Let $X$, $Y$ and $Z$ be iid with $P(X>t)=e^{-t}$ for $t>0$. Let $U$, $V$ be independent uniform on $[0,1]$. Let $A=\min(U,V)$ and $B=\max(U,V)$. Show that $(A,B),$ and $(X/(X+Y+Z), (X+Y)/(X+Y+Z))$ have the same joint distribution.
So, the joint distribution for A and B should be:
$f_{A,B}(a,b)=f_{A|B}(a|b)f_B(b)=f_{B|A}(b|a)f_a(a).$ (*)
$f_{A|B}(a|b)=I_{[0,B]}/B$,
$f_{B|A}(b|a)=I_{[a,1]}/(1-a)$,
$f_{a}(a)=f_{b}(b)=I_{[0,1]}$.
However, using these, the equality above (*) does not hold. Where is the mistake here?