Starting from \(n\) random variables, \(x=\left(x_{1}, \ldots, x_{n}\right)\), the following technique can be used to determine the joint p.d.f of \(n\) linearly independent functions \(a_{i}(\mathbf{x})\), with \(i=1, \ldots, n \). Assuming the functions \(a_{1}, \ldots, a_{n}\) can be inverted to give \(x_{i}\left(a_{1}, \ldots, a_{n}\right), i=1, \ldots, n\), the joint p.d.f. for the \(a_{i}\) is given by
$$
g\left(a_{1}, \dots, a_{n}\right)=f\left(x_{1}, \dots, x_{n}\right)|J|
$$
where \(|J|\) is the absolute value of the Jacobian determinant for the transformation. To determine the marginal p.d.f. for one of the functions (say \(g_{1}\left(a_{1}\right)\) ) the joint p.d.f. \(g\left(a_{1}, \ldots, a_{n}\right)\) must be integrated over the remaining \(a_{i}\).