1. EGR 252 - 51 Joint Probability Distributions The probabilities associated with two things both happening, e.g. … –probability associated with the hardness and tensile strength of an alloy f(h, t) = P(H = h, T = t) –probability associated with the lives of 2 different components in an electronic circuit f(a, b) = P(A = a, B = b) –probability associated with the diameter of a mold and the diameter of the part made by that mold f(d 1, d 2 ) = P(D 1 = d 1, D 2 = d 2 )

2. EGR 252 - 52 Example Statistics were collected on 100 people selected at random who drove home after a college football game, with the following results: Accident No accident TOTAL Drinking72330 Not drinking 66470 TOTAL1387100

3. EGR 252 - 53 Joint Probability Distribution The joint probability distribution or probability mass function for this data: –The marginal distributions associated with drinking and having an accident are calculated as: _______________________________________ Accident No accident g(x) Drinking0.070.230.3 Not drinking 0.060.640.70 h(y)0.130.871

4. EGR 252 - 54 Conditional Probability Distributions The conditional probability distribution of the random variable Y given that X = x is For our example, the probability that a driver who has been drinking will have an accident is f(accident | drinking) = ___________________ (note: refer to page 95 of your textbook for the conditional probability that a variable falls within a range.)

5. Statistical Independence If f(x | y) doesn’t depend on y, then f(x | y) = g(x) and f(x, y) = g(x) h(y) X and Y are statistically independent if and only if this holds true for all (x, y) within their range. For our example, are drinking and being in an accident statistically independent? Why or why not? EGR 252 - 55

6. Continuous Random Variables f(x, y) is the joint density function of the continuous variables X and Y, and the associated probability is for any region A in the xy plane. The marginal distributions of X and Y alone are See examples 3.15, 3.17, 3.19, 3.20, and 3.22 (starting on pg. 93) EGR 252 - 56

7. 7 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$50200350 P(X = x)0.30.20.5

8. EGR 252 - 58 Expected value μ = E(X) –μ = mean of the probability distribution For discrete variables, μ = E(X) = ∑ x f(x) So, for our example, E(X) = ________________________

9. EGR 252 - 59 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?

10. EGR 252 - 510 Expected value of Continuous Variables For continuous variables, μ = E(X) = _______ Example: Recall from last time, problem 3.7 (pg. 88) x, 0 < x < 1 f(x) =2-x,1 ≤ x < 2 0, elsewhere (in hundreds of hours.) What is the expected value of X? {

11. EGR 252 - 511 E(X) = ∫ x f(x) dx = ________________________

12. EGR 252 - 512 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) = ____________________ = _______________________________ x456789 P(X = x)1/12 1/4 1/6