# Question

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.

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.

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