Let there be a set of random variables $\left\{ X_0, \ldots, X_{N} \right\}$, that are independent but not identically distributed.
Will the equality:
$$F(X_0 \leq x) F(X_1 \leq x) \cdots F(X_N \leq x) = P( X_0 \leq x, X_1 \leq x, \ldots, X_{N} \leq x) = P( X_{\left( N \right)} \leq x ) $$
always hold?
I think I'm being confused by the definition of the lower order statistics requiring us to sum over all the permutations of the joint probabilities and their possible sequencings.
So it makes sense to me that for the maximum order statistic we should also permute over all possible sequencings of orderings.
So the equality should hold only when the random variables are exchangeable right? But I haven't found any good examples of a set of random variables that are both independent and not exchangeable. Or a set of INID random variables whose maximum order statistic is not equal to the CDF of the joint probability.
Maybe someone can help me with a counter example in either case?