I am working on a problem that says
Prove or disprove that $R=\sqrt{X_1^2+X_2^2}$ and $W=\arctan{(X_1/X_2)}$ are independent.
There is no other information given so I will assume a natural situation where $X_1$ and $X_2$ are the randomly chosen abscissa and the ordinate on the Cartesian coordinate plane.
Intuitively I know that they are independent because $R$ and $W$ represents the distance from the origin and the angle between the $y$-axis and the line do not affect each other.
In other words, we can change the value of $R$ while holding $W$ constant and vice versa.
I doubt that this argument is rigorous enough to show that this is true, and I am also not convinced because I don't know if there is a particular distribution of $X_1$ and $X_2$ that actually makes $R$ and $W$ dependent.
So, here is what I would like.
1), If I were to make a more rigorous proof of the argument that I made, how would it go?
2), Are there situations where $X_1$ and $X_2$ can be chosen so that $R$ and $W$ are dependent?
If this question is too open I would like to apologize, but I would like to have some opinion.
One approach is to start with the joint density of $(X_1, X_2)$, and use a change of variables (may involve a Jacobian) to obtain the joint density of $(R, W)$. If the joint density is separable (can be written as the product of two marginal densities), then you will have shown independence.
Definitely.