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

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

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 4000 -1000 P(X) 0.3 0.2 E(X) = 4000 (0.3) -1000(0.7) = 500

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

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

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

Your Turn Problem 20, pg. 95