Suppose X and Y are two independent random variables each distributed as $Uniform[0,1)$.
1) Find the joint distribution of X and Y.
2) Let $U = \cos(2\piY)\sqrt{-2\ln(X)}$ and $V = \sin(2\piY)\sqrt{-2\ln(X)}$. Find the joint distribution
of U and V assuming that the transformation is one-to-one?
3) Find the marginal distributions of U and V?
For 1, I got that my distribution is 1 by multiplying the two distributions together. For 2, I have been getting stuck trying to get my equations in terms of X and Y to perform the transformation. I simplified and got $Y = \frac{1}{2}\tan^{-1}(\frac{V}{U})$ but something seems wrong about this and am having trouble trying to get X. For 3, I am obviously stuck and even not totally sure on the support. Any help is appreciated.
1) Find the joint distribution of X and Y.
2) Let $U = \cos(2\piY)\sqrt{-2\ln(X)}$ and $V = \sin(2\piY)\sqrt{-2\ln(X)}$. Find the joint distribution
of U and V assuming that the transformation is one-to-one?
3) Find the marginal distributions of U and V?
For 1, I got that my distribution is 1 by multiplying the two distributions together. For 2, I have been getting stuck trying to get my equations in terms of X and Y to perform the transformation. I simplified and got $Y = \frac{1}{2}\tan^{-1}(\frac{V}{U})$ but something seems wrong about this and am having trouble trying to get X. For 3, I am obviously stuck and even not totally sure on the support. Any help is appreciated.