How do I find the inverse of the below using inverse mapping?
$X,Y$ are independent random variables.
$U = [x/(x+y)], V = [(x+y)]$
An example of a solution to another problem is here. However, I do not understand how the inverse is made. The answer in that question bypasses the actual process of the mapping.
This is my incorrect solution:
Inverse of $U$:
1) $U = X/V$ Substituting V for the (x+y) term.
2) $X = U/V$ Swap U and X to create the inverse function
3) According to the process of solving inverses, you solve for X. It's already solved for X so no further work necessary.
The same process applies to $V$:
1) $V = (U/V)+Y$ Substituting U for V in the X term.
2) $Y = (U/V)+V$ Swap V with Y to create the inverse.
3) As before, a third step would apply if variable Y was not isolated, however it is and no further work is necessary as we have the inverse.
Inverse of U and V are: $$ X=UV$$ $$ Y=V(1-U)$$
Then, you have to calculate the Jacobian determinant of this inverse and evaluate the integral, in the domain determinated for U and V, of the pdf of X and Y calculated in the function inverse.
This is the theorem.