1. PROBABILITY AND STATISTICS WEEK 8 Onur Doğan

2. Skewness Onur Doğan

3. Pearson's skewness coefficient Onur Doğan

4. Example It’s been understood that, in a hosptial patients’ average hospital stay is 28, median is 25 and mode is 23 (days). And the standard deviation calculated as 4,2. Define the skewness type, find the pearson coefficient and interpret it. Onur Doğan

5. Joint Probability Distributions Discrete Random Variables (two and multiple) Continious Random Variables (two and multiple) Onur Doğan

6. Two Discrete Random Variables Onur Doğan

7. Example A large insurance agency services a number of customers who have purchased both a homeowner’s policy and an automobile policy from the agency. For each type of policy, a deductible amount must be specified. For an automobile policy, the choices are $100 and $250, whereas for a homeowner’s policy, the choices are 0, $100, and $200. Suppose an individual with both types of policy is selected at random from the agency’s files. Let X the deductible amount on the auto policy and Y the deductible amount on the homeowner’s policy. Suppose the joint pmf is given in the accompanying joint probability table: Onur Doğan

8. Example P(Y ≥100)=? Onur Doğan

9. Marginal Probability Mass Function Onur Doğan

10. Example Let us obtain the marginal pmf of X evaluated at, x=100 and x=250. Let us obtain the marginal pmf of Y. Onur Doğan

11. Remember We already know! Onur Doğan

12. Conditional Probability Mass Function Recall: P(A/B)=P(A  B)/P(B) Onur Doğan

13. Example Opinion about school (X) and teachers (Y) surveyed for students as follows: a)Determine the joint probability distribution of X and Y. b)Find the probability student give 1 point for school and 2 or above point for teachers? c)Find the marginal probability distribution, mean and variance of school scores. d) Determine the conditional probability distribution of teacher scores given that school score is 2. And find the (conditional) mean and variance of Y given X=2. e)Check if school opinions and teacher opinions are independent. Onur Doğan Teachers Scores School Scores 123 1301020 2503010 3 30

14. Two Continuous Random Variables Onur Doğan

15. Example A bank operates A and B facilities. On a randomly selected day, let X the proportion of time that A in use and Y the proportion of time that the B in use. Then the set of possible values for (X, Y) is the rectangle D = {(x, y): 0≤ x ≤ 1, 0 ≤ y ≤ 1}. Suppose the joint pdf of (X, Y) is given by, a) Verify that this is a legitimate pdf. b) Find the probability that neither facility is busy more than one- quarter of the time. Onur Doğan

16. Marginal Probability Density Function Onur Doğan

17. Example Find the marginal pdf of X and Y for previous question. Find P(1/4≤Y ≤3/4)=? Onur Doğan

18. Example A nut company markets cans of deluxe mixed nuts containing almonds, cashews, and peanuts. Suppose the net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two gives all necessary information about the weight of the third type. Let X = the weight of almonds in a selected can and Y = the weight of cashews. Verify that this is a legitimate pdf. To compute the probability that the two types of nuts together make up at most 50% of the can, Onur Doğan

19. Conditional Probability Mass Function Onur Doğan

20. Example The sizes of college books are under study in terms of width (X) and height (Y). Suppose that the joint p.d.f of X and Y is modeled as f (x,y)=cxy, 2<x<5 and x+ 1 <y<x+3 (unit:inch) Determine the value of c. Determine the marginal probability distribution of X. Find also the mean and variance of X. Onur Doğan

21. Covariance Covariance and correlation between x and y? Onur Doğan

22. Covariance Shortcut formula: Onur Doğan

23. Correlation Onur Doğan

24. Example Onur Doğan