1. Lecture 15: Expectation for Multivariate Distributions Probability Theory and Applications Fall 2008 Those who ignore Statistics are condemned to reinvent it. Brad Efron

2. Outline Correlation Expectations of Functions of R.V. Covariance Covariance and Independence Algebra of Covariance

3. Correlation Intuition Covariance is a measure of how much RV vary together. Wife’s Age and Husband’s Age Correlation.97 Example from http://cnx.org/content/m10950/latest/

4. Sometimes not so perfect Arm Strength Versus Grip Strength Pearson’s Correlation R=.63

5. Negative Correlation Child Labor versus GDP

6. Extreme Correlation 1 Linear relation with positive slope

7. Extreme: Correlation -1 Linear relation with negative slope

8. Zero Correlation Independent Random

9. Guess Covariance??? Positive, Negative, 0 Crime Rate, Housing Price SAT Scores, GPA Freshman Year Weight and SAT Score Average Daily Temperature, Housing Price GDP, Infant Mortality Life Expectancy, Infant Mortality

10. Expectations of Functions of R.V. NOTE substitute appropriate summation for discrete

11. Variance and Covariance Univariate becomes variance Multivariate becomes covariance Note: NOTE substitute appropriate integral for continuous

12. Calculating Covariance Can simplify

13. Correlation of X and Y Definition The correlation always falls in [ -1, 1] It a measure of the linear relation between X and Y

14. Extreme Cases If X=Y then ρ=1. If X=-Y then ρ=-1. If X and Y independent, then ρ=0. If X=-2Y then ρ=?.

15. Example Joint is Find correlation of X and Y

16. Example Joint is Find correlation of X and Y

17. Properties of Covariance a) b) cov(aX+bY)=ab[cov(X,Y)]

18. Properties of Covariance c) d)

19. Properties of Covariance e)If X and Y are independent Proof:

20. Find Covariance X\Y01 0.3 0.40 10.3

21. Are X and Y independent X\Y01 0.3 0.40 10.3

22. Note Cov(X,Y)=0 does not imply independence of X and Y Independence of X and Y implies cov(X,Y)=0 In this case Y=X 2 so the variables are definitely not independent but their covariance is 0 because they have no linear relation.

23. Algebra of variance/covariance/correlation Given: Calculate mean of Z=2X-3Y+5 variance of Z=2X-3Y+5

24. Long steps

25. Working Rules for linear combinations Write formula Discard Constants Square it Replace squared R.V with var and crossterms with cov

26. Example Given var(X)=4 var(Y)=10 ρ(X,Y)=1/2 Find variance of X-5Y+6?

27. Given same facts as previous problem Find covariance x-5y+6 and -4X+3Y+2

28. Working rule works also for more than two variables Find variance of W=2x-3Y+5Z+1