Let X and Y be random variable with correlation coefficient prove : ((|)) ≤ (1 − ^2 ) ()

Question

Let X and Y be random variable with correlation coefficient prove that :

$((|)) ≤ (1 − ^2 ) ()$

My attempt:

We know that $Var(Y|X) = E(Y^2|X) - E(Y|X)^2$

$E[Var(Y|X)] = E[E(Y^2|X)] - E[E(Y|X)^2] = E[Y^2] - E[E(Y|X)^2]$

I'm stuck here.

Answer 1

Wlog $\mathbb E X = \mathbb E Y = 0$ and $Var(X) = Var(Y)=1$.

Then, $\mathbb E[Y-\rho X]=0$ and $Var(Y-\rho X) = 1+\rho^2-2\rho^2 = 1-\rho^2$. Moreover, $Cov(X,Y-\rho X) = 0$.

Let's pose $Z:=Y-\rho X$, then $Y = Z+\rho X$ and $Var(Y|X) = Var(Z|X)$. We need $$ \mathbb E[Var(Z|X)] \stackrel ?\leq 1-\rho^2 = Var(Z)$$

However, $$\mathbb E[Var(Z|X)] = \mathbb E[Z^2] - \mathbb E[\mathbb E[Z|X]^2] = Var(Z) +\mathbb E[Z]^2- \mathbb E[\mathbb E[Z|X]^2] = Var(Z)+\mathbb E[\mathbb E[Z|X]]^2- \mathbb E[\mathbb E[Z|X]^2] = Var(Z)-Var(\mathbb E[Z|X]) \leq Var(Z)$$