I came across this problem from an exam about recursive estimation I found online:
Let $\newcommand{\R}{\mathbb{R}}$
$x\in\R^n$ and $y\in\R^n$ with a joint Gaussian distribution,
$\begin{bmatrix}
x\\
y\\
\end{bmatrix}$ ~ $\mathcal{N}$ $\begin{pmatrix}\begin{bmatrix} \mu_x\\
\mu_y\\
\end{bmatrix}\
,\begin{bmatrix}\sum_x & \sum_{xy} \\
\sum_{xy}^T & \sum_y \\
\end{bmatrix}\end{pmatrix} $
Find the variance of z = x – y
I couldn't find a change of variable formula for multivariate gaussian distributions online and I really don't know how to possibly solve this. Any help is appreciated
Best Answer
Lets rewrite your variables, $$ q = \begin{bmatrix} x\\ y\\\end{bmatrix}\\ \mu = \begin{bmatrix} \mu_x\\ \mu_y\\\end{bmatrix}\\ \Sigma = \begin{bmatrix} \Sigma_{xx} & \Sigma_{xy}\\ \Sigma_{yx} & \Sigma_{xy} \end{bmatrix} $$ Then what you are referring to is an affine transformation $$ {z} = B q\\ B = \begin{bmatrix} {I} & -{I}\end{bmatrix}\\ $$ According to wikipedia, after an affine transformation, $z$ will be distributed according to a normal distribution $$ z \sim N\left( B\mu, B \Sigma B^T \right). $$