1. Chapter 4 Joint Distribution & Function of rV

2. Joint Discrete Distribution Definition

3. Joint discrete CDF

4. Example for joint distributions Consider the following table: Using the table, we have Y=0 Y=3 Y=4 X=5 1/7 1/7 1/7 3/7 X=8 3/7 0 1/7 4/7 4/7 1/7 2/7 pXpX pYpY

5. The Marginal PDF

6. Example : Air Conditioner Maintenance –A company that services air conditioner units in residences and office blocks is interested in how to schedule its technicians in the most efficient manner –The random variable X, taking the values 1,2,3 and 4, is the service time in hours –The random variable Y, taking the values 1,2 and 3, is the number of air conditioner units

7. Expected Values for Jointly Distributed Random Variables Let X and Y be discrete random variables with joint probability density function p(x, y). Let the sets of values of X and Y be A and B, resp. We define E(X) and E(Y) as For the random variables X and Y from the previous slide example,

8. Joint p.d.f Joint cdf Y= number of units X=service time 1234 10.120.080.070.05 20.080.150.210.13 30.01 0.020.07 Find E[X] and E[Y] !!

9. Previously Example –Marginal p.d.f of X –Marginal p.d.f of Y

10. Joint continuous distribution

11. Th

12. Ex a. Then find the joint CDF !

13. The Marginal Continuous PDF FindFrom previously example

14. Ex

15. Independent rV

16. Conditional pdf

17. Joint MGF

18. Group Discuss the exercise bellow ! BAIN, page : 166- No 7, 9, 21, 30 Time : 30’