{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-prob-gaussian",
    "area": "mathematics",
    "topic": "probability",
    "origin": "original",
    "title": "Linear transform of a multivariate Gaussian",
    "skills": [
      "multivariate-gaussian",
      "transformations",
      "covariance-correlation"
    ],
    "priority": "core",
    "difficulty": "medium",
    "estimated_minutes": 8,
    "prerequisites": [
      "matrix-multiplication",
      "gaussian-distribution"
    ],
    "prompt": "Let $X \\sim \\mathcal{N}(\\mu,\\Sigma)$, where\n$$\n\\mu = \\begin{bmatrix} 1 \\\\ 2 \\end{bmatrix},\n\\qquad\n\\Sigma = \\begin{bmatrix} 2 & 1 \\\\ 1 & 3 \\end{bmatrix},\n\\qquad\nY = \\begin{bmatrix} 1 & -1 \\end{bmatrix}X + 2.\n$$\nFind the distribution of $Y$.\n",
    "answer": "$Y$ is Gaussian with mean\n$$\n\\begin{bmatrix} 1 & -1 \\end{bmatrix}\\mu + 2 = 1\n$$\nand variance\n$$\n\\begin{bmatrix} 1 & -1 \\end{bmatrix}\n\\Sigma\n\\begin{bmatrix} 1 & -1 \\end{bmatrix}^{\\mathsf T} = 3.\n$$\nTherefore, $Y \\sim \\mathcal{N}(1,3)$.\n",
    "check": {
      "kind": "numeric",
      "id": "original-prob-gaussian",
      "values": [
        1,
        3
      ]
    }
  }
}