I have a some questions that have been bothering for a while now. First, how does one obtain the joint probability distribution function of $X_{1},\cdots ,X_{n}$? Would it be $\prod\limits_{i=1}^n F_{X_{i}}$? What about the marginal probability distributions? Is it just $F_{X}$ for all $i$?
Second, given two random variables, $X$ and $Y$, is it it true that $X$ and $Y$ are independent if and only if $F_{X}=F_{Y}$? I think it is not true, but I can't readily find counterexamples.
Any form of help would be appreciated.
Thanks.