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The distributions of sum, minima and maxima of generalized geometric random variables

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  • Fatih Tank
  • Serkan Eryilmaz

Abstract

Geometric distribution of order $$k$$ k as one of the generalization of well known geometric distribution is the distribution of the number of trials until the first $$k$$ k consecutive successes in Bernoulli trials with success probability $$p$$ p . In this paper, it is shown that this generalized distribution can be represented as a discrete phase-type distribution. Using this representation along with closure properties of phase-type distributions, the distributions of sum, minima and maxima of two independent random variables having geometric distribution of order $$k$$ k are obtained. Numerical results are presented to illustrate the computational details. Copyright Springer-Verlag Berlin Heidelberg 2015

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  • Fatih Tank & Serkan Eryilmaz, 2015. "The distributions of sum, minima and maxima of generalized geometric random variables," Statistical Papers, Springer, vol. 56(4), pages 1191-1203, November.
    • Handle: RePEc:spr:stpapr:v:56:y:2015:i:4:p:1191-1203
    DOI: 10.1007/s00362-014-0632-4
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    References listed on IDEAS

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    1. Philippou, Andreas N. & Georghiou, Costas & Philippou, George N., 1983. "A generalized geometric distribution and some of its properties," Statistics & Probability Letters, Elsevier, vol. 1(4), pages 171-175, June.
    2. Muselli, Marco, 1996. "Simple expressions for success run distributions in bernoulli trials," Statistics & Probability Letters, Elsevier, vol. 31(2), pages 121-128, December.
    3. Frosso Makri & Zaharias Psillakis, 2013. "Exact distributions of constrained (k, ℓ) strings of failures between subsequent successes," Statistical Papers, Springer, vol. 54(3), pages 783-806, August.
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    Cited by:

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    3. Anna Dembińska & Krzysztof Jasiński, 2021. "Maximum likelihood estimators based on discrete component lifetimes of a k-out-of-n system," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(2), pages 407-428, June.
    4. repec:bpj:demode:v:6:y:2018:i:1:p:131-138:n:8 is not listed on IDEAS
    5. Krzysztof Jasiński, 2023. "On conditional residual lifetimes of coherent systems consisting of components with discrete lifetimes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(2), pages 205-218, February.
    6. Eryilmaz, Serkan, 2017. "On compound sums under dependence," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 228-234.
    7. Krzysztof Jasiński, 2021. "The number of failed components in a coherent working system when the lifetimes are discretely distributed," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(7), pages 1081-1094, October.
    8. Paula R. Bouzas & Nuria Ruiz-Fuentes & Carmen Montes-Gijón & Juan Eloy Ruiz-Castro, 2021. "Forecasting counting and time statistics of compound Cox processes: a focus on intensity phase type process, deletions and simultaneous events," Statistical Papers, Springer, vol. 62(1), pages 235-265, February.
    9. Eryilmaz, Serkan, 2016. "A new class of lifetime distributions," Statistics & Probability Letters, Elsevier, vol. 112(C), pages 63-71.

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