I have derived the density for the ratio of two independent random variables,via the transformation formula. In this way: $V = X/Y $ and $ U = X $ inversion yields: $Y = U/V$ och $X =U$ , the jacobian: $J(x,y)/J(u,v) = \begin{vmatrix} 1 & 0 \\ 1/v & -u/v^{2} \end{vmatrix} = -u/v^2$, so the joint density $f_{U,V}(u,v) = f_{X,Y}(u,u/v) u/v^{2} $ now, i can obtain the density for $V$ like this $f_{V= X/Y}(v) = \int_{-\infty}^{\infty} f_X(u)f_Y(u/v )u/v^2 du $ ,where i have also used that X and Y are independent.
first of all, did my calculations turn out to be right?, secondly, i want to apply this when X and Y are independent $N(0,1)$ to show that the ratio X/Y is C(0,1) - Cauchy. In that case,using what i have obtained above, we would get:
$$f_{V= X/Y}(v)= \int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}} e^{-u^2/2} \frac{1}{\sqrt{2\pi}} e^{-u^2/2v^2} u/v^2 du$$
but Iam stuck this doesnt work i don't know where Iam wrong. anyone?
Use the transformation
$$U = \frac{X}{Y}\,\,,V = Y\\X = UV\,\,,Y=V$$
The Jacobian is
$$J = \frac{ \partial (x,y)}{\partial (u,v)}= v$$
The joint density is:
$$ f_{UV}(u,v)=f_{XY}(X(u,v),Y(u,v))|J|= \frac{|v|}{2\pi}\exp[-v^2(u^2+1)/2].$$
Integrate to find the marginal density for $U$,
$$ f_{U}(u)= \frac{1}{2\pi}\int_{-\infty}^{\infty}|v|\exp[-v^2(u^2+1)/2]dv= \frac{1}{\pi}\int_{0}^{\infty}v\exp[-v^2(u^2+1)/2]dv.$$
Use the substitution $s^2 = v^2(u^2+1)$
$$ f_{U}(u)= \frac{1}{\pi(u^2+1)}\int_{0}^{\infty}s\exp[-s^2/2]ds=\frac{1}{\pi(u^2+1)}.$$