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?