This really does not seem right to me, but according to a study manual I purchased to study for SOA exam P, this is a valid identity. Could someone confirm for me that this is false, or if true, show me a derivation of the identity?
If it were not for the double expectation, it would obviously be false. Consider the formula for conditional variance:
Var(Y|X) = E[Y^2|X] - E[Y|X]^2
If the two terms on the right were equivalent, all conditional variances would be zero, which is absurd.
The actual identity in the manual I refer to is this:
Var(E[Y|X]) = E[E[Y|X]^2] - E[E[Y|X]]^2
= E[E[Y^2|X]] - E[Y]^2