Combinatorics • Topic 24

Expected Value

The expected value is the "long-run average" of a random variable. It is the single most important statistic in probability.

Definition

Definition (Expected Value)

For a discrete random variable

X

, the expected value

E [X]

is the weighted average of all possible values:

E [X] = x \sum x \cdot P (X = x)

If you place a weight of mass

P (X = x)

at position

x

on a number line,

E [X]

is the center of mass.

Example (Fair Die)

Let

X

be the roll of a die.

E [X] = 1 (\frac{1}{6}) + 2 (\frac{1}{6}) + \dots + 6 (\frac{1}{6}) = \frac{21}{6} = 3.5

. Note: You can never actually roll a 3.5.

E [a X + b] = a E [X] + b

Exercise (Problem 1)

In a raffle, there is 1 prize of

1000 an d 999 l os in g t i c k e t s . A t i c k e t cos t s

2. What is the expected profit of buying one ticket?

Exercise (Problem 2)

Find the expected value of

X^{2}

X

is the roll of a fair die. (Note:

E [X^{2}] \neq = (E [X])^{2}

Exercise (Problem 3)

A game ends when you roll a 6. What is the expected number of rolls required? (Recall the Geometric series).