Coupon collector's problem: Difference between revisions
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==External links== |
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* "[http://demonstrations.wolfram.com/CouponCollectorProblem/ Coupon Collector Problem]" by [[Ed Pegg, Jr.]], the [[Wolfram Demonstrations Project]]. Mathematica package. |
* "[http://demonstrations.wolfram.com/CouponCollectorProblem/ Coupon Collector Problem]" by [[Ed Pegg, Jr.]], the [[Wolfram Demonstrations Project]]. Mathematica package. |
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* [http://www.mathematik.uni-stuttgart.de/~riedelmo/papers/coupon-stirling.pdf Coupon collector problem and varieties of Stirling numbers]—Computational examples of computing coupon collector statistics with Stirling numbers and related |
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* ''[http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/coupon.html How Many Singles, Doubles, Triples, Etc., Should The Coupon Collector Expect?]'', a short note by [[Doron Zeilberger]]. |
* ''[http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/coupon.html How Many Singles, Doubles, Triples, Etc., Should The Coupon Collector Expect?]'', a short note by [[Doron Zeilberger]]. |
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Revision as of 02:59, 2 May 2017
In probability theory, the coupon collector's problem describes the "collect all coupons and win" contests. It asks the following question: Suppose that there is an urn of n different coupons, from which coupons are being collected, equally likely, with replacement. What is the probability that more than t sample trials are needed to collect all n coupons? An alternative statement is: Given n coupons, how many coupons do you expect you need to draw with replacement before having drawn each coupon at least once? The mathematical analysis of the problem reveals that the expected number of trials needed grows as .[1] For example, when n = 50 it takes about 225[2] trials to collect all 50 coupons.
Solution
Calculating the expectation
Let T be the time to collect all n coupons, and let ti be the time to collect the i-th coupon after i − 1 coupons have been collected. Think of T and ti as random variables. Observe that the probability of collecting a new coupon is pi = (n − (i − 1))/n. Therefore, ti has geometric distribution with expectation 1/pi. By the linearity of expectations we have:
Here Hn is the n-th harmonic number. Using the asymptotics of the harmonic numbers, we obtain:
where is the Euler–Mascheroni constant.
Now one can use the Markov inequality to bound the desired probability:
Calculating the variance
Using the independence of random variables ti, we obtain:
The is a value of the Riemann zeta function (see Basel problem).
Now one can use the Chebyshev inequality to bound the desired probability:
Tail estimates
A different upper bound can be derived from the following observation. Let denote the event that the -th coupon was not picked in the first trials. Then:
Thus, for , we have .
Extensions and generalizations
- Paul Erdős and Alfréd Rényi proved the limit theorem for the distribution of T. This result is a further extension of previous bounds.
- Donald J. Newman and Lawrence Shepp found a generalization of the coupon collector's problem when m copies of each coupon need to be collected. Let Tm be the first time m copies of each coupon are collected. They showed that the expectation in this case satisfies:
- Here m is fixed. When m = 1 we get the earlier formula for the expectation.
- Common generalization, also due to Erdős and Rényi:
- In the general case of a nonuniform probability distribution, according to Philippe Flajolet,[3]
See also
Notes
- ^ Here and throughout this article, "log" refers to the natural logarithm rather than a logarithm to some other base. The use of Θ here invokes big O notation.
- ^ E(50) = 50(1 + 1/2 + 1/3 + ... + 1/50) = 224.9603, the expected number of trials to collect all 50 coupons. The approximation for this expected number gives in this case .
- ^ Flajolet, Philippe; Gardy, Danièle; Thimonier, Loÿs (1992), "Birthday paradox, coupon collectors, caching algorithms and self-organizing search", Discrete Applied Mathematics, 39 (3): 207–229
References
- Blom, Gunnar; Holst, Lars; Sandell, Dennis (1994), "7.5 Coupon collecting I, 7.6 Coupon collecting II, and 15.4 Coupon collecting III", Problems and Snapshots from the World of Probability, New York: Springer-Verlag, pp. 85–87, 191, ISBN 0-387-94161-4, MR 1265713.
- Dawkins, Brian (1991), "Siobhan's problem: the coupon collector revisited", The American Statistician, 45 (1): 76–82, doi:10.2307/2685247, JSTOR 2685247.
- Erdős, Paul; Rényi, Alfréd (1961), "On a classical problem of probability theory" (PDF), Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei, 6: 215–220, MR 0150807.
- Newman, Donald J.; Shepp, Lawrence (1960), "The double dixie cup problem", American Mathematical Monthly, 67: 58–61, doi:10.2307/2308930, MR 0120672
- Flajolet, Philippe; Gardy, Danièle; Thimonier, Loÿs (1992), "Birthday paradox, coupon collectors, caching algorithms and self-organizing search", Discrete Applied Mathematics, 39 (3): 207–229, doi:10.1016/0166-218X(92)90177-C, MR 1189469.
- Isaac, Richard (1995), "8.4 The coupon collector's problem solved", The Pleasures of Probability, Undergraduate Texts in Mathematics, New York: Springer-Verlag, pp. 80–82, ISBN 0-387-94415-X, MR 1329545.
- Motwani, Rajeev; Raghavan, Prabhakar (1995), "3.6. The Coupon Collector's Problem", Randomized algorithms, Cambridge: Cambridge University Press, pp. 57–63, MR 1344451.
External links
- "Coupon Collector Problem" by Ed Pegg, Jr., the Wolfram Demonstrations Project. Mathematica package.
- How Many Singles, Doubles, Triples, Etc., Should The Coupon Collector Expect?, a short note by Doron Zeilberger.