ML Prep

Linear transform of a multivariate Gaussian

Problem

Let X∼N(μ,Σ)X \sim \mathcal{N}(\mu,\Sigma), where

μ=[12],Σ=[2113],Y=[1−1]X+2.\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 YY.

Reveal answer or reference solution

YY is Gaussian with mean

[1−1]μ+2=1\begin{bmatrix} 1 & -1 \end{bmatrix}\mu + 2 = 1

and variance

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

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

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