Multivariate Transformations (One-to-One)

Note that in this extension, we will use $X_1,\,X_2,\,\dots ,\, X_n$ for the `original' continuous variables (rather than $X$ and $Y$ as we had for 2 variables) and $U_1,\,U_2,\,\dots ,\,U_n$ or $Y_1,\,\dots ,\,Y_n$ are used for the `new' variables (rather than $U$ and $V$).

Given random variables $X_1,\,X_2,\,\dots ,\, X_n$ with joint pdf $f_{\bf X} (x_1,\,x_2,\,\dots ,\,x_n)$ which is non-zero on the $n$-dimensional space ${\cal A}$. Define

$$\left. \begin{array}{l} u_1 = g_1(x_1,\,x_2,\,\dots ,\,x_n... ...\\ u_n =g_n(x_1,\,x_2,\,\dots ,\,x_n) \end{array} \right\}$$ (3.3)

and suppose this is a one-to-one transformation mapping ${\cal A}$ onto a space ${\cal B}$. Extending (2.3) to this case we have, for the joint pdf of $U_1,U_2,\ldots,U_n$,

$$f_{\bf U}(u_1,u_2,\ldots,u_n)=f_{\bf X}(x_1,x_2, \ldots,x_n)... ...box{ where } J=\left(\frac{\partial x_i}{\partial u_j}\right).$$ (3.4)

[Note that $J$ is the matrix of partial derivatives.]