Let $\Omega$ = [-1/2 ; 1/2], $\mathcal{F} = \mathcal{B}([-1/2 ; 1/2])$ be the Borel $\sigma$– algebra on $\Omega$ and P be the Lebesgue measure. Let X, Y $\mapsto \mathcal{B}(R) $ be two random variables defined as:
$ X(\omega) = \omega^2$ and $Y(\omega) = \omega^3$ for all $ \omega \in \Omega$.
I need to calculate the following conditional expectations:
$E[X|Y] := E[X|\sigma(Y)]$
$E[Y|X] := E[Y|\sigma(X)]$.
For the first one I know it is just X as X is $\sigma(Y)$-measurable. I also believe that Y is $\sigma(X)$– measurable based on the criterion whether the set
$\ \left\{ Y < \alpha \right\}\ \ \ \forall \alpha \in \Omega$ is measurable or not. However I am not almost sure on that argument.
Best Answer
$E(Y|X)=0$ by asymmetry argument. Can you apply the definition of conditional expectation to prove this? Hint: every set in $\sigma (X)$ is a symmetric set.