What is $E(\min\{X,Y\}\vert Y)$ where $X\sim\text{Exp}_\alpha$ and $Y$ is an arbitrary positive-real-valued RV?
It is clear that $\min\{X,Y\}$ is integrable, and also the following holds $$E(\min\{X,Y\}\vert Y) = E(1_{X\le Y} X\vert Y)+E(1_{X>Y}Y\lvert Y)$$ but I don't see how I can proceed without any knowledge what so ever about $Y$.