Let $X,Y \sim U[0,1]$ ($X,Y$ are independent), we want to find $E[X|X>Y].$
I tried a few approaches to the above problem, but am not confident in my answer. One approach is as follows. Note that
$$f_{x|y}(x)=\frac{f_{xy}(x,y)}{f_{y}(y)}=1, x\in [0,1].$$
Hence
$$E[X|X>Y] = \int_{y}^{1}xdx = \frac{1-y^2}{2}.$$
My above answer is a function of $y$, and hence is not making sense to me.
Note: I put my latest thoughts on the question in the comments below glens post, can someone confirm if I am correct or not?