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@matejker
Last active September 11, 2020 08:52
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Cupon collector's problem / The Double Dixie Cup Problem
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Cupon collector's problem\n",
"How many cupons do you need to purchase to complete a set of $n$ pictures$^1$? On average you need $E(n) = n \\big(1 + \\frac{1}{2} + \\ldots + \\frac{1}{n}\\big)$ *purchases*, which can be turn into: $$E(n) = n log(n) + \\gamma n + \\frac{1}{2} + O(1/n),$$\n",
"where $\\gamma \\approx 0.577216$ is **Euler–Mascheroni** constant [2].\n",
"\n",
"In order to complete $m$ full collections, we need to find $E_m(n,\\ k)$ given by Newman and Shepp formula [1]: \n",
"$$E_m(n,\\ k) = n\\int_{0}^{\\infty} 1 - \\Bigg( 1 - e^{-x}\\bigg(1\\ + \\frac{x}{1!}\\ +\\ \\ldots\\ + \\frac{x^{m-1}}{(m-1)!}\\bigg)\\Bigg)^kdx, $$\n",
"where $m$ is the number of complete sets of $n$ pictures collection. Parameter $k$ stays for number of *desired picures* in the set, if the whole set is *desired* then $k=n$. This 👆 formula can be simplify for $n \\rightarrow \\infty$ and fixed $m$:\n",
"$$E_m(n) = n\\ log(n) + (m-1)\\ n\\ log\\ log(n) + O(n)$$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import typing as t\n",
"import matplotlib.pyplot as plt\n",
"from scipy.integrate import quad\n",
"from scipy.special import factorial\n",
"import numpy as np\n",
"import warnings\n",
"warnings.filterwarnings('ignore')"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"def E_m(x: t.Union[np.array, float, int], n: int, m: int, k: t.Optional[int] = None) -> float:\n",
" k = k or n\n",
" assert k <= n, f\"Parameter k={k} cannot be k > n={n}\"\n",
" def exp_series() -> float:\n",
" powers = x**np.arange(m)\n",
" fact = factorial(np.arange(m))\n",
" return np.sum(powers / fact)\n",
" \n",
" return n * (1 - (1 - exp_series() / (np.exp(1) ** x)) ** k)\n",
"\n",
"def E_m_limit(n: int, m: int) -> float: \n",
" return n * log(n) + (m - 1) * n * np.log(np.log(n))"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"n = 25\n",
"m = np.arange(1, 70)\n",
"y = np.array([np.round(quad(E_m, 0, np.inf, args=(n, i)), 4)[0] for i in m])"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [
{
"data": {
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\n",
"text/plain": [
"<Figure size 1080x720 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(15, 10))\n",
"plt.title(f\"Expected number of purchases to collect m sets: $E_m(n, k)$, $n={n}$\", fontsize=15)\n",
"plt.ylabel(f\"$E_m(n, k)$\", fontsize=12)\n",
"plt.xlabel(f\"$m$\", fontsize=12)\n",
"plt.scatter(m, y, linewidths=0.1, alpha=0.8, marker='X', label=r'$E_m(n)$')\n",
"plt.plot(m, y, alpha=0.1)\n",
"plt.plot(m, m * n, 'r--', alpha=0.3, label=r'$m*n$')\n",
"plt.legend()\n",
"plt.show()"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 1080x720 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(15, 10))\n",
"plt.title(f\"Expected number of purchases for m$-th$ set: $E_m(n, k) / m$, $n={n}$\", fontsize=15)\n",
"plt.ylabel(f\"$E_m(n, k) / m$\", fontsize=12)\n",
"plt.xlabel(f\"$m$\", fontsize=12)\n",
"plt.scatter(m, y / m, linewidths=0.1, alpha=0.8, marker='X', label=r'$E_m(n)/m$')\n",
"plt.plot(m, y / m, alpha=0.1)\n",
"plt.plot([1, 70], [n, n], 'r--', alpha=0.3, label=f'$n={n}$')\n",
"plt.legend()\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$E_m(n)$ for $m = 1,2, \\ldots, 69$"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 95.399 , 142.3677, 183.6583, 222.2086, 259.0621, 294.7315,\n",
" 329.5142, 363.6003, 397.1206, 430.1693, 462.8171, 495.1185,\n",
" 527.1169, 558.8471, 590.3377, 621.6128, 652.6924, 683.5938,\n",
" 714.3318, 744.9193, 775.3675, 805.6863, 835.8847, 865.9703,\n",
" 895.9503, 925.8309, 955.6179, 985.3165, 1014.9314, 1044.467 ,\n",
" 1073.9271, 1103.3154, 1132.6354, 1161.89 , 1191.0822, 1220.2146,\n",
" 1249.2898, 1278.31 , 1307.2774, 1336.194 , 1365.0618, 1393.8825,\n",
" 1422.6578, 1451.3893, 1480.0784, 1508.7267, 1537.3353, 1565.9057,\n",
" 1594.4389, 1622.9361, 1651.3984, 1679.8268, 1708.2223, 1736.5858,\n",
" 1764.9182, 1793.2203, 1821.493 , 1849.7371, 1877.9532, 1906.1421,\n",
" 1934.3045, 1962.441 , 1990.5523, 2018.6388, 2046.7013, 2074.7403,\n",
" 2102.7563, 2130.7498, 2158.7213])"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"y"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$E_m(n) / m$ for $m = 1,2, \\ldots, 69$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([95.399 , 71.1839, 61.2194, 55.5522, 51.8124, 49.1219, 47.0735,\n",
" 45.45 , 44.1245, 43.0169, 42.0743, 41.2599, 40.5475, 39.9176,\n",
" 39.3558, 38.8508, 38.3937, 37.9774, 37.5964, 37.246 , 36.9223,\n",
" 36.6221, 36.3428, 36.0821, 35.838 , 35.6089, 35.3933, 35.1899,\n",
" 34.9976, 34.8156, 34.6428, 34.4786, 34.3223, 34.1732, 34.0309,\n",
" 33.8948, 33.7646, 33.6397, 33.5199, 33.4048, 33.2942, 33.1877,\n",
" 33.0851, 32.9861, 32.8906, 32.7984, 32.7093, 32.623 , 32.5396,\n",
" 32.4587, 32.3804, 32.3044, 32.2306, 32.159 , 32.0894, 32.0218,\n",
" 31.956 , 31.892 , 31.8297, 31.769 , 31.7099, 31.6523, 31.5961,\n",
" 31.5412, 31.4877, 31.4355, 31.3844, 31.3346, 31.2858])"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.round(y / m, 4)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"> For large $m$, by the law of large numbers, the number required is asymptotic to $mn$ [1]\n",
"\n",
"In theory, the limit $\\lim_{m\\to\\infty} E_m(n)$ *should* go to $mn$ as there infinitely many pictures of each kind and to compleate a new set, you just need to buy *another* $n$ cupons."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## References\n",
"[1] Donald J. Newman and Shepp L. (1960), *The Double Dixie Cup Problem*, The American Mathematical Monthly, Vol. 67, No. 1 (Jan., 1960), pp. 58-61, https://faculty.wharton.upenn.edu/wp-content/uploads/2012/04/Double-dixie-cup-problem.pdf \n",
"[2] Ferrante M. and Saltalamacchia M. (2014) *The Coupon Collector’s Problem*, MATerials MATemàticsVolum 2014, treball no. 2, 35 pp. ISSN: 1887-1097, http://mat.uab.cat/matmat_antiga/PDFv2014/v2014n02.pdf"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$^1$ Pictures, Dixie cups, figures or any other shining things 😀"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.9"
}
},
"nbformat": 4,
"nbformat_minor": 4
}
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