{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-prob-total-variance",
    "area": "mathematics",
    "topic": "probability",
    "origin": "original",
    "title": "Conditional expectation and total variance",
    "skills": [
      "conditional-expectation",
      "total-expectation-variance",
      "expectation",
      "variance"
    ],
    "priority": "core",
    "difficulty": "medium",
    "estimated_minutes": 8,
    "prerequisites": [
      "conditional-probability"
    ],
    "prompt": "Let $Z \\sim \\operatorname{Bernoulli}(1/2)$. Conditional on $Z=0$, $X$ has mean $0$ and variance $1$; conditional on $Z=1$, $X$ has mean $2$ and variance $3$. Compute $\\mathbb{E}[X]$ and $\\operatorname{Var}(X)$.\n",
    "answer": "We have $\\mathbb{E}[X]=1$. Also,\n$$\n\\mathbb{E}[\\operatorname{Var}(X\\mid Z)] = \\frac{1+3}{2} = 2\n$$\nand\n$$\n\\operatorname{Var}(\\mathbb{E}[X\\mid Z]) = 1,\n$$\nbecause $\\mathbb{E}[X\\mid Z]$ equals $0$ or $2$ with equal probability. Therefore, $\\operatorname{Var}(X)=3$.\n",
    "check": {
      "kind": "numeric",
      "id": "original-prob-total-variance",
      "values": [
        1,
        3
      ]
    }
  }
}