Let X and Y be random variables with means µX and µY . Thecovariance of X and Y is given by, Cov(X, Y ) = E[(X ? µX)(Y ? µY)]
a) Prove the following three equalities: Cov(X, Y ) = E[(X ?µX)Y ] = E[X(Y ? µY )] = E(XY ) ? µXµY
b) Suppose that E(Y |X) = E(Y ). Show that Cov(X, Y ) = 0 (hint:use the law of interated expectations to show that E(XY ) = µXµY ).In this case, what is the correlation coefficient ? between X and Y, equal to?
c) Suppose that Cov(X, Y ) = 0. Does this imply that X and Y areindependent? Explain your reasoning.
