Counter Example for a couple for random variables $(U,V)$ and $(X,Y).$

Question

The goal is to find random variables $U, V, X$ and $Y$ such that the join distribution of $(U,V)$ and $(X,Y)$ is not the same, however, $U=X$ and $V=Y.$ From the statement, we see that all the random variables take values in the same in the set. I was thinking of working on the set $\Omega = \{0,1\}.$ Then perhaps we could take $U\sim B(p)$ and $X\sim B(p)$ Bernoulli and $V$ and $Y$ to be Bernoulli with parameter $q.$ Then we have the second condition satisfied. However, I want the joint distribution to be not equal and for that, I need to create some dependence between the variables $U$, $V$ and $X$, $Y$. This I am not sure of. Any hints will be much appreciated.

Answer 1

Take $\varepsilon$, a random variable taking the values $-1$ and $1$ with probability $1/2$, $U=X=\varepsilon=V$ and $Y=-\varepsilon$. The marginals of the couples $(U,V)$ and $(X,Y)$ are the same, but the couple do not have the same distribution: the first one takes the values $(1,1)$ and $(-1,-1)$ with probability $1/2$; the second one the values $(1,-1)$ and $(-1,1)$ with probability $1/2$.