91Ó°ÊÓ

For the random variables x,y ,z \(Cov\left( {X,Y|z} \right)\)is the covariance of x and yin their cdf given Z=z

\(\begin{aligned}{c}Cov\left( {X,Y} \right) = E\left( {\left( {X - \mu x} \right)\left( {Y - \mu y} \right)} \right)\\ = E\left\{ {\left( {X - E\left( {X|Z} \right) + E\left( {X|Z} \right) - \mu x} \right) \times \left( {Y - E\left( {Y|Z} \right) + E\left( {Y|Z} \right) - \mu y} \right)} \right\}\\ = E\left\{ {\left( {X - E\left( {X|Z} \right)} \right)\left( {Y - E\left( {Y|Z} \right)} \right)} \right\} + E\left\{ {\left( {X - E\left( {X|Z} \right)} \right)\left( {E\left( {Y|Z} \right) - \mu y} \right)} \right\}\\ + E\left\{ {\left( {E\left( {X|Z} \right) - \mu x} \right)\left( {Y - E\left( {Y|Z} \right)} \right)} \right\} + E\left\{ {\left( {E\left( {X|Z} \right) - \mu x} \right)\left( {E\left( {Y|Z} \right) - \mu y} \right)} \right\}\end{aligned}\).......(1)

In the given equation taking the last 4 expectations.

In the 1^st expectation:

if we 1^st calculate the conditional expectation given Z and then take the expectation over Z, we get

\(E\left( {Cov\left( {X,Y|Z} \right)} \right)\)

In the 2^nd and 3^rd expectation:

We obtain value 0 when we take the conditional expectation given Z.

In the 4^th expectation:

\(Cov\left( {E\left( {X|Z} \right),E\left( {Y|Z} \right)} \right)\)

Hence

\(Cov\left( {X,Y} \right) = E\left( {Cov\left( {X,Y|z} \right)} \right) + Cov\left( {E\left( {X|Z} \right),E\left( {Y|Z} \right)} \right)\)