$X$ 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:
$ \begin{eqnarray} \mathbb{P}\left(CX\right) \\ &=& 1-\mathbb{P}\left(Y>2A-X+t \Big| Y>X\right) \\ &=& 1-\mathbb{P}\left(Y>2A-2X-t\right) \end{eqnarray} $ because exponential is memoryless. So: $ = \mathbb{P}\left(2X+Y-2A So $C$ and $A$ are not independent...Something's wrong!
2° Demonstration:
$\begin{eqnarray} \mathbb{P}\left(C by conditioning $ \ldots = \mathbb{P}\left(A+B-2A < t\right) = \mathbb{P}\left(B-A < t\right) = \mathbb{P}\left(C This seems correct, but that would mean that this property is true for every $X$, $Y$. But this is not true. So where's the mistake?
Thanks