I'm looking for some help with this probability problem.

Here's the question:

> 
> Suppose that $X$ and $Y$ are independent standard normal random variables. Show that the probability density function of $Z = X / |Y|$ is given by
>  $$
> f(t) = \frac{1}{\pi(1+t^2)}, \quad (-\infty < t < \infty).
> $$
> 
> 
> 

Thanks for looking through my question.

-updated-
Hi guys, I've looked through the solution and am still thinking, i'm also thinking if its possible to log the expressions and apply convolution theorem... thanks for the answers once again.

-updated-
tried the solution using first method but couldn't figured out how to complete the last step.

Heres how i did it:

$f(x,y) = f\_{x}(x)f\_{y}(y) = \frac{1}{2\pi}e^{-\frac{1}{2}(x^2 + y^2)}$

let Z = X/|Y|, V = X then x = v, y = v/z

the Jacobian J = $x\_{z}y\_{v}-x\_{v}y\_{z} = \frac{v}{z^2}$ 

by Transformation theorem,

$w(z,v)=f(v,\frac{v}{z})|J| = ???$ 

Will someone please point out to me how do i proceed from here to obtain f(t)? thanks a lot!

-update 3- this is actually a Cauchy density!! thanks guys, i think i got it figured out.