{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-stats-mom-map",
    "area": "mathematics",
    "topic": "statistics",
    "origin": "original",
    "title": "Method of moments and Bayesian updating",
    "skills": [
      "method-of-moments",
      "map",
      "bayesian-updating",
      "likelihood"
    ],
    "priority": "core",
    "difficulty": "medium",
    "estimated_minutes": 10,
    "prerequisites": [
      "exponential-distribution",
      "beta-distribution"
    ],
    "prompt": "For $\\operatorname{Exponential}(\\lambda)$ data under the rate parameterization, derive the method-of-moments estimate. Separately, with a $\\operatorname{Beta}(2,2)$ prior and $7$ successes in $10$ Bernoulli trials, give the posterior and its MAP.\n",
    "answer": "Matching $\\mathbb{E}[X]=1/\\lambda$ to the sample mean gives\n$$\n\\widehat{\\lambda} = \\frac{1}{\\overline{X}}.\n$$\nThe posterior is $\\operatorname{Beta}(9,5)$, whose MAP is\n$$\n\\frac{9-1}{9+5-2} = \\frac{2}{3}.\n$$\n",
    "check": {
      "kind": "numeric",
      "id": "original-stats-mom-map",
      "values": [
        9,
        5,
        0.6666666666666666
      ]
    }
  }
}