<s>[INST] Showing that two random variables are independent I have the following problem: 
Given two independent standard normal random variables, call them $X$ and $Y$, how can I show that $Z = X^2 + Y^2$ and $W=\frac{X}{Y}$ are also independent?
I know that because $X$ and $Y$ are standard normal I can write their distributions as $$f_X(x) = \frac{1}{2\pi}e^\frac{-(x)^2}{2}$$ $$f_Y(y) = \frac{1}{2\pi}e^\frac{-(y)^2}{2}$$
Then because they are independent I can write their joint probability distribution as $$f_{X,Y}(x,y) = \frac{1}{2\pi}e^\frac{-(x^+y^2)}{2}$$
Now I know that to show $Z$ and $W$ are independent, I need to show that their joint probability distribution is equal to the product of the marginal distribution functions. But I'm not sure how to find 
$f_{Z,W}(z,w)$, $f_Z(z)$ and $f_W(w)$. 
Any help is much appreciated. 
 [/INST] The joint distribution for $(X^2,Y^2)$ is 
$$\frac{x^{-1/2}e^{-x/2}}{\sqrt{2}\Gamma(1/2)}1_{(0,+\infty)}(x)\times \frac{y^{-1/2}e^{-y/2}}{\sqrt{2}\Gamma(1/2)}1_{(0,+\infty)}(x)=\frac{(xy)^{-1/2}e^{-(x+y)/2}}{2(\Gamma(1/2))^2}1_{(0,+\infty)}(x)1_{(0,+\infty)}(y)$$
Since $X^2=Z-\frac{Z}{W^2+1}=\frac{ZW^2}{W^2+1},Y^2=\frac{Z}{W^2+1}$, the Jacobian matrix is $$
\left|\begin{array}{cc} 
   \frac{W^2}{W^2+1}& \frac{1}{W^2+1}\\ 
    \frac{2ZW}{(W^2+1)^2}  & -\frac{2ZW}{(W^2+1)^2}
\end{array}\right| =\frac{2ZW}{(W^2+1)^2}
$$
Therefore, the joint pdf of $(Z,W)$ is 
$$\frac{(\frac{ZW^2}{W^2+1}\frac{Z}{W^2+1})^{-1/2}e^{-(\frac{ZW^2}{W^2+1}+\frac{Z}{W^2+1})/2}}{2(\Gamma(1/2))^2}\times \frac{2ZW}{(W^2+1)^2}1_{(0,+\infty)}(z)=\frac{1}{w^2+1}\frac{e^{-z/2}}{\pi}1_{(0,+\infty)}(z).$$
Hence, the pdf of $W$ and $Z$ are $\frac{1}{\pi (1+w^2)}$ and $\frac{1}{2}e^{-z/2}1_{(0,+\infty)}(z)$.
For now, one can see the independence of $W$ and $Z$.
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