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Then I have Y=X1^2+ X2^2

and

Z=X1/(X1^2+x2^2)

I'm supposed to find the distribution of Y and Z and then determine if they are independent.

Clearly Y is chi squared with degrees of freedom 2.

But I have no idea how to find the distribution of Z. I know there is a shortcut without using the jacobian, like I did with the Chi Squares, but I'm not sure how to do it.

I know Y and Z are not independent because with some algebra,

Y=Y*Z^(2)+X2

So Y depends on Z and vice versa, therefore they cannot be independent. Is that true?