Abstract
Recently, the degenerate Poisson random variable with parameter \(\alpha > 0\), whose probability mass function is given by \(P_{\lambda}(i) = e_{\lambda}^{-1} (\alpha) \frac{\alpha^{i}}{i!} (1)_{i,\lambda}\) \((i = 0,1,2,\dots)\), was studied. In probability theory, the zero-truncated Poisson distributions are certain discrete probability distributions whose supports are the set of positive integers. These distributions are also known as the conditional Poisson distributions or the positive Poisson distributions. In this paper, we introduce the degenerate zero-truncated Poisson random variables whose probability mass functions are a natural extension of the zero-truncated Poisson distributions, and investigate various properties of those random variables.
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References
L. Carlitz,, “Degenerate Stirling, Bernoulli and Eulerian Numbers”, Util. Math., 15 (1979), 51тАУ-88.
L. Comtet, “Nombres de Stirling generaux et fonctions symetriques”, C. R. Acad. Sci. Paris. (Series A), 275 (1972), 747–750.
B. S. El-Desouky,, “A Generalization of Stirling, Lah and Harmonic Numbers with Some Computational Applications”, Ars Combin., 130 (2017), 333–356.
B. S. El-Desouky,, “The Multiparameter Non-Central Stirling Numbers”, Fibonacci Quart., 32 (1994), 218–225.
W. Feller,, An Introduction to Probability Theory and Its Applications, Vol. II., 2nd ed. John Wiley & Sons, Inc., New York-London-Sydney,, 1971.
Y. He, J. Pan, “Some Recursion Formulas for the Number of Derangements and Bell Numbers”, J. Math. Res. Appl., 36:1 (2016), 15–22.
F. T. Howard, “Bell Polynomials and Degenerate Stirling Numbers”, Rend. Sem. Mat. Univ. Padova., 61 (1979), 203–219.
T. Kim, “\(\lambda\)-Analogue of Stirling Numbers of the First Kind”, Adv. Stud. Contemp. Math. (Kyungshang), 27:3 (2017), 423–429.
T. Kim, “A Note on Degenerate Stirling Polynomials of the Second Kind”, Proc. Jangjeon Math. Soc., 20:3 (2017), 319–331.
T. Kim and D. S. Kim, “Extended Stirling Numbers of the First Kind Associated with Daehee Numbers and Polynomials”, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas: RACSAM, 113:2 (2019), 1159–1171.
T. Kim, D. S. Kim, L.-C. Jang, and H.-Y. Kim, “A Note on Discrete Degenerate Random Variables”, Proc. Jangjeon Math. Soc., 23:1 (2020), 125–135.
T. Kim, D. S. Kim, H. Lee, and J. Kwon, “A Note on Some Identities of New Type Degenerate Bell Polynomials”, Mathematics, 7:11 (2019).
T. Kim, Y. Yao, D. S. Kim, and H.-I. Kwon, “Some Identities Involving Special Numbers and Moment of Random Variables”, Rocky Mountain J. Math., 49:2 (2019), 521–538.
M. Koutras, “Non-Central Stirling Numbers and Some Applications”, Discrete Math., 42 (1982), 73–89.
A. Leon-Garcia, Probability and Random Processes for Electronical Engineering, 3rd ed., Addition-Wesley, Massachusetts, 2008.
S. Roman, The Umbral Calculus, Pure and Applied Mathematics 111, Academic Press. Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984.
S. M. Ross, Introduction to Probability Models, 4th ed., Elsevier/Academic Press, Amsterdam, 2014.
Y. Simsek, “Identities and Relations Related to Combinatorial Numbers and Polynomials”, Proc. Jangjeon Math. Soc., 20:1 (2017), 127–135.
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Kim, T., Kim, D.S. Degenerate Zero-Truncated Poisson Random Variables. Russ. J. Math. Phys. 28, 66–72 (2021). https://doi.org/10.1134/S1061920821010076
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DOI: https://doi.org/10.1134/S1061920821010076