{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-prob-distributions",
    "area": "mathematics",
    "topic": "probability",
    "origin": "original",
    "title": "Recognize distribution moments",
    "skills": [
      "discrete-random-variables",
      "continuous-random-variables",
      "named-distributions",
      "sums-of-random-variables",
      "expectation",
      "variance"
    ],
    "priority": "foundation",
    "difficulty": "easy",
    "estimated_minutes": 8,
    "prerequisites": [
      "random-variables"
    ],
    "prompt": "Give the mean and variance of $\\operatorname{Binomial}(n,p)$, $\\operatorname{Poisson}(\\lambda)$, $\\operatorname{Exponential}(\\lambda)$ under the rate parameterization, and $\\mathcal{N}(\\mu,\\sigma^2)$. Then identify the distribution of a sum of independent Poisson variables.\n",
    "answer": "- $\\operatorname{Binomial}(n,p)$: mean $np$ and variance $np(1-p)$.\n- $\\operatorname{Poisson}(\\lambda)$: mean $\\lambda$ and variance $\\lambda$.\n- $\\operatorname{Exponential}(\\lambda)$: mean $1/\\lambda$ and variance $1/\\lambda^2$.\n- $\\mathcal{N}(\\mu,\\sigma^2)$: mean $\\mu$ and variance $\\sigma^2$.\n\nIndependent Poisson variables sum to a Poisson variable with rate equal to the sum of the rates.\n",
    "check": {
      "kind": "symbolic",
      "id": "original-prob-distributions",
      "values": [
        1,
        1
      ]
    }
  }
}