The Conditional Covariance Formula. The conditional covariance of X and Y, given Z, is defined by
Cov(X, Y|Z) = E[(X − E[X|Z])(Y − E[Y|Z])|Z]
(a) Show that
Cov(X, Y|Z) = E[XY|Z] − E[X|Z]E[Y|Z]
(b) Prove the conditional covariance formula
Cov(X, Y) = E[Cov(X, Y|Z)] + Cov(E[X|Z],E[Y|Z])
(c) Set X = Y in part (b) and obtain the conditional variance formula.