New easy to compute formulas for the moments of random variables appearing in the coupon collectorproblem

• Autorzy: León-García Amaryani , Pérez Aroldo , Bolívar-Cimé Addy

Identyfikatory

DOI

10.37190/0208-4147.00130

• Języki publikacji: EN

Abstrakty

EN

Assuming that there are N types of coupons, where the prob- ability that the ith coupon appears is pi ≥ 0 for i = 1, . . . , N , with [formula], we consider the variable T^k which represents the number of acquisitions needed to obtain k ≤ N different coupons, and the variable Yn which represents the number of different coupons obtained in n acquisitions. In the coupon collector problem it is of interest to obtain the expected value of these random variables, as well as their rth moments. We provide new expressions for the rth moments of T_kand Y_n, and we give expressions for their moment generating functions. Unlike known formulas, our formula for the rth moment of T_k is given in terms of recursive expressions and that of Y_n is given in terms of finite sums, so that they can be easily implemented computationally. Furthermore, our formulas allow obtaining simplified expressions of the first few moments of the variables.

Słowa kluczowe

EN

coupon collector problem; combinatorial probability; moments; moment generating function

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2023
• Tom: Vol. 43, Fasc. 2
• Strony: 155--164
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

León-García Amaryani

• División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco, CP. 86690, Cunduacán, Tabasco, Mexico

autor

Pérez Aroldo

• División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco, CP. 86690, Cunduacán, Tabasco, Mexico
• [e-mail]

autor

Bolívar-Cimé Addy

• División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco, CP. 86690, Cunduacán, Tabasco, Mexico

Bibliografia

• [1] S. Aki and K. Hirano, Coupon collector’s problems with statistical applications to rankings. Ann. Inst. Statist. Math. 65 (2013), 571-587.
• [2] E. Anceaume, Y. Busnel and B. Sericola, New results on a generalized coupon collector problem using Markov chains, J. Appl. Probab. 52 (2015), 405-418.
• [3] E. Anceaume, Y. Busnel, E. Schulte-Geers and B. Sericola, Optimization results for a generalized coupon collector problem, J. Appl. Probab. 53 (2016), 622-629.
• [4] A. Boneh and M. Hofri, The coupon-collector problem revisited-a survey of engineering problems and computational methods, Stoch. Models 13 (1997), 39-66.
• [5] K. N. Boyadzhiev, Close encounters with the Stirling numbers of the second kind, Math. Mag. 85 (2012), 252-266.
• [6] A. V. Doumas and V. G. Papanicolaou, The coupon collector’s problem revisited: generalizing the double Dixie cup problem of Newman and Shepp, ESAIM Probab. Statist. 20 (2016), 367-399.
• [7] M. Ferrante and N. Frigo, A note on the coupon-collector’s problem with multiple arrivals and random sampling, arXiv:1209.2667 (2012).
• [8] M. Ferrante and M. Saltalamacchia, The coupon collector’s problem, Materials Mathemàtics, Univ. Autònoma de Barcelona 2014, art. 2, 35 pp.
• [9] P. Flajolet, D. Gardy and L. Thimonier, Birthday paradox, coupon collectors, caching algorithms and self-organizing search, Discrete Appl. Math. 39 (1992), 207-229.
• [10] H. W. Gould, Euler’s formula for nth differences of powers, Amer. Math. Monthly 85 (1978), 450-467.
• [11] J. E. Kobza, S. H. Jacobson and D. E. Vaughan, A survey of the coupon collector’s problem with random sample sizes, Methodol. Comput. Appl. Probab. 9 (2007), 573-584.
• [12] S. N. Luko, The “coupon collector’s problem” and quality control, Quality Engrg. 21 (2009), 168-181.
• [13] D. J. Newman and L. Shepp, The double dixie cup problem, Amer. Math. Monthly 67 (1960), 58-61.
• [14] C. E. G. Pineda, S. M. García and L. E. O. Acevedo, La serie geométrica y su derivada, Scientia et Technica 17 (2011), 196-200.
• [15] H. von Schelling, Coupon collecting for unequal probabilities, Amer. Math. Monthly 61 (1954), 306-311.
• [16] N. Zoroa, E. Lesigne, M. J. Fernández-Sáenz, P. Zoroa and J. Casas, The coupon collector urn model with unequal probabilities in ecology and evolution, J. Roy. Soc. Interface 14 (2017), art. 20160643, 14 pp.