Given the sample space $\Omega=[(x,y)\in R^2,x>0, x^2+y^2<1]$. Let $(\Omega,F,P)$ be a probability space with the stochastic variables $X$ and $X$ given as $X(x,y)=x$ and $Y(x,y)=y$.
I want to determine the conditional expectation $E[X|Y]$. I have simply stated that $X$ and $Y$ are each uniformly distributed, where $X\sim U(0,1)$ and $Y\sim U(-1,1)$, and then for any fixed $y\in (-1,1)$ found that $X|Y=y \sim U(0,\sqrt{1-y^2})$, which then has mean $E[X|Y]=\sqrt{1-y^2}/2$.
Does this seem reasonable?
Thanks in advance
Try and go back to look at how you define your marginal distributions. Are they really uniformly distributed? Remember the geometry of the problem and the relation between the variables. The consideration about the conditional expectation is true, but I don't think it follows from your other assumptions. This being homework, I will not give you the answer, but you can ask tomorrow in class.