I have two random variables $x_1,x_2$ whose distributions are unknown. I define $y_1=g(x_1,x_2)$ and $y_2=f(x_1,x_2)$ where $f(\cdot)$ and $g(\cdot)$ are known and the probability distributions $p(y_1)$ and $p(y_2)$ are known. How could I obtain the joint probability?
One proposed: $$ P(x_1,x_2)=\int\int P(y_2)P(y_1)\delta(y_2 -f(x_1,x_2))\;\delta(y_1 -g(x_1,x_2))dy_1dy_2 $$ However I am unable to derive the formula nor the reason behind it. Thank you very much for your help! Any idea highly appreciated.