Linear transform of a multivariate Gaussian

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Problem

Let $X \sim \mathcal{N}(\mu,\Sigma)$ , where

\mu = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \qquad \Sigma = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}, \qquad Y = \begin{bmatrix} 1 & -1 \end{bmatrix}X + 2.

Find the distribution of $Y$ .

Reveal answer or reference solution

$Y$ is Gaussian with mean

\begin{bmatrix} 1 & -1 \end{bmatrix}\mu + 2 = 1

and variance

\begin{bmatrix} 1 & -1 \end{bmatrix} \Sigma \begin{bmatrix} 1 & -1 \end{bmatrix}^{\mathsf T} = 3.

Therefore, $Y \sim \mathcal{N}(1,3)$ .

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Probability Practice