# Linear image of a multivariate Gaussian

> ML Prep screening item.

- Stable ID: `math-prob-09-gaussian-linear-map`
- Collection: screening
- Area: Mathematics
- Topic: Probability
- Origin: original
- Difficulty: hard
- Expected time: 5 minutes
- Skills: multivariate-gaussian, transformations, covariance
- Item JSON: https://mlprep.iwase.dev/item/math-prob-09-gaussian-linear-map.json

## Problem

Let
$$
X\sim\mathcal N\left(\begin{bmatrix}1\\-1\end{bmatrix},
\begin{bmatrix}2&1/2\\1/2&1\end{bmatrix}\right),\qquad
Y=\begin{bmatrix}1&2\end{bmatrix}X+3
$$
Find the distribution of $Y$.


## Answer policy

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## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `math-prob-09-gaussian-linear-map`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.