I've been working on a two part question on bivariate transformations and marginal densities but am having difficulty finding where I have made a mistake as the final answer for the marginal density is giving an answer of zero.
The first part of the question states that Y ~ N(0,1), Z ~ N(0,1) and that S = $\frac{Y}{Z}$ and Z = T.
In this part I aim to find the joint distribution of S and T,
$f_{ST}= f_{XY}$ |detJ|.
I find the determinant to equal T and $f_{XY} = \frac{1}{2\pi}$exp${-\frac{(x^2+y^2)}{2}}$, (assuming independence of Y and Z) and thus I get that
$f_{ST} = \frac{T}{2\pi}$exp${-\frac{(T^2+S^2T^2)}{2}}$,
replacing Z and Y with S and T.
However, in the second part of the question, it asks me to find the marginal distribution of S, which upon integrating from negative infinity to infinity comes out to be zero.
I don't think this is a correct answer as the area under $f_S(s)$ should be 1 and thus it can't be zero everywhere, and therefore I was wondering where I went wrong?
Thanks in advance, any help will be greatly appreciated!
Mark