covariance correlation conditional expectation and variance

Var(Y|X) = E([Y - E(Y|X)]2 | X)
(Note that both expected values here are conditional expected values.)
Conditional Variance as a Random Variable: As with E(Y|X), we can consider
Var(Y|X) as a random variable. 2
Expected Value of the Conditional Variance: Since Var(Y|X) is a random variable, we
can talk about its expected value. Using the formula Var(Y|X) = E(Y2|X) - [E(Y|X)]2, we
have
E(Var(Y|X)) = E(E(Y2|X)) - E([E(Y|X)]2)
We have already seen that the expected value of the conditional expectation of a random
variable is the expected value of the original random variable, so applying this to Y2
gives
(*) E(Var(Y|X)) = E(Y2) - E([E(Y|X)]2)
Variance of the Conditional Expected Value: For what comes next, we will need to
consider the variance of the conditional expected value. Using the second formula for
variance, we have
Var(E(Y|X)) = E([E(Y|X)]2) - [E(E(Y|X))]2
Since E(E(Y|X)) = E(Y), this gives
(**)Var(E(Y|X)) = E([E(Y|X)]2) - [E(Y)]2.
Putting It Together:
Note that (*) and (**) both contain the term E([E(Y|X)]2), but with opposite signs. So
adding them gives:
E(Var(Y|X)) + Var(E(Y|X)) = E(Y2) - [E(Y)]2,