# Linear transform of a multivariate Gaussian

> ML preparation item.

- Stable ID: `original-prob-gaussian`
- Area: Mathematics
- Topic: Probability
- Origin: original
- Expected time: 8 minutes
- Skills: multivariate-gaussian, transformations, covariance-correlation
- Item JSON: https://mlprep.iwase.dev/data/items/original-prob-gaussian.json

## 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 $Y$.


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Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `original-prob-gaussian`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.