Proof: Probability density function of a linear function of a continuous random vector
Theorem: Let $X$ be an $n \times 1$ random vector of continuous random variables with possible outcomes $\mathcal{X} \subseteq \mathbb{R}^n$ and let $Y = \Sigma X + \mu$ be a linear transformation of this random variable with constant $n \times 1$ vector $\mu$ and constant $n \times n$ matrix $\Sigma$. Then, the probability density function of $Y$ is
\[\label{eq:pdf-linfct} f_Y(y) = \left\{ \begin{array}{rl} \frac{1}{\left| \Sigma \right|} f_X(\Sigma^{-1}(y-\mu)) \; , & \text{if} \; y \in \mathcal{Y} \\ 0 \; , & \text{if} \; y \notin \mathcal{Y} \end{array} \right.\]where $\lvert \Sigma \rvert$ is the determinant of $\Sigma$ and $\mathcal{Y}$ is the set of possible outcomes of $Y$:
\[\label{eq:Y-range} \mathcal{Y} = \left\lbrace y = \Sigma x + \mu: x \in \mathcal{X} \right\rbrace \; .\]Proof: Because the linear function $g(X) = \Sigma X + \mu$ is invertible and differentiable, we can determine the probability density function of an invertible function of a continuous random vector using the relation
\[\label{eq:pdf-invfct} f_Y(y) = \left\{ \begin{array}{rl} f_X(g^{-1}(y)) \, \left| J_{g^{-1}}(y) \right| \; , & \text{if} \; y \in \mathcal{Y} \\ 0 \; , & \text{if} \; y \notin \mathcal{Y} \end{array} \right. \; .\]The inverse function is
\[\label{eq:g-inv} X = g^{-1}(Y) = \Sigma^{-1}(Y-\mu) = \Sigma^{-1} Y - \Sigma^{-1} \mu\]and the Jacobian matrix is
\[\label{eq:J-g-inv} J_{g^{-1}}(y) = \left[ \begin{matrix} \frac{\mathrm{d}x_1}{\mathrm{d}y_1} & \ldots & \frac{\mathrm{d}x_1}{\mathrm{d}y_n} \\ \vdots & \ddots & \vdots \\ \frac{\mathrm{d}x_n}{\mathrm{d}y_1} & \ldots & \frac{\mathrm{d}x_n}{\mathrm{d}y_n} \end{matrix} \right] = \Sigma^{-1} \; .\]Plugging \eqref{eq:g-inv} and \eqref{eq:J-g-inv} into \eqref{eq:pdf-invfct} and applying the determinant property $\lvert A^{-1} \rvert = \lvert A \rvert^{-1}$, we obtain
\[\label{eq:pdf-linfct-qed} f_Y(y) = \frac{1}{\left| \Sigma \right|} f_X(\Sigma^{-1}(y-\mu)) \; .\]- Taboga, Marco (2017): "Functions of random vectors and their distribution"; in: Lectures on probability and mathematical statistics, retrieved on 2021-08-30; URL: https://www.statlect.com/fundamentals-of-probability/functions-of-random-vectors.
Metadata: ID: P255 | shortcut: pdf-linfct | author: JoramSoch | date: 2021-08-30, 07:46.