The question is:
Let $X,Y,Z$ be 3 random variables ; show that
$$Cov(X,Y∣Z) = E(XY∣Z)-E(X∣Z)E(Y∣Z)$$
So I started from the LHS :
$Cov(X,Y∣Z) = E[(X-E(X∣Z))(Y-E(Y∣Z))∣Z] = E[[XY+E(X∣Z)E(Y∣Z)-XE(X∣Z)-YE(Y∣Z)] ∣ Z] = E(XY∣Z)+E[E(X∣Z)E(Y∣Z)∣Z]-E[XE(Y∣Z)∣Z]-E[YE(X∣Z)∣Z]$
and i'm lost here
Is that $E[XE(Y∣Z)∣Z] = E[X∣Z]E[Y∣Z]$? and $E[E(X∣Z)E(Y∣Z)∣Z] = E[X∣Z]E[Y∣Z]$?