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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.


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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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Thanks for your answer. I do not understand your last paragraph. Could you please elaborate a bit? Thanks. – Jack Sep 19 '11 at 23:49
In standard usage, $F_{X_i}(x)$ means $\Pr(X_i \le x)$. So one could say $F_{X_i}$ is the same as $x\mapsto \Pr(X_i\le x)$. – Michael Hardy Sep 19 '11 at 23:49
@Jack: Suppose I choose a number by throwing a fair die with values 1 to 6 equally likely, and call the result $Y$. My friend also looks at my throw and chooses the same number as $X$; as they are always the same, they have the same distribution, but they are not independent. If on the other hand we each throw our own fair dice then our values will have the same distribution as each other and they will be independent (and more often than not will differ). Wikipedia has a rather more technical discussion. – Henry Sep 20 '11 at 0:02
@Michael: Indeed, and $\Pi_{i=1}^n F_{X_{i}}$ might be read as applying to something in $\mathbb{R}^n$ but I am not sure that would be clear here. – Henry Sep 20 '11 at 0:06
@Henry: Thanks once again. How about a counterexample for the other direction? – Jack Sep 20 '11 at 1:05

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