1. 4.3 Covariance ﹠Correlation

2. 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,

3. Proof

4. 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

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

6. 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)

7. (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 P84

8. 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 .

9. 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

10. D 1

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

12. Proof

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

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