I would like to know whether it is possible to obtain the joint density function $p(x,y)$ if I know one marginal, which is Gaussian $p(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp(-\frac{1}{2} (\frac{x - \mu}{\sigma})^2)$ and the relation between variables $y = \frac{1}{(1+x^{2})}$.
I know that virtually one can always obtain the marginal probability density function $p(y)$ if $p(x)$ and the dependence between variables $y=f(x)$ are known, via the distribution function technique. However I don't know how one would construct from there, if possible, the joint density.
I am aware that this is doable numerically, but I am looking for a method to obtain an analytical expression. Thank you!
There is no joint density for $x$ and $y$. The range of $(x,y)$ lies in a set of measure zero so the density function does not exist.