The following question is from the exercises in a course on Stochastic Differential equations and I am confused as to the terminology and the last part. We use "Stochastic Differential Equations" by Oksendal as a textbook.
An experiment consists of choosing a point in the following set.
$\Omega$={$(x,y):x>0,x^2+y^2<1\}$
Let the $\sigma$-algebra $\mathcal{F}$ be the least $\sigma$-algebra such that all sets $\{(x,y)\in \Omega: a \leq x \leq b \}$ and $\{(x,y)\in \Omega: a \leq y \leq b \}$ are events, for any real $a,b$. Let $\mathcal{P}(A)$ of any $A\in\mathcal{F}$ be proportional to its area. Define random variables $X(x,y)=x$ and $Y(x,y)=y$.
- Sketch the sample space $\Omega$ and events $X\leq a$ and $Y\leq b$ for $0<a,b<1$. Then find and plot the cdf of $X$ and $Y$.
- State the conditional mean $Z={E}\{X|Y\}$. Give an example of a Y-measurable event $H$ with $0<\mathcal{P}(H)<1$ and verify that the defining property of the conditional expectation holds, i.e. $E\{X\cdot 1_H\}=E\{Z\cdot 1_H\}$
Now I think I get most of question 1, my problem comes in question 2. I am not sure how I would go about calculating $Z$ as it is not discrete. I am unsure as well how you would find $H$ and finally, the equality of the expectations is not something I have been able to wrap my head around.