I am trying to work out how to generate numbers from a bivariate distribution (any one but the normal distribution) while still being able to control the correlation between the two variables (let's call them $X$ and $Y$). When generating numbers from a normal distribution it is quite easy, as you just specify the covariance (correlation) in the covariance matrix. I have done this in the normal case and compared the power of three correlation tests for various values of rho, the true correlation.

Now I want to do this for some other bivariate distribution, but I don't know how to do it while still knowing the true correlation. One suggestion I was offered was to generate $Z$ and $e$, and then let $V = b*Z + e$, but I tried calculating the correlation when $Z$ and $e$ were both chi square and uniform, but could not figure anything out.