1. 4.1 Mathematical Expectation
Example: Repair costs for a particular machine are represented by the following probability distribution:
What is the expected value of the repairs?
That is, over time what do we expect repairs to cost on average? x
$50
200
350
P(X = x)
0.3
0.2
0.5

2. Expected value μ = E(X) For discrete variables, μ = E(X) = ∑ x f(x)
μ = mean of the probability distribution
For discrete variables,
μ = E(X) = ∑ x f(x)
So, for our example,
E(X) = ________________________
E(x) <weighted average>
E(X) = 50(0.3) + 200(0.2) + 350(0.5) = $230

3. Your turn …
By investing in a particular stock, a person can take a profit in a given year of $4000 with a probability of 0.3 or take a loss of $1000 with a probability of 0.7. What is the investor’s expected gain on the stock? (problem 7, pg. 94) X
P(X)
E(X) = 4000 (0.3) -1000(0.7) = 500

4. Expected value of Continuous Variables
For continuous variables,
μ = E(X) = _______
Example: Recall from last time, problem 7 (pg. 73)
x, 0 < x < 1
f(x) = 2-x, 1 ≤ x < 2
0, elsewhere
(in hundreds of hours.)
What is the expected value of X? {
E(X) = ∫xf(x)dx

5. = ________________________
E(X) = ∫ x f(x) dx
= ________________________
∫01 x2 dx + ∫12x(2-x)dx = x3/3 |10 + (x2 – x3/3)|12 = 1 *100 = 100 hours

6. Functions of Random Variables
Example 4.4. Probability of X, the number of cars passing through a car wash in one hour on a sunny Friday afternoon, is given by
Let g(X) = 2X -1 represent the amount of money paid to the attendant by the manager. What can the attendant expect to earn during this hour on any given sunny Friday afternoon?
E[g(X)] = Σ g(x) f(x) = ____________________
= _______________________________ x 4 5 6 7 8 9
P(X = x)
1/12
1/4
1/6
Σ (2x-1) f(x) = 7(1/12) + 9(1/12) … + 17(1/6) = $12.67

7. Your Turn
Problem 20, pg. 95