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:

YX YXEYXCov ,

YEXEXYEYXCov ,

Variance of a Sum

YXCovYVarXVarYXVar ,2

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!

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.

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

YSDXSD

YXCovYXCorr

,,

1,1 YXCorr

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!

• 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?

• Let T1 and T3 be the times of the first and third arrivals in a Poisson process with rate .– Find Corr(T1,T3).

– What is the variance of T1 + T3?