Editing Multivariate random variable (section)

===Invertible mappings===
More generally we can study invertible mappings of random vectors.<ref name=Lapidoth>{{cite book |last=Taboga |first=Marco |title=Lectures on Probability Theory and Mathematical Statistics |publisher= CreateSpace Independent Publishing Platform |year=2017 |isbn=978-1981369195 }}</ref>{{rp|p.284–285}}

Let <math>g</math> be a one-to-one mapping from an open subset <math>\mathcal{D}</math> of <math>\mathbb{R}^n</math> onto a subset <math>\mathcal{R}</math> of <math>\mathbb{R}^n</math>, let <math>g</math> have continuous partial derivatives in <math>\mathcal{D}</math> and let the [[Jacobian matrix and determinant|Jacobian determinant]] <math>\det\left (\frac{\partial \mathbf{y}}{\partial \mathbf{x}}\right )</math> of <math>g</math> be zero at no point of <math>\mathcal{D}</math>. Assume that the real random vector <math>\mathbf{X}</math> has a probability density function <math>f_{\mathbf{X}}(\mathbf{x})</math> and satisfies <math> P(\mathbf{X} \in \mathcal{D}) = 1</math>. Then the random vector <math>\mathbf{Y}=g(\mathbf{X})</math> is of probability density

:<math>\left. f_{\mathbf{Y}}(\mathbf{y})=\frac{f_{\mathbf{X}}(\mathbf{x})}{\left |\det\left (\frac{\partial \mathbf{y}}{\partial \mathbf{x}}\right )\right |} \right |_{\mathbf{x}=g^{-1}(\mathbf{y})} \mathbf{1}(\mathbf{y} \in R_\mathbf{Y})</math>

where <math>\mathbf{1}</math> denotes the [[indicator function]] and set <math>R_\mathbf{Y} = \{ \mathbf{y} = g(\mathbf{x}): f_{\mathbf{X}}(\mathbf{x}) > 0 \} \subseteq \mathcal{R} </math> denotes support of <math>\mathbf{Y}</math>.