I kindly ask for your help to solve this problem. Consider two standard uniform random variables $X_1,X_2\sim U[0,1]^2$. Then, the questions are: 1) is it possible to find the explicit form of joint distribution of $(X_1,X_2)$ with given linear correlation $\rho$, with $-1\leq \rho \leq 1$? 2) if yes, is it possible to have this family of densities parametrized by $\rho$? That is, $\rho \rightarrow f(x_1,x_2;\rho)$ where $f(x_1,x_2;\rho)$ is the joint distribution with uniform marginals and correlation $\rho$?