TITLE: Random variables with equal joint distributions have equal marginal distributions?
QUESTION [4 upvotes]: We are given two vectors $X=(X_1,X_2, . . . ,X_n)$ and $Y=(Y_1, Y_2, . . . , Y_n)$ with equal joint distributions. Do their marginal distributions $P_{X_i}$ and $ P_{Y_i}$ have to be equal?
I have no idea so far how to approach this problem. I'm sure there is a simple counterexample with a small $n$. I suppose I should look for dependent random variables.
Could you help me a bit?

REPLY [4 votes]: If $$P(X_1 = z_1, \ldots, X_j = z_j,\ldots, X_n = z_n)=P(Y_1 = z_1, \ldots, Y_j = z_j,\ldots, Y_n = z_n)$$ for all $z_1,\ldots,z_j,\ldots,z_n$  is what is meant by "equal joint distributions" then you might say the marginal probabilities are $$\displaystyle P(X_i=z_i) = \sum_{z_1}\cdots \sum_{z_j, j\not = i} \cdots \sum_{z_n} P(X_1 = z_1, \ldots, X_i = z_i,\ldots, X_j = z_j,\ldots, X_n = z_n)$$ and $$\displaystyle P(Y_i=z_i) = \sum_{z_1}\cdots \sum_{z_j, j\not = i} \cdots \sum_{z_n} P(Y_1 = z_1, \ldots, Y_i = z_i, \ldots, Y_j = z_j, \ldots, Y_n = z_n)$$ and these are clearly equal to each other since they are sums of equal probabilities over the same indicies.
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