4.3 Covariance ﹠Correlation

1.Covariance That is to say If X and Y are discrete random variables, Definition 4.3 That is to say If X and Y are discrete random variables, If X and Y are continuous random variables,

Proof

Example Suppose that (X, Y) is uniformly distributed on D={(X, Y)：x2+y21} .Prove that X and Y are uncorrelated but not independent. Proof

Thus X and Y are uncorrelated. Since Thus, X is not independent of Y.

2. Properties of covariance:P82 (1) Cov(X, Y)=Cov(Y, X); (2) Cov(aX, bY)=abCov(X, Y), where a, b are constants Proof Cov(aX, bY)=E(aXbY)-E(aX)E(bY) =abE(XY)-aE(X)bE(Y) =ab[E(XY)-E(X)E(Y)] =abCov(X,Y)

(3) Cov(X+Y，Z)=Cov(X, Z)+Cov(Y, Z); Proof Cov(X+Y，Z)= E[(X+Y)Z]-E(X+Y)E(Z) =E(XZ)+E(YZ)-E(X)E(Z)-E(Y)E(Z) =Cov(X,Z)+Cov(Y,Z) (4) D(X+Y)=D(X)+D(Y)+2Cov(X, Y). Remark D(X-Y)=D[X+(-Y)]=D(X)+D(Y)-2Cov(X,Y) Example 4.15----P84

3.Correlation Coefficients Definition4.4 Suppose that r.v. X，Y has finite variance, dentoed by DX>0,DY>0，respectively, then, is name the correlation coefficients of r.v. X and Y .

EX Properties of coefficients x=y D 1 (1) |XY|1； (2) |XY|=1There exists constants a, b such that P {Y= aX+b}=1； (3) X and Y are uncorrelated XY; 1. Suppose that (X,Y) are uniformly distributed on D:0<x<1,0<y<x, try to determine the coefficient of X and Y. EX x=y Answer D 1

D 1

What does Example 2 indicate？ Answer 1) 2) What does Example 2 indicate？

Proof

Note P86 Thus, if （X，Y）follow two-dimensional distribution, then “X and Y are independent” is equvalent to “X and Y are uncorrelated

Example 4.16—4.18 (P86) Exercise:P90—11 Find Cov(X,Y),12 Homework:P91—16,17