# Expectation of Squared Condtitional expectation and the tower property

What can I say about $$E(X \hat{X})$$ where $$\hat{X}$$ is a version of $$E(X|\mathcal{G})$$, where $$X \in \mathcal{L}^2(\Omega,\mathcal{F},\mathbb{P})$$ and $$\hat{X} \in \mathcal{L}^2(\Omega,\mathcal{G},\mathbb{P})$$ and $$\mathcal{G}$$ is a sub-$$\sigma$$-algebra of $$\mathcal{F}$$?

I know I have to use the tower property of expectations to make this collapse, but two things confuse me - the power (is then inside the conditional or outside the conditional expectation) - and independence - nothing is stated about $$\sigma(X)$$ being independent of $$\mathcal{G}$$, so I'm not sure if I can collapse them then.

Any suggestions?

$$E(X\hat {X} |\mathcal G)=\hat {X} E(X|\mathcal G)=\hat {X}^{2}$$ because $$\hat {X}$$ is already measurable with respect to $$\mathcal G$$. Take expectation to get $$EX\hat {X}=E\hat {X}^{2}$$.

• Thanks, that makes sense. I just miss one step inbetween - is it always justified to say that $E(X \hat{X}) = E(X\hat {X} |\mathcal G)$? – Daniel Oct 3 '18 at 10:20
• No, the left side in your equation is a number and the right side is a random variable. You have to take expectation again on the right side to make this equation correct. – Kavi Rama Murthy Oct 3 '18 at 10:23
• Got it! Now it all makes sense. Thanks a lot – Daniel Oct 3 '18 at 10:29