Probability | Part IA, 2006

Let X,YX, Y be independent random variables with values in (0,)(0, \infty) and the same probability density 2πex2\frac{2}{\sqrt{\pi}} e^{-x^{2}}. Let U=X2+Y2,V=Y/XU=X^{2}+Y^{2}, V=Y / X. Compute the joint probability density of U,VU, V and the marginal densities of UU and VV respectively. Are UU and VV independent?

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