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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.

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I assume $F_{X_i}$ is a shorthand for the cumulative distribution function $\Pr(X_i \le x_i)$ or something similar. As it is a function it would be better to show what it is a function of, for example $F_{X_i}(x_i)$.

If $X_{1},\ldots ,X_{n}$ are independent then the probability they are each less than the respective $x_i$ is indeed the product.

The marginal cumulative distribution functions are still $F_{X_i}(x_i)$. If the distributions are identical, you might consider dropping the $i$s.

Your line on $F_{X}=F_{Y}$ seems to confuse identical and independent. For independence, you want something like $F_{X|Y=y}(x)=F_{X}(x)$ for all $x$ and $y$: in other words, knowledge of $Y$ does not affect the distribution of $X$.

As a counter example just take any (non-singular) distribution for $Y$ and let $X=Y$. Then they obviously have the same cumulative distribution function but are not independent as $Y$ determines $X$ precisely.

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    @Jack: now you independent but not identical? I will throw two dice and add them up, which you can throw fifteen dice independently of me and add them up. The distributions of our scores will be different (and in this example you will always score more than me)2011-09-20