Independence of max and min of a set of random variables. Suppose $X_1,\ldots,X_n$ are independent and identically distributed random variables with cdf $F_X(x)$. Define $U$ and $L$ as $U=\max\{ X_1, \ldots ,X_n\}$ and $L = \min\{X_1,\ldots,X_n\}$.
Are $U$ and $L$ independent?
I believe they are I just don't know how to prove it.

No, they are not. If we know that $L=x$, then the probability that $U<x$ is zero (while it is well possible that the prior probability was non-zero).
