Finding joint density of dependent variables

Question

I've got two random variables, $x$ and $y$, where $x=u(y)$ and $y=v(x)$. How do I find the joint density of the two variables?

The eventual purpose is to find the integral

$\int^a\int^b f(x,y) dydx$,

where $f(.)$ is said joint density function.

This question is related (the joint distribution of dependent random variables) but I'm still a bit confused by the answer. I am indeed effectively looking for $Prob(x \leq a, y \leq b)$ but don't understand how I can get there without knowing the PDF first.

Any advice is much appreciated. Thanks a lot in advance!

Answer 1

...two random variables, $X$ and $Y$, where $X=u(Y)$ and $Y=v(X)$.

This is a highly peculiar setting since it may happen that the distribution of $(x,y)$ is actually concentrated on one point. For example, if the functions are $u:y\mapsto3y-1$ and $v:x\mapsto2x-3$, the only $(x,y)$ such that $x=u(y)$ and $y=v(x)$ is $(2,1)$.

The set of solutions of $x=u(y)$ and $y=v(x)$ might even be empty...