Suppose $X$ is a uniformly distributed random variable on $[0,1]$ and given $X=x,$ a number $Y$ is chosen at random between $0$ and $x$. Suppose that you only know the value $y$ of $Y$ and you don't know $x$. We guess $x$ to be $\mathbb{E}(X|Y=y)$. Find this.
How would I start this?
This is what I did:
$Y|X \sim Unif(0,x)$.
So $$f_{X|Y}(x|y) = f_{Y|X}(y|x)f_{X}(x)/f_{Y}(y) \\ = \frac{\frac{1}{x}I_{[0,x]}(y)I_{[0,1]}(x)}{f_{Y}(y)}$$
and I'm stuck here.