Query regarding a transformation of continuous random variables question

Asked Mar 23 '18 at 18:13

Active Mar 23 '18 at 18:13

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I have a query related to a transformation of variables question; the images of which I have attached here.

I have solved the question almost, but there seems to be a problem with the distribution that I have derived in the last line of the solution.

I would greatly appreciate any kind of help. Thanks in advance.

N.B.: In the question, $U$ should follow $N(\mu = 0, \sigma^2 = \frac{1}{4} = 0.25)$.

asked Mar 23 '18 at 18:13

Dwaipayan Gupta

1

My post here seems relevant: https://math.stackexchange.com/a/2647462/321264. – StubbornAtom Mar 23 '18 at 18:25
Hi @StubbornAtom. Thanks for the link. Can you please explain as to why the transformation (x,y) -> (u,v) not one-to-one ? Also, we are mapping from X and Y. Why have you considered here -X and -Y ? – Dwaipayan Gupta Mar 23 '18 at 19:25
Check the general change of variables (transformation) formula. – StubbornAtom Mar 23 '18 at 19:33
The mapping is not one to one precisely because you get two values of $x,y$ when written in terms of $u,v$ (i.e. two preimages). – StubbornAtom Mar 23 '18 at 19:38

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