I have two random variables $X$ and $Y$. The marginal distribution of them are both uniform. Specifically, $X\sim~U[a,b]$ and $Y\sim~U[c,d]$.
If I specify a correlation between $X$ and $Y$ to be $\rho$, then is the joint distribution of $(X,Y)$ uniquely determined? If not, can one give an example of the specific form of joint distribution of $(X,Y)$, $f(x,y)$ such that their correlation is $\rho$ and the marginal distribution is $X\sim~U[a,b]$ and $Y\sim~U[c,d]$?
Here are two joint cdfs with marginals as given in your question:
$$F_1(x,y)=\left(\max\left\{\left(\frac{x-a}{b-a}\right)^{-\theta_1} + \left(\frac{y-c}{d-c}\right)^{-\theta_1} - 1,0\right\}\right)^{-1/\theta_1} \; \text{ for } \; \theta_1 \in [-1,\infty)\setminus\{0\} \; ,$$
and
$$F_2(x,y)=\exp\left(\left\{\left(-\log\left(\frac{x-a}{b-a}\right)\right)^{-\theta_2} + \left(-\log\left(\frac{y-c}{d-c}\right)\right)^{-\theta_2} \right\}^{-1/\theta_2}\right) \; \text{ for } \; \theta_2 \in [1,\infty) \; .$$
It is clear that both cdfs will represent different dependencies between $X$ and $Y$, it is harder to compute the Pearson correlations. But I think you can toy around with the parameters $\theta_1$ and $\theta_2$ and find by numeric integration or some other numerical scheme some choices for the parameters such that the correlations are equal.
