Question
Let $X$ and $Y$ be independent uniform random variables on $[1, 2]$. Define $U = \min\{ X, Y \}$ and $V = max \{X, Y \}$.
Show that the joint PDF of $U$ and $V$ is given by $$ f_{U,V}(u,v) = cI(-1 \leq u \leq v \leq 2),$$ where $I(\cdot)$ is an indicator function and $c$ is a constant to be determined.
Can we determine whether or not $U$ and $V$ are independent just by looking at the joint PDF? If yes, what do you conclude about their independence and why?
My thoughts
I think I can show the joint PDF as requested but I have some doubts about my answer to the second part of the question. In particular, I think we are able to conclude that $U$ and $V$ are not independent just by looking at the joint PDF because the upper bound of $u$ is determined by $v$ as seen in the range of the indicator function (and conversely the lower bound of $v$ is determined by $u$).
Would my conclusion and reasoning be correct? Any intuitive explanations will be greatly appreciated.