there is a 10% tourism tax for each room. What is the expected cost for the 30 hotel rooms?