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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.

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    $\begingroup$ Are $X_1$ and $X_2$ independent normal rvs with mean zero and same variance? $\endgroup$ Sep 9 '19 at 5:21
  • $\begingroup$ 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. $\endgroup$
    – hyg17
    Sep 9 '19 at 6:43
  • $\begingroup$ How can one possibly answer the question in the title without the distribution of $(X_1,X_2)$? $\endgroup$ Sep 10 '19 at 6:48
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  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.

  2. Definitely.

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  • $\begingroup$ 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? $\endgroup$
    – hyg17
    Sep 9 '19 at 6:44

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