Let there be two random variables and with a certain joint copula. Is it always true that there is another random variable independent from such as the vectors $(X,Y)$ and $(X,Z)$ have the same law?



The existence of a copy of a random variable with conditional expectation constraint
No, suppose $X=Y$ a.s. and that they are non-degenerate. If we want $(X, Z)$ to have the same joint distribution as $(X, Y)$, we must also have $X=Z$ a.s. and hence $Y=Z$ a.s. Then $Y$ and $Z$ can obviously not be independent.


