1. Definition of Covariance The covariance of X & Y, denoted Cov(X,Y), is the number where  X = E(X) and  Y = E(Y). Computational Formula:

2. Variance of a Sum

3. Covariance and Independence If X & Y are independent, then Cov(X,Y) = 0. If Cov(X,Y) = 0, it is not necessarily true that X & Y are independent!

4. The Sign of Covariance If the sign of Cov(X,Y) is positive, above-average values of X tend to be associated with above-average values of Y and below-average values of X tend to be associated with below- average values of Y. If the sign of Cov(X,Y) is negative, above-average values of X tend to be associated with below-average values of Y and vice versa. If the Cov(X,Y) is zero, no such association exists between the variables X and Y.

5. Correlation The sign of the covariance has a nice interpretation, but its magnitude is more difficult to interpret. It is easier to interpret the correlation of X and Y. Correlation is a kind of standardized covariance, and

6. Conditions for X & Y to be Uncorrelated The following conditions are equivalent: Corr(X,Y) = 0 Cov(X,Y) = 0 E(XY) = E(X)E(Y) in which case X and Y are uncorrelated. Independent variables are uncorrelated. Uncorrelated variables are not necessarily independent!

7. Let (X, Y) have uniform distribution on the four points (-1,0), (0,1), (0,-1) and (1,0). Show that X and Y are uncorrelated but not independent. What is the variance of X + Y?

8. Let T 1 and T 3 be the times of the first and third arrivals in a Poisson process with rate. –Find Corr(T 1,T 3 ). –What is the variance of T 1 + T 3 ?