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Compound Geometric Distribution of Order k

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  • Markos V. Koutras

    (University of Piraeus)

  • Serkan Eryilmaz

    (Atilim University)

Abstract

The distribution of the number of trials until the first k consecutive successes in a sequence of Bernoulli trials with success probability p is known as geometric distribution of order k. Let T k be a random variable that follows a geometric distribution of order k, and Y 1,Y 2,… a sequence of independent and identically distributed discrete random variables which are independent of T k . In the present article we develop some results on the distribution of the compound random variable S k = ∑ t = 1 T k Y t $S_{k} =\sum_{t=1}^{T_{k}}Y_{t}$ .

Suggested Citation

  • Markos V. Koutras & Serkan Eryilmaz, 2017. "Compound Geometric Distribution of Order k," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 377-393, June.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:2:d:10.1007_s11009-016-9482-y
    DOI: 10.1007/s11009-016-9482-y
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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. Stavros Papastavridis & Markos Koutras, 1992. "Consecutive k-out-of-n systems with maintenance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(4), pages 605-612, December.
    3. M. Koutras, 1997. "Waiting Time Distributions Associated with Runs of Fixed Length in Two-State Markov Chains," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(1), pages 123-139, March.
    4. Philippou, Andreas N. & Makri, Frosso S., 1986. "Successes, runs and longest runs," Statistics & Probability Letters, Elsevier, vol. 4(2), pages 101-105, March.
    5. Philippou, Andreas N. & Makri, Frosso S., 1986. "Successes, runs and longest runs," Statistics & Probability Letters, Elsevier, vol. 4(4), pages 211-215, June.
    6. Sigeo Aki & Katuomi Hirano, 1995. "Joint distributions of numbers of success-runs and failures until the first consecutivek successes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(2), pages 225-235, June.
    7. Koutras, M. V. & Alexandrou, V. A., 1997. "Non-parametric randomness tests based on success runs of fixed length," Statistics & Probability Letters, Elsevier, vol. 32(4), pages 393-404, April.
    8. Eisele, Karl-Theodor, 2006. "Recursions for compound phase distributions," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 149-156, February.
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    Cited by:

    1. Coskun Kus & Altan Tuncel & Serkan Eryilmaz, 2022. "Assessment of Shock Models for a Particular Class of Intershock Time Distributions," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 213-231, March.
    2. Eryilmaz, Serkan, 2021. "Revisiting discrete time age replacement policy for phase-type lifetime distributions," European Journal of Operational Research, Elsevier, vol. 295(2), pages 699-704.
    3. Xian Zhao & Yanbo Song & Xiaoyue Wang & Zhiyue Lv, 2022. "Distributions of $$({k}_{1},{k}_{2},\dots ,{k}_{m})$$ ( k 1 , k 2 , ⋯ , k m ) -runs with Multi-state Trials," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2689-2702, December.
    4. Vasileios M. Koutras & Markos V. Koutras, 2020. "Exact Distribution of Random Order Statistics and Applications in Risk Management," Methodology and Computing in Applied Probability, Springer, vol. 22(4), pages 1539-1558, December.
    5. Vasileios M. Koutras & Markos V. Koutras & Spiros D. Dafnis, 2022. "A Family of Induced Distributions," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1833-1848, September.

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