$X$ and $Y$ are two independent variables with an exponential distribution with parameters $\lambda$ and $\mu$; $A = \min(X,Y)$, $B = \max(X,Y)$ and $C = B-A$. I want to prove that $A$ and $C$ are independent. I have provided two different demonstrations, but they are both wrong and i don't know why:
1° Demonstration: 2° Demonstration: Thanks
$$ \begin{eqnarray}
\mathbb{P}\left(C
$$\begin{eqnarray}
\mathbb{P}\left(C