Suppose X,Y are inddependent standard normal rndom variable . Let $Z= \frac{|X|}{|Y|}$
I am asked to find the pdf of Z
$Z= \frac{X}{Y} \quad X\ge 0 ; Y\ge 0 \\ =\frac{-X}{Y} \quad X\le 0 ; Y\ge 0 \\ = \frac{-X}{Y} \quad X\ge 0 ; Y\le 0\\ = \frac{X}{Y} \quad X\le 0 ; Y\le 0\\ $
Can I apply Jacobian method for this transformation? (I dont think jacobian can be applied since Z is not one-one function of x,y)
$Z= \frac{X}{Y} \quad ; \quad V=X\\ J= \begin{vmatrix} \frac{\partial X}{\partial Z} &\frac{\partial X}{\partial V}\\ \frac{\partial Y}{\partial Z} & \frac{\partial Y}{\partial V} \end{vmatrix}$
$|J|= \frac{1}{z^2}$
$f_{X,Y}(X,Y)= \frac{1}{2\pi} e^{\frac{x^2 + y^2}{2}}$
$\therefore f_{V,Z}(v,z)= \frac{1}{2\pi} e^{\frac{v^2}{2} + \frac{v^2}{2z^2}}$
$f_Z(z)= \int_V f_{V,Z}(v,z)dz$
this is the part i am stuck at, What are the limits of V?
I am not even sure if my method is correct , but this is the only way i can think of .
Any help ?? Thank You
Hint: Easier to find $X/Y$ first, then take it's absolute value. Show that $X/Y$ is standard Cauchy, so that $|X/Y| $ is 'folded' Cauchy.
You are headed in the right direction but you forgot to multiply the |Jacobian| in the joint density of $(V,Z)$, which is actually $v/z^2$. Support of both $X$ and $Y$ is $\mathbb R$. So both $V$ and $Z$ also have support $\mathbb R$. The integration limits should be clear now. The transformation is one-to-one.
(For deriving the distribution of $X/Y$ , an easier choice of $V$ is $Y$.)