{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-stats-estimator",
    "area": "mathematics",
    "topic": "statistics",
    "origin": "original",
    "title": "Bias, variance, MSE, and consistency",
    "skills": [
      "samples-statistics",
      "sampling-distributions",
      "estimator-bias-variance-mse",
      "consistency",
      "standard-error"
    ],
    "priority": "foundation",
    "difficulty": "easy",
    "estimated_minutes": 7,
    "prerequisites": [
      "expectation",
      "variance"
    ],
    "prompt": "For i.i.d. observations with mean $\\mu$ and variance $\\sigma^2$, analyze the sample mean $\\overline{X}_n$ as an estimator of $\\mu$: give its bias, variance, MSE, standard error, and whether it is consistent.\n",
    "answer": "The bias is $0$; the variance and MSE are $\\sigma^2/n$; and the standard error is $\\sigma/\\sqrt{n}$. It is consistent because the variance tends to zero and the estimator is unbiased.\n",
    "check": {
      "kind": "symbolic",
      "id": "original-stats-estimator",
      "values": [
        0,
        1
      ]
    }
  }
}