# Joint distribution of transformed variables

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 $f_{Y_1,Y_2}(y_1,y_2) = f_{X_1,X_2}(x_1,x_2) \cdot ({\partial g_1\over\partial y_1} \cdot {\partial g_2\over\partial y_2} - {\partial g_2\over\partial y_1} \cdot {\partial g_1\over\partial y_2})$

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Are your "$*$" just ordinary multiplication? That is usually not written as a star in properly typeset mathematics, but as nothing (juxtaposition) or a centered dot (\cdot in TeX). –  Henning Makholm Jan 15 '12 at 21:05
yes just ordinary multiplication.... –  dasman Jan 15 '12 at 21:10
Note that you need the absolute value of the Jacobian, not the Jacobian. The basic idea is that the probability that $(Y_1,Y_2)$ is in a small neighborhood (of area $(\Delta y_1 \Delta y_2)$) of $(y_1,y_2)$ equals the probability that $(X_1,X_2)$ is in a small neighborhood (of area $(\Delta x_1 \Delta x_2)$) of $(x_1,x_2)$. These probabilities are approximately $f_{Y_1,Y_2}(y_1,y_2)(\Delta y_1 \Delta y_2)$ and $f_{X_1,X_2}(x_1,x_2)(\Delta x_1 \Delta x_2)$ and the ratio of the areas is the absolute value of the Jacobian. –  Dilip Sarwate Jan 15 '12 at 21:14