1. Chap 10 More Expectations and Variances Ghahramani 3rd edition
2017/4/17

2. Outline 10.1 Expected values of sums of random variables
10.2 Covariance
10.3 Correlation
10.4 Conditioning on random variables
10.5 Bivariate normal distribution

3. 10.1 Expected values of sums of random variables

4. Expected values of sums of random variables

5. Expected values of sums of random variables

6. Expected values of sums of random variables

7. Expected values of sums of random variables

8. Expected values of sums of random variables

9. Expected values of sums of random variables

10. Expected values of sums of random variables

11. Expected values of sums of random variables

12. Expected values of sums of random variables

13. Expected values of sums of random variables

14. Expected values of sums of random variables

15. Expected values of sums of random variables

16. Expected values of sums of random variables

17. Expected values of sums of random variables

18. Expected values of sums of random variables

19. 10.2 Covariance Var(aX+bY)=E[(aX+bY)-E(aX+bY)]2 =E[(aX+bY)-aEX-bEY]2
Motivation:
Var(aX+bY)=E[(aX+bY)-E(aX+bY)]2
=E[(aX+bY)-aEX-bEY]2
=E[a[X-EX]+b[Y-EY]]2
=E[a2[X-EX]2+b2[Y-EY]2
+2ab[X-EX][Y-EY]]

20. Covariance
Def Let X and Y be jointly distributed r. v.’s; then the covariance of X and Y is defined by
Cov(X, Y)=E[(X-EX)(Y-EY)]
Note that Cov(X, X)=Var(X),
and also by Cauchy-Schwarz inequality

21. Covariance Thm 10.4 Var(aX+bY) =a2Var(X)+b2Var(Y)+2abCov(X,Y).
In particular, if a=b=1,
Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)

22. Covariance

23. Covariance 1. X and Y are positively correlated if Cov(X,Y) > 0.
2. X and Y are negatively correlated if
Cov(X,Y) < 0.
3. X and Y are uncorrelated if
Cov(X,Y) = 0.

24. Covariance If X and Y are independent then Cov(X,Y)=EXY-EXEY=0.
But the converse is not true
Ex 10.9 Let X be uniformly distributed over (-1,1) and Y=X2. Then
Cov(X,Y)=E(X3)-EXE(X2)=0.
So X and Y are uncorrelated but surely X and Y are dependent.

25. Covariance
Ex Let X be the lifetime of an electronic system and Y be the lifetime of one of its components. Suppose that the electronic system fails if the component does (but not necessarily vice versa). Furthermore, suppose that the joint density function of X and Y (in years) is given by

26. Covariance
(a) Determine the expected value of the remaining lifetime of the component when the system dies.
(b) Find the covariance of X and Y.
Sol:

27. Covariance

28. Covariance

29. Covariance
Ex Let X be the number of 6’s in n rolls of a fair die. Find Var(X).

30. Covariance
Sol:

31. Covariance
Ex X ~ B(n,p). Find Var(X).
Sol:

32. Covariance
Ex X ~ NB(r,p). Find Var(X).
Sol:

33. 10.3 Correlation
Motivation:
Suppose X and Y, when measured in centimeters, Cov(X,Y)= But if we change the measurements to millimeters, the X1=10X and Y1=10Y and
Cov(X1,Y1)=Cov(10X,10Y)=100Cov(X,Y)=15
This shows that Cov(X,Y) is sensitive to the units of measurement.

34. Correlation

35. Correlation

36. Correlation

37. Correlation

38. Correlation

39. Correlation

40. Correlation

41. Correlation

42. 10.4 Conditioning on random variables
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10.4 Conditioning on random variables
10.5 Bivariate normal distribution