1. Statistics 270 - Lecture 20

2. Last Day…completed 5.1 Today Section 5.2 Next Day: Parts of Section 5.3 and 5.4

3. Expected Value for Joint Distributions If X and Y are jointly distributed rv’s, then the expected value of a function, h(X,Y), is given by Continuous Discrete

4. Example An instructor has given a short test consisting of two parts For a randomly selected student, let X = # points earned on the first part and Y =#points earned on the second part Suppose that the joint pmf of X and Y is given by

5. Example If the score recorded in the grade book is the total number of points earned on the two parts, what is the expected recorded score?

6. Show that if X and Y are independent random variables, then E(XY)=E(X)E(Y)

7. Covariance Suppose X and Y are jointly distributed rv’s If X and Y are dependent, one is frequently interested in the strength of their relationship This can be measured by the covariance Gives both direction and magnitude and direction of association

8. Covariance If X and Y are jointly distributed rv’s, then the covariance between X and Y is Continuous Discrete

9. Covariance Visually

10. Covariance – Short-cut If X and Y are jointly distributed rv’s, then the covariance between X and Y is Cov(X,Y)=E(XY)-E(X)E(Y)

11. Example An instructor has given a short test consisting of two parts For a randomly selected student, let X = # points earned on the first part and Y =#points earned on the second part Suppose that the joint pmf of X and Y is given by

12. Example Find the covariance between X and Y

13. Correlation Covariance has the drawback that the value depends on the units of measurement…hard to interpret A related, unit free, measure is the correlation coefficient

14. Correlation Properties:

15. Example Find the correlation between X and Y in the previous example