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]Cov(X,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.