Let $X\sim \operatorname{Exp}(\lambda)$ and $Y\sim \operatorname{U}(0,1)$, I need to build 3 joint distributions, one where they are independent (This is trivial, it's just the product of tha marginals), one where they are positive correlated and another where they are negative correlated.
I'd like to employ copulas instead of simple variable transformations, but I don't know how to proceed. Can i choose any type of copula? How i do choose a certain level of correlation? I'd like some insight on the problem, please.