1. ## iid Normal

If $\displaystyle X_1 \;,\; X_2$ are iid $\displaystyle N (0, \sigma^2)$ random variables with $\displaystyle Y_1 = {X_1}^2+{X_2}^2 \; and\; Y_2 = \dfrac{X_1}{\sqrt{{X_1}^2+{X_2}^2}}$. The book shows that $\displaystyle Y_1\;and\;Y_2$ are independent. But how would I find the pdf of $\displaystyle \dfrac{X_1}{X_2}$??

I tried doing $\displaystyle Let\; U = \dfrac{X_1}{X_2} \;and\; V = {X_2} \implies X_1=UV$

then $\displaystyle h_{X_1}(u,v) = uv \;and\;h_{X_2}(u,v)=v$

thus, $\displaystyle J = \left|\begin{array}{cc}v&u\\0&1\end{array}\right|\ ;=\;v$

so, $\displaystyle f(u,v) = \dfrac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{(uv)^2}{2\s igma^2}} \times \dfrac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{(v)^2}{2\si gma^2}} \times v$

is this correct? thanks.

2. Since $\displaystyle X_1 \mbox{and} X_2$ are independent:

$\displaystyle f(x_1,x_2)= \dfrac{1}{2\pi\sigma^2} e^{\frac{{-({x_1}^2+{x_2}^2)}}{2\sigma^2}}$

since $\displaystyle x_1 = uv\;,\; x_2=v\;,\;and\; |J|=v$

$\displaystyle f(u,v)= 2f(x_1,x_2) \; |J| = ....$

Note that you need to find the pdf of $\displaystyle {X_1}/{X_2} = U$. You can do so by finding the marginal pdf of $\displaystyle f(u,v)$