Going through my notes on a course on probability I found excercises of the following form:
"Let $(X,Y)$ be a random vector with density $f_{X,Y}$ which is non-zero on $A$. Given the functions $U,V:\mathbb{R}^2\rightarrow \mathbb{R}^2$, determine whether the random vector $(U,V)$ has a joint density function $f_{U,V}$ and if so, find it."
Solving the exercise requires one to go through the following steps:
- Define $g:A\rightarrow B\subseteq \mathbb{R}^2:(x,y)\mapsto(u,v)$ where $(u,v)=\big(U(x,y),V(x,y)\big)$ and $B$ is the subset of $\mathbb{R}^2$ where $(u,v)$ is nonzero.
- Find $g^{-1}$.
- Find $B$.
- Compute the Jacobian of $g^{-1}$.
Once all of this is done, we have
$$f_{U,V}(u,v)=f_{X,Y}(g^{-1}(x,y))\mid \det J_{g^{-1}}(u,v)\mid 1_B(u,v) \tag{1}$$
where $J_{g^{-1}}$ is the Jacobian of $g^{-1}$, and $1_B(u,v)=1$ for any $(u,v)\in B$ and $0$ otherwise.
How could one prove equation $(1)$?