# Show independence of random variables $R=\sqrt{X_1^2+X_2^2}$ and $W=\arctan{X_1/X_2}$

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.

• Are $X_1$ and $X_2$ independent normal rvs with mean zero and same variance? Sep 9 '19 at 5:21
• That's the thing, there is no assumption written so I am thinking that the problem depends on the choice of distribution. I would love to see the argument when it is though. Sep 9 '19 at 6:43
• How can one possibly answer the question in the title without the distribution of $(X_1,X_2)$? Sep 10 '19 at 6:48

1. 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.
• Thank you for your help. Just out of curiosity, would there be a uniform $(-\infty, \infty)$ distribution so that my choice of $X_1$ and $X_2$ can equally likely be any real number? Sep 9 '19 at 6:44