1. Chapter Five Joint Probability Distributions & Random Samples

2. Joint Probability Mass Function For Discrete RV’s X & Y: p(x,y) = P(X=x and Y=y) For any set A consisting of pairs of (x, y) values: P[(X,Y)  A] =   p(x,y) (x,y)  A

3. Joint pmf Example In an automobile plant two tasks are performed by robots. The first entails welding 2 joints; the second, tightening 3 bolts. RV X denotes the number of defective welds and RV Y the number of improperly tightened bolts produced per car. Past data indicates the following joint pmf for (X,Y). x/y Is this a valid pmf? What is the probability of exactly one error?

4. Joint pmf Example You’re working on a project at your local supermarket. Let RV X denote the number of customers in line at the express checkout at a particular time of day and let RV Y denote the number of customers in line at the super-express checkout at the same time. The joint pmf of X & Y follows: Y X What is the probability that there is exactly 1 customer in each line? What is the probability that the number of customers in the 2 lines are identical? What is the probability that the total number of customers in the 2 lines is exactly four?

5. Joint pmf Example Find the joint pmf and joint CDF with the experiment of tossing 2 dice where RV X represents the number appearing on the up face of one black die and RV Y represents the number appearing on the up face of a red die.

6. Joint Discrete pmf Example Two students have just submitted reports on final projects. RV X denotes the number of typographical errors on the first student’s report and RV Y denotes the number of errors on the second report. Assume both X and Y are independent and both have a Poisson distribution with parameter  = 1. What is the joint pmf of X and Y? What is the probability that at most one error is made on both reports combined?

7. Joint Probability Density Function For Continuous RV’s X & Y with Joint pdf f(x,y): b d P(axb, cyd) =   f(x,y) dxdy a c

9. Joint pdf Example In a healthy individual age 20 to 29 years, the calcium level in the blood, X, is usually between 8.5 and 10.5 milligrams per deciliter (mg/dl) and the cholesterol level, Y, is usually between 120 and 240 mg/dl. Assume that for a healthy individual in this age group the RV (X,Y) is uniformly distributed and the joint pdf for (X,Y) = k. What is the value of k for this to be a valid pdf? What is the probability that an individual’s calcium level will lie between 9 & 10 mg/dl, whereas the cholesterol level is between 125 &140 mg/dl?

10. Example: Joint PDF Before a polymer can be used to make products, it is usually necessary to mix it with added ingredients. Plasticizers are used to increase flexibility while antioxidants are used to protect against atmospheric oxidation. (Many polymers have sites on their molecular chains which are susceptible to attack by oxygen). A supplier of polymer-additives stocks both a plasticizer and an antioxidant. Let RV X denote the amount (tons) of plasticizer in stock and RV Y the amount (tons) of antioxidant in stock. The joint pdf of X and Y is f(x,y) = k*(x)*(y) for x ≥ 0 and y ≥ 0. Inventory is controlled by keeping the added amounts of both additives between 1 and 2 tons; therefore, 1 ≤ (x + y) ≤ 2 also. What is the value of k that makes this a valid joint pdf?

11. Joint pdf Example In the behavior of air support roofs, the RV X, the inside barometric pressure (inches of mercury), and RV Y, the outside pressure have a joint pdf as follows: f(X,Y) = c/x where 27yx33. What is the value for c? What is the probability that the inside pressure is at most 30 & the outside pressure is at most 28?

12. Joint pdf Continuous Example The joint pdf for the times to failure (in years) of two transistors is given by: f(X,Y) = 2e-(x+y) where  = 2. What is the probability that one will fail before 2 years and one will fail before 3 years?

13. Marginal Probability Mass Function For Discrete RV’s X & Y:
Marginal Probability Mass Function For Discrete RV’s X & Y: px(x) =  p(x,y) y py(y) =  p(x,y) x

14. Joint Discrete pmf Example A restaurant serves 3 dinners costing 7, 9, & 10 dollars. For a randomly selected couple, let RV X denote the cost of the man’s dinner & RV Y denote the cost of the lady’s dinner. The joint pmf of X and Y follows: Y $7 $9 $10 X $ $ $ What is the marginal pmf’s of X and Y? What is the probability that the man’s & his date’s dinner cost at most $9 each? Are RV X & RV Y independent?

15. Marginal Probability Density Function For Continuous RV’s X & Y:  px(x) =  f(x,y) dy for -<x<   py(y) =  f(x,y) dx for -<y< 

16. Marginal Continuous pdf Example Let RV X denote an individual’s blood calcium level and RV Y his or her blood cholesterol level. The joint pdf is f(X,Y) = 1/240 for 8.5  x  and 120  y  240. What are the marginal pdf’s for X &Y? What is the probability that a healthy individual has a cholesterol level between 150 and 200?

17. Marginal pdf Example The joint pdf of RV’s X and Y is as follows: f(x,y) = axy for 1  x  3, 2  y  4 and 0 elsewhere. What is the value of a? What is the marginal pdf of X? What is the marginal CDF of Y?

18. Marginal pdf Example The joint pdf of RV’s X and Y is as follows: f(x,y) = a(x+y) for 0  x  1, 0  y  2 and 0 elsewhere. What is the value of a? What is the marginal pdf of Y? What is the CDF of X,Y?

19. Expected Values of Jointly Distributed RV’s For Discrete X and Y: E[h(X,Y)] =   h(x,y) p(x,y) x y For Continuous X and Y:  + E[h(X,Y)] =   h(x,y)f(x,y)dxdy  - With Joint Distributions pmf p(x,y) or pdf f(x,y).

20. Example Expected Value Joint pmf You continue work on your supermarket project. What is the expected difference between the number of customers in line at the express checkout and the number in line at the super-express checkout?

21. Expected Joint Example In an automobile plant two tasks are performed by robots. The first entails welding 2 joints; the second, tightening 3 bolts. RV X denotes the number of defective welds and RV Y the number of improperly tightened bolts produced per car. Past data indicates the following joint pmf for (X,Y). x/y | fx(x) | | | fy(y) | What is the expected number mistakes?

22. Joint Expected Value Example RV X and Y have joint pdf: f(x,y) = 2y for 0  x   y  elsewhere What is the E(XY) value of the joint pdf?

23. Joint Expected Value Example For design of air support roofs, it is important to know the expected lift pressure (inches of mercury) of your system. For the system where the joint pdf was f(X,Y) = c/x where 27yx33, find the expected lift pressure. (c = 0.15)

24. Example Joint Expected Value You have agreed to meet your date for dinner between 6 & 7 at the local Red Lobster. Your arrival time, independent of your date’s arrival time, is f(x) = 3x2. Your date’s arrival time is f(y) = 2y. What is the expected amount of time that the one who arrives first must wait for the other person?

25. Covariance Cov(X,Y) = E[(X-ux)(Y-uy)] Discrete Case:   (x – ux)(y – uy) p(x,y) X Y Continuous Case: +   (x – ux)(y – uy) f(x,y) dxdy -

26. Covariance Short Cut Method Cov(X,Y) = E(XY) - uxuy

27. Covariance Example In an automobile plant two tasks are performed by robots. The first entails welding 2 joints; the second, tightening 3 bolts. RV X denotes the number of defective welds and RV Y the number of improperly tightened bolts produced per car. Past data indicates the following joint pmf for (X,Y). x/y | fx(x) | | | fy(y) | What is the Covariance between X and Y?

28. Covariance Example RV’s X and Y have joint pdf:
Covariance Example RV’s X and Y have joint pdf: f(x,y) = 2x for 0  x  for 0  y  elsewhere What is the Covariance of RV’s X & Y?

29. Correlation Coefficient Corr(X,Y) = Cov(X,Y)
Correlation Coefficient Corr(X,Y) = Cov(X,Y) x y Properties:  Corr(X,Y)  +1 For any two RV’s X and Y: Corr(aX+b, cY+d) = Corr(X,Y)

30. Covariance & Correlation Example RV’s X and Y have joint pmf: p(x,y) = ¼ for (x,y) = (- 4,1),(4,-1),(2,2),(-2,-2) elsewhere What is the Covariance and Correlation Coefficient for X, Y?

31. Covariance & Correlation Example A maker of lawnmowers found the following joint pmf for RV X (the months of warranty) and RV Y (the engine horsepower) for lawnmowers customers were buying: Y > X What is the Covariance & Correlation between X and Y?

32. Correlation Example Continuous RV’s X and Y have the following values: E(X) = E(Y) = -1 E(X2) = E(Y2) = 10 E(XY) = 6 Find the Correlation Coefficient for X and Y? Are X and Y independent?

33. Correlation Example The joint pdf for (X,Y) is given by:
Correlation Example The joint pdf for (X,Y) is given by: f(x,y) = x3y3 /  x   y  2 What is the correlation between RV’s X and Y?