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Let $X,Y\sim N(0,1)$. Are $X^2 +Y^2$ and $X/Y$ independent?

I found that $X^2 + Y^2$ is $\mathrm{Exp}(1/2)$ but I am having trouble finding the distribution of $X/Y$—it seems like it is impossible to integrate.

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1 Answer 1

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Let $X = R \cos \Theta$, $Y = R \sin \Theta$ for random variables $(R, \Theta)$ supported on $\mathbb{R}_+ \times [0,2\pi)$. It's easy to see that this covers the entire support of $(X,Y)$.

The determinant of the Jacobian for $(x,y) \mapsto (r\cos \theta, r\sin \theta)$is $r$.

$$f_{R\Theta}(r,\theta) = r \frac{1}{2 \pi} \exp(- r^2 \cos^2 \theta - r^2 \sin \theta) 1_{\{ r \ge 0\}} 1_{\{\theta \in [0,2\pi)\}} $$ $$ ~~~~~~~~~~~= \frac{r}{2 \pi} \exp(-r^2) 1_{\{ r \ge 0\}} 1_{\{\theta \in [0,2\pi)\}} = f_R(r) f_\Theta(\theta)$$

Since $R$ and $\Theta$ are independent, so are any deterministic functions of them. Consequently, $R^2 $ and $\tan \Theta$ are independent.

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