Novel formulas of moments of negative binomial distribution connected with Apostol-Bernoulli numbers of higher order and Stirling numbers. (English) Zbl 07885770
Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 118, No. 4, Paper No. 142, 12 p. (2024).
Summary: The main subject of this article is to present and reveal some new relationships between the moment generating functions of the Negative Binomial distribution and the generating functions for the Apostol-Bernoulli numbers and polynomials. By the help of these relations and Binomial series, we derive many computation formulas. These formulas give relations among moments, factorial moments, and the Apostol-Bernoulli numbers and polynomials, the Stirling numbers, and also other special functions related to zeta functions. By using these formulas, we give some numerical values of moments, expected value, and variance. Finally, we give some observations on formulas for the moments involving binomial series and zeta functions.
MSC:
| 60E05 | Probability distributions: general theory |
| 05A15 | Exact enumeration problems, generating functions |
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
Keywords:
probability distribution; negative binomial distribution; moment generating function; Apostol Bernoulli numbers and polynomials of higher order; Stirling numbers; binomial coefficientsReferences:
| [1] | Apostol, TM, On the Lerch zeta function, Pac. J. Appl., 1, 161-167, 1951 · Zbl 0043.07103 |
| [2] | Bertsekas, DP; Tsitsiklis, SN, Introduction to Probability, 2008, Belmont Massachusetles, Second edition: Athena Scientific, Belmont Massachusetles, Second edition |
| [3] | Bury, K.: Statistical Distributions in Engineering. Cambridge University Press (1999) |
| [4] | Chattamvelli, R., Shanmugam, R.: Discrete distributions in engineering and the applied sciences, Morgan & Claypool Publishers series (2020) · Zbl 1450.62001 |
| [5] | Charalambides, CA, Combinatorial Methods in Discrete Distributions, 2015, Publication: A John Wiley and Sons Inc., Publication |
| [6] | Charalambides, CA, Enumerative Combinatorics, 2002, London, New York: Chapman and Hall/Crc Press Company, London, New York · Zbl 1001.05001 |
| [7] | Evans, M., Hastings, N., Peacock, B.: Statistical Distributions, Wiley-Interscience, (2000) · Zbl 0960.62001 |
| [8] | Degroot, M.H., Schervish, M.J.: Probability and Statistics. 4. addition, New York, London, Tokyo: Addison-Wesley, (2012) |
| [9] | Luo, QM; Srivastava, HM, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308, 290-302, 2005 · Zbl 1076.33006 · doi:10.1016/j.jmaa.2005.01.020 |
| [10] | Luo, QM; Srivastava, HM, Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput., 217, 12, 5702-5728, 2011 · Zbl 1218.11026 · doi:10.1016/j.amc.2010.12.048 |
| [11] | Oldford, R.W., Shuldiner, P.: Bernoulli Sums: The only random variables that count. doi:10.48550/arXiv.2110.02363 |
| [12] | Simsek, B., New moment formulas for moments and characteristic function of the geometric distribution in terms of Apostol?Bernoulli polynomials and numbers, Math. Meth. Appl. Sci., 47, 9169-9179, 2024 · Zbl 07872023 · doi:10.1002/mma.10066 |
| [13] | Simsek, B.: Certain mathematical formulas for moments of Geometric distribution by means of the Apostol-Bernoulli polynomials, preprint (2023) |
| [14] | Simsek, B., Formulas derived from moment generating functions and Bernstein polynomials, Appl. Anal. Discrete Math., 13, 839-848, 2019 · Zbl 1499.11154 · doi:10.2298/AADM191227036S |
| [15] | Simsek, B.; Simsek, B., The computation of expected values and moments of special polynomials via characteristic and generating functions, AIP Conf. Proceed., 1863, 300012-1-300012-5, 2017 · doi:10.1063/1.4992461 |
| [16] | Srivastava, HM; Choi, J., Zeta and \(q\)-zeta functions and associated series and integrals, 2012, Amsterdam, London and New York: Elsevier Science Publishers, Amsterdam, London and New York · Zbl 1239.33002 |
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