Suppose that $X$ and $Y$ are independent random variables, $X$ is uniformly distributed on $[0,1]$, and $Y$ is uniformly distributed on $\{ 1,2,3\}$. How to get $E((X+Y)^2 | Y)$?
It seems to me that the conditional density $p_{X|Y}(x|y)$ is equal to $p_X(x)$, so what is the role of $Y$ here?