$X$ and $Y$ are independent $N(0,1)$ random variables: What is the distribution, of $$\frac{2XY}{\sqrt{X^{2}+Y^{2}}}?$$ I know the numerator is a difference of two independent $\chi^2$ random variables, and the denominator is a $\chi_{n=2}$ distributed random variable, but I don't know how to calculate the distribution of the ratio.
If $X=\left(X_{1},X_{2},…,X_{n}\right)$ is a random vector with $X_i\sim N(0,1)$ independent coordinates, than what is the distribution of $Y_{1}=\frac{X}{\left\Vert X\right\Vert }$ and $Y_{2}=\left\Vert X\right\Vert ^{2}$ where the norm is the eucledian norm? Are they ($Y_1$ and $Y_2$ ) independent?
I think $Y_1$ has some kind of $t$-distribution, and $Y_2$ has some kind of $\chi^2_n$ distribution, and I dont think they are independent, but I'm not sure.
Best Answer
Use polar coordinates $X = R \cos(\Theta)$, $Y = R \sin(\Theta)$, $R \ge 0$, $0 \le \Theta \le 2\pi$. Then $R$ and $\Theta$ are independent with $R$ having density $f_R(r) = r e^{-r^2/2}$ and $\Theta$ uniform in $[0,2\pi]$. Your random variable is $$ V = 2 R \sin(\Theta)\cos(\Theta) = R \sin(2\Theta)$$ I don't know if there's a nice formula for the PDF or CDF of $V$, but for the MGF I get $$ \mathbb E [\exp(sV)] = \frac{e^{s^2/2}}{4} \left(1 + \text{erf}(s/\sqrt{2})\right)$$
EDIT: This is wrong, see comments.