Definition of Covariance The covariance of X Y

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

Variance of a Sum

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 belowaverage 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

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