I was wondering if there is a way to calculate the joint distribution of two fully correlated variables, both with known distributions, expected value and variance, without knowing the conditional distribution?

If this is not possible, is there a way of finding \(Var(X,Y) = E[(XY)^2] - E[XY]^2\) when knowing that \(Cor(X,Y) = 1\)? I can't seem to find an expression for \( E[(XY)^2] \)...

Thanks!