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

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

10.1 Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

Expected values of sums of random variables

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]]

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

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)

Covariance

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.

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.

Covariance Ex 10.12 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

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:

Covariance

Covariance

Covariance Ex 10.13 Let X be the number of 6’s in n rolls of a fair die. Find Var(X).

Covariance Sol:

Covariance Ex 10.15 X ~ B(n,p). Find Var(X). Sol:

Covariance Ex 10.16 X ~ NB(r,p). Find Var(X). Sol:

10.3 Correlation Motivation: Suppose X and Y, when measured in centimeters, Cov(X,Y)=0.15. 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.

Correlation

Correlation

Correlation

Correlation

Correlation

Correlation

Correlation

Correlation

10.4 Conditioning on random variables Skip 10.4 Conditioning on random variables 10.5 Bivariate normal distribution