Let $Y$ follow the distribution described by the PDFs $f_Y(y)=2y$ on $(0,1)$. Conditionally on $Y=y$, $X$ follows a uniform distribution on $(0,y)$. Compute $E(X)$ and $E(X/Y)$.
I have calculated the expectation of $X$, which I make as $E(X)=2$, using the formula for iterated conditional expectation. However, I don't know how to proceed from here, since I calculate the combined pdf to also equal $2$, and the pdf of $X$ to equal $2$. Therefore I think I have made a mistake somewhere, but I don't know where, or in reality, how to even tackle that second part of the question. If someone can confirm or refute my answers so far, and show me how to calculate $E(X/Y)$, that would be amazing.