I have been reviewing this notes:
In section 4.1 Order statistics it is stated that:
Suppose $X$, $Y$ have joint density $f_{XY} (x, y)$, then $U = \min(X, Y )$, $V = \max(X, Y )$, then:
$f_{UV} (u, v) = f_{XY} (u, v) + f_{XY} (v, u)$ for , $ u < v $; and $ 0$ else.
I have been trying to prove that using the Jacobian, but I think I'm missing something.
So, What I see is that we have two possible transformations:
for $U = \min(X,Y)$ we can do, $x = u $ so $y=v$,
and the proper for, $V = \max(X,Y)$ we can do, $x = v $ so $y=u$.
Then, I what I guess is that we may have two Jacobians, $J_1$ and $J_2$, and both are $|J_1| = |J_2| =1$.
What I cannot see is how to combine them to get $f_{UV}$.
Some guidance please?
Thanks