{
  "aliases": [
    "coupon-collector-problem"
  ],
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  "body": "\n## Problem\n\nThere are $n \\ge 1$ different types of coupons. Each draw gives a coupon of a uniform random type, independently of the other draws.\nLet $T$ be the number of draws needed to collect all $n$ types. Find $E[T]$.\n\n## Remarks\n\nSolution: write $T = t_1 + \\dots + t_n$, where $t_i$ is the number of draws to get a new type when $i - 1$ types are known.\nEach $t_i$ counts trials up to the first success, with success probability $p_i = (n - i + 1)/n$. So $E[t_i] = 1/p_i$ by the [[mean of the geometric distribution|0DB0R4N5XX]].\nBy [[linearity of expectation|0P7PTC8DDE]], $E[T] = n \\left(\\frac{1}{1} + \\frac{1}{2} + \\dots + \\frac{1}{n}\\right) = n H_n$, where $H_n$ is the $n$-th harmonic number. So $E[T] = n \\ln n + \\gamma n + \\frac{1}{2} + o(1)$, with the Euler–Mascheroni constant $\\gamma \\approx 0.5772$.\nThe $t_i$ are independent, so $\\operatorname{Var}(T) = \\sum_i (1 - p_i)/p_i^2 \u003c \\pi^2 n^2 / 6$.\n\n## Explanations\n\n### L1 human\n\nTo collect all $n$ pictures of a series, you need about $n \\ln n$ purchases, not $n$.\nThe last pictures are the slow ones: when only one type is missing, you wait on average $n$ draws for it. For $n = 50$, $E[T] = 50 H_{50} \\approx 225$.\n",
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    "aliases": [
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    "name": "Coupon collector's problem",
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      {
        "locator": "Calculating the expectation",
        "url": "https://en.wikipedia.org/wiki/Coupon_collector's_problem"
      },
      {
        "locator": "Calculating the variance",
        "url": "https://en.wikipedia.org/wiki/Coupon_collector's_problem"
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  "msc": [
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  "name": "Coupon collector's problem",
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    "tag": "0MXCP7MXWJ",
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      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
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