# Joint probability distribution of functions of random variables

Let $X_1$ and $X_2$ be jointly continuous random variables with joint probability density function $f(x_1,x_2)$.It is sometimes necessary to obtain the joint distribution of the random variables $Y_1$ and $Y_2$ that arise as functions of $X_1$ and $X_2$.Specifically, suppose that $Y_1=g_1(X_1,X_2)$ and $Y_2=g_2(X_1,X_2)$ for some functions $g_1$ and $g_2$

Assume that the functions $g_1$ and $g_2$ satisfy the following conditions

1)The equations $y_1=g_1(x_1,x_2)$ and $y_2=g_2(x_1,x_2)$ can be uniquely solved for $x_1$ and $x_2$ in terms of $y_1$ and $y_2$ with solutions given by, say , $x_1=h_1(y_1,y_2),x_2=h_2(y_1,y_2)$

2) the functions $g_1$ and $g_2$ have continuous partial derivatives at all points $(x_1,x_2)$ and are such that the following $2\times 2$ determinant

$j(x_1,x_2)=\left|\begin{matrix}\frac{dg_1}{dx_1}&\frac{dg_1}{dx_2}\\ \frac{dg_2}{dx_1}&\frac{dg_2}{dx_2}\end{matrix}\right| \equiv \frac{dg_1}{dx_1}*\frac{dg_2}{dx_2}-\frac{dg_1}{dx_2}*\frac{dg_2}{dx_1}\not=0$

at all points $(x_1,x_2)$.

Under these two conditions it can be shown that the random variables $Y_1$ and $Y_2$ are jointly continuous with joint density function given by

$f_{Y_1,Y_2}(y_1,y_2)=f_{X_1,X_2}(x_1,x_2)|J(x_1,x_2)|^{-1}$...(1)

where $x_1=h_1(y_1,y_2),x_2=h_2(y_1,y_2).$

I want to find out the proof of equation (1).How shall I calculate it? Your help is required.

## migrated from stats.stackexchange.comFeb 25 '16 at 1:57

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