I'm hoping to get a hint on a problem. The problem formulation is: there are two random variables X and Y, both of which are a Uniform RV on (0,1). Let x be values on (0,1) for X and y be values on (0,1) for Y. Then define a function g(x,y) = (XY, X). What is the inverse of g?
For clean enough variables, I believe the analytic inverse of g is the answer to this question. When I look at g, and want and inverse, I'm thinking an analytic inverse of the PDF of g. However -- up to this point, I'm used to calculating joint PDFs using the independence of the random variables. But XY and X can't be independent.
How would I approach trying to calculate the PDF of g for two non-independent random variables?
$$f_{W,X}(w,x)=f_{W\mid X}(w\mid x)\cdot f_X(x)=\frac{\mathbf 1_{0\leqslant w\leqslant x}}x\cdot\mathbf 1_{0\leqslant x\leqslant 1}=\frac{\mathbf 1_{0\leqslant w\leqslant x\leqslant 1}}x$$