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On waiting time distribution of runs of ones or zeros in a Bernoulli sequence

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  • Kim, Sungsu
  • Park, Chongjin
  • Oh, Jungtaek

Abstract

Consider an infinite sequence of Bernoulli trials {Xi|i=1,2,…}. Let W(k) denote the waiting time, the number of trials needed, to get either consecutive k ones or k zeros for the first time. The probability distribution of W(k) is derived for both independent and homogeneous two-state Markovian Bernoulli trials, using a generalized Fibonacci sequence of order k. For independent Bernoulli trials, a special case of symmetric trial with p=12 is considered.

Suggested Citation

  • Kim, Sungsu & Park, Chongjin & Oh, Jungtaek, 2013. "On waiting time distribution of runs of ones or zeros in a Bernoulli sequence," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 339-344.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:1:p:339-344
    DOI: 10.1016/j.spl.2012.10.001
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    References listed on IDEAS

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    1. M. Koutras, 1996. "On a waiting time distribution in a sequence of Bernoulli trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(4), pages 789-806, December.
    2. Qing Han & Sigeo Aki, 2000. "Sooner and Later Waiting Time Problems Based on a Dependent Sequence," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(3), pages 407-414, September.
    3. S. Aki & N. Balakrishnan & S. Mohanty, 1996. "Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(4), pages 773-787, December.
    4. Masayuki Uchida & Sigeo Aki, 1995. "Sooner and later waiting time problems in a two-state Markov chain," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(3), pages 415-433, September.
    5. Ebneshahrashoob, M. & Sobel, Milton, 1990. "Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas," Statistics & Probability Letters, Elsevier, vol. 9(1), pages 5-11, January.
    6. Balasubramanian, K. & Viveros, R. & Balakrishnan, N., 1993. "Sooner and later waiting time problems for Markovian Bernoulli trials," Statistics & Probability Letters, Elsevier, vol. 18(2), pages 153-161, September.
    7. Sigeo Aki & Katuomi Hirano, 2007. "On the Waiting Time for the First Success Run," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(3), pages 597-602, September.
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    Cited by:

    1. 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.
    2. Kim, Bara & Kim, Jeongsim, 2019. "Sooner waiting time problems in a sequence of multi-state trials with random rewards," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 171-179.

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