1
$\begingroup$

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.

$\endgroup$

migrated from stats.stackexchange.com Feb 25 '16 at 1:57

This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.