Let $X, Y$ - independent random events which have exponential distribution with parameter $1$. Are the given pairs independent?
a) $\max \{X,Y\}, \min \{X,Y\}$
b) $\max\{X,Y\}, \mathbb{1}_{X<Y}$
I suspect that the answer is a) yes, b) no but I don't have some idea how to prove it
$\max(X,Y) $ and $\min(X,Y)$ can't be independent because $\min(X,Y)$ cannot exceed $\max(X,Y)$
$\max(X,Y)$ and $1_{X<Y}$ can't be independent because if $X<Y$ then $1_{X<Y} = 1$ and $\max(X,Y) = Y$, otherwise $1_{X<Y} = 0$ and $\max(X,Y) = X$. If you tell me the $1_{X<Y} = 1$, I will know that $\max(X,Y) = Y$. Thus they are dependent.