$$ \newcommand{\qed}{\tag*{$\square$}} \newcommand{\span}{\operatorname{span}} \newcommand{\dim}{\operatorname{dim}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\norm}[1]{\|#1\|} \newcommand{\grad}{\nabla} \newcommand{\prox}[1]{\operatorname{prox}_{#1}} \newcommand{\inner}[2]{\langle{#1}, {#2}\rangle} \newcommand{\mat}[1]{\mathcal{M}[#1]} \newcommand{\null}[1]{\operatorname{null} \left(#1\right)} \newcommand{\range}[1]{\operatorname{range} \left(#1\right)} \newcommand{\rowvec}[1]{\begin{bmatrix} #1 \end{bmatrix}^T} \newcommand{\Reals}{\mathbf{R}} \newcommand{\RR}{\mathbf{R}} \newcommand{\Complex}{\mathbf{C}} \newcommand{\Field}{\mathbf{F}} \newcommand{\Pb}{\operatorname{Pr}} \newcommand{\E}[1]{\operatorname{E}[#1]} \newcommand{\Var}[1]{\operatorname{Var}[#1]} \newcommand{\argmin}[2]{\underset{#1}{\operatorname{argmin}} {#2}} \newcommand{\optmin}[3]{ \begin{align*} & \underset{#1}{\text{minimize}} & & #2 \\ & \text{subject to} & & #3 \end{align*} } \newcommand{\optmax}[3]{ \begin{align*} & \underset{#1}{\text{maximize}} & & #2 \\ & \text{subject to} & & #3 \end{align*} } \newcommand{\optfind}[2]{ \begin{align*} & {\text{find}} & & #1 \\ & \text{subject to} & & #2 \end{align*} } $$
Let be a random variable with support in and density , and let , where is a one-to-one, continuously differentiable mapping such that its Jacobian determinant is nonzero for all . Then the density of , which we denote , is given by
or equivalently
We can sketch a proof of this result by appealing to the change of variables theorem. For all subsets in the support of ,
On the other hand, under the change of variables , the integral in (2) is equal to
Since this is true for all subsets , we can conclude that the integrands in (1) and (3) are equal, giving
as desired.