Change of Variable Formula

We first focus on 1D variable case. If \( X=f(Z) \) and \( f(\cdot) \) is monotone with inverse \( Z=f^{-1}(X)=h(X) \), then

\begin{equation} p_{X}(x)=p_{Z}(h(x)) |h'(x)| \end{equation}

To generalize the above to multivariable cases, we use Jacobian of \( \mathbf{f}(\cdot) \). The mapping between \(Z\) and \(X\) given by \(\mathbf{f}:\mathbb{R}^{n} \mapsto\mathbb{R}^{n}\) is invertible such that \( X=\mathbf{f}(Z) \) and \( Z=\mathbf{f}(X) \)

\begin{equation} p_{X}(\mathbf{x}) = p_{Z}(\mathbf{f}^{-1}(\mathbf{x})) \left\vert \det \left( \frac{\partial \mathbf{f}^{-1}(\mathbf{x})}{\partial \mathbf{x}} \right) \right\vert \end{equation}

Note that \(\mathbf{x},\mathbf{z}\) need to be continuous and have same dimension.