I have a problem in deriving the transformed joint distribution for continuous random variables. The textbook says use jacobian which makes sense but I wanted to go from first principles like below...
Let $x_1$, $x_2$ be continuous random variables with distributions $X_1$, $X_2$
Transformed as $y_1 = f_1(x_1,x_2)$; $y_2 = f_2(x_1,x_2)$
and the inverses $x_1 = g_1(y_1,y_2)$; $x_2 = g_2(y_1,y_2)$ assume monotonic functions, one - one mapping etc....
Let CDF of the Joint dist of $x_1$, $x_2$ be $F_{X_1,X_2}(x_1,x_2)$
Let CDF of the Joint dist of $y_1$, $y_2$ be $F_{Y_1,Y_2}(y_1,y_2)$
we can write Joint PDF of $y_1,y_2$ as $f_{Y_1,Y_2}(y_1,y_2)$= $\partial^2 {F_{Y_1,Y_2}(y_1,y_2)}\over \partial{y_1}\partial{y_2}$
which can be written as $\partial {\partial F_{Y_1,Y_2}(y_1,y_2)\over\partial y_2} \over\partial y_1$
$\partial F_{Y_1,Y_2}(y_1,y_2)\over\partial y_2$ = $_{\delta y_2\to 0} {F_{Y_1,Y_2}(y_1,y_2 + \delta y_2) - F_{Y_1,Y_2}(y_1,y_2)} \over \delta y_2$ ...(1)
$\partial^2 {F_{Y_1,Y_2}(y_1,y_2)}\over \partial{y_1}\partial{y_2}$ = $_{\delta y_1\to 0} {{_{\delta y_2\to 0}{F_{Y_1,Y_2}(y_1+\delta y_1,y_2 + \delta y_2) - F_{Y_1,Y_2}(y_1+\delta y_1,y_2)} \over \delta y_2} - {_{\delta y_2\to 0} {F_{Y_1,Y_2}(y_1,y_2 + \delta y_2) - F_{Y_1,Y_2}(y_1,y_2)} \over \delta y_2}}\over {\delta y_1} $ ...(2)
Now we can replace $F_{Y_1,Y_2}(y_1,y_2)$ by $F_{X_1,X_2}(x_1,x_2)$
and $\delta x_1 = \delta y_1 \cdot {\partial{g_1}\over\partial{y_1}} + \delta y_2 \cdot { \partial{g_1}\over\partial{y_2}};\delta x_2 = \delta y_1 \cdot {\partial{g_2}\over\partial{y_1}} + \delta y_2 \cdot { \partial{g_2}\over\partial{y_2}}$ ...(3)
so that we can write $F_{Y_1,Y_2}(y_1+\delta y_1,y_2 + \delta y_2)$ = $F_{X_1,X_2}(x_1+\delta y_1 \cdot {\partial{g_1}\over\partial{y_1}} + \delta y_2 \cdot { \partial{g_1}\over\partial{y_2}},x_2 + \delta y_1 \cdot {\partial{g_2}\over\partial{y_1}} + \delta y_2 \cdot { \partial{g_2}\over\partial{y_2}})$ ...(4)
The above term can be written as $\int\limits_{}^{[x_1+\delta y_1 \cdot {\partial{g_1}\over\partial{y_1}} + \delta y_2 \cdot { \partial{g_1}\over\partial{y_2}}]} \int\limits_{}^{[x_1+\delta y_1 \cdot {\partial{g_1}\over\partial{y_1}} + \delta y_2 \cdot { \partial{g_1}\over\partial{y_2}}]} f_{X_1,X_2}(x_1,x_2) dx_1\,dx_2$
using the same substitution as (4) for other terms in (2) and doing some basic arithmetic, I get an answer that is a bit different than whats written in text books $f_{Y_1,Y_2}(y_1,y_2) = f_{X_1,X_2}(x_1,x_2) \cdot ({\partial g_1\over\partial y_1} \cdot {\partial g_2\over\partial y_2} + {\partial g_2\over\partial y_1} \cdot {\partial g_1\over\partial y_2})$
The text book says using Jacobian $f_{Y_1,Y_2}(y_1,y_2) = f_{X_1,X_2}(x_1,x_2) \cdot ({\partial g_1\over\partial y_1} \cdot {\partial g_2\over\partial y_2} - {\partial g_2\over\partial y_1} \cdot {\partial g_1\over\partial y_2})$