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Successes, runs and longest runs

Author

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  • Philippou, Andreas N.
  • Makri, Frosso S.

Abstract

The probability distribution of the numbeer of success runs of length k ( >/ 1) in n ( [greater-or-equal, slanted] 1) Bernoulli trials is obtained. It is noted that this distribution is a binomial distribution of order k, and several open problems pertaining to it are stated. Let Sn and Ln, respectively, denote the number of successes and the length of the longest success run in the n Bernoulli trials. A formula is derived for the probability P(Ln [less-than-or-equals, slant] k Sn = r) (0 [less-than-or-equals, slant] k [less-than-or-equals, slant] r [less-than-or-equals, slant] n), which is alternative to those given by Burr and Cane (1961) and Gibbons (1971). Finally, the probability distribution of Xn, Ln(k) is established, where Xn, Ln(k) denotes the number of times in the n Bernoulli trials that the length of the longest success run is equal to k.

Suggested Citation

  • Philippou, Andreas N. & Makri, Frosso S., 1986. "Successes, runs and longest runs," Statistics & Probability Letters, Elsevier, vol. 4(2), pages 101-105, March.
    • Handle: RePEc:eee:stapro:v:4:y:1986:i:2:p:101-105
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    Citations

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    Cited by:

    1. Yong Kong, 2017. "The mth longest runs of multivariate random sequences," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(3), pages 497-512, June.
    2. Spiros Dafnis & Andreas Philippou & Demetrios Antzoulakos, 2012. "Distributions of patterns of two successes separated by a string of k-2 failures," Statistical Papers, Springer, vol. 53(2), pages 323-344, May.
    3. Han, Qing & Aki, Sigeo, 1998. "Formulae and recursions for the joint distributions of success runs of several lengths in a two-state Markov chain," Statistics & Probability Letters, Elsevier, vol. 40(3), pages 203-214, October.
    4. Serkan Eryilmaz, 2005. "On the distribution and expectation of success runs in nonhomogeneous Markov dependent trials," Statistical Papers, Springer, vol. 46(1), pages 117-128, January.
    5. Muselli, Marco, 1996. "Simple expressions for success run distributions in bernoulli trials," Statistics & Probability Letters, Elsevier, vol. 31(2), pages 121-128, December.
    6. Dafnis, Spiros D. & Makri, Frosso S. & Philippou, Andreas N., 2019. "The reliability of a generalized consecutive system," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 186-193.
    7. Eryilmaz, Serkan, 2018. "On success runs in a sequence of dependent trials with a change point," Statistics & Probability Letters, Elsevier, vol. 132(C), pages 91-98.
    8. Sungsu Kim & Chong Jin Park, 2021. "An Asymptotic Conditional Test of Independence in Bernoulli Sequences Using the Number of Runs Given the Number of Successes," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 143-154, February.
    9. 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.
    10. Frosso Makri & Andreas Philippou, 2005. "On binomial and circular binomial distributions of orderk forl-overlapping success runs of lengthk," Statistical Papers, Springer, vol. 46(3), pages 411-432, July.
    11. Anant Godbole & Stavros Papastavridis & Robert Weishaar, 1997. "Formulae and Recursions for the Joint Distribution of Success Runs of Several Lengths," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(1), pages 141-153, March.
    12. Upadhye, N.S. & Kumar, A.N., 2018. "Pseudo-binomial approximation to (k1,k2)-runs," Statistics & Probability Letters, Elsevier, vol. 141(C), pages 19-30.
    13. S. Aki & K. Hirano, 1989. "Estimation of parameters in the discrete distributions of order k," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 41(1), pages 47-61, March.
    14. Spiros D. Dafnis & Frosso S. Makri & Markos V. Koutras, 2021. "Generalizations of Runs and Patterns Distributions for Sequences of Binary Trials," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 165-185, March.
    15. K. K. Kamalja, 2017. "Markov binomial distribution of order k and its applications," Statistical Papers, Springer, vol. 58(3), pages 831-853, September.
    16. Sigeo Aki & Katuomi Hirano, 2000. "Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(4), pages 767-777, December.
    17. M. Koutras & V. Alexandrou, 1995. "Runs, scans and URN model distributions: A unified Markov chain approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(4), pages 743-766, December.
    18. Demetrios Antzoulakos & Stathis Chadjiconstantinidis, 2001. "Distributions of Numbers of Success Runs of Fixed Length in Markov Dependent Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 599-619, September.

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