ML Prep

Linear image of a multivariate Gaussian

Problem

Let

X∼N([1−1],[21/21/21]),Y=[12]X+3X\sim\mathcal N\left(\begin{bmatrix}1\\-1\end{bmatrix}, \begin{bmatrix}2&1/2\\1/2&1\end{bmatrix}\right),\qquad Y=\begin{bmatrix}1&2\end{bmatrix}X+3

Find the distribution of YY.

Reveal answer or reference solution

An affine image of a Gaussian is Gaussian. Its mean is

[1 2](1,−1)T+3=2[1\ 2](1,-1)^{\mathsf T}+3=2

Its variance is

[1 2][21/21/21][12]=8[1\ 2]\begin{bmatrix}2&1/2\\1/2&1\end{bmatrix} \begin{bmatrix}1\\2\end{bmatrix}=8

Therefore Y∼N(2,8)Y\sim\mathcal N(2,8).

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