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Binomial Distribution of Order k in a Modified Binary Sequence

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Abstract

Let us consider a sequence of n binary trials (signals). A counter registers successes, but once a success is registered the mechanism is locked for a number of trials following each registration. Under this framework the observed sequence of outcomes turns to a dependent sequence with non-identical success probabilities even if the original trials were independent and identically distributed. In the present paper, we study the distribution of the number of success runs registered by the counter after the completion of the n signals. Our study covers the general case where the original trials are independent but not necessarily identically distributed. The special case of identically distributed trials gives birth to the modified binomial distribution of order k, which generalizes binomial distributions extensively studied in the literature. In this case, we derive neat recursive relations for the probability mass function, the probability generating function and the moments. The applicability of the modified binomial distribution of order k in several research areas is highlighted and after developing theoretical results we discuss how they can be exploited to study a biomedical engineering problem.

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Acknowledgements

The authors wish to thank Dr. Aris Dermitzakis, researcher at Department of Medicine, University of Patras, for his valuable help regarding the proposed application of the current work in Biomedical Engineering. The authors wish to thank the referees for their comments and suggestions which helped to improve the article.

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Authors and Affiliations

  1. Department of Crop Science, Agricultural University of Athens, Iera Odos 75, 11855, Athens, Greece

    Spiros D. Dafnis

  2. Department of Statistics and Insurance Science, University of Piraeus, Karaoli and Dimitriou 80, 18534, Piraeus, Greece

    Markos V. Koutras

  3. Department of Mathematics, University of Patras, Rio Campus, 26504, Patras, Greece

    Frosso S. Makri

Authors
  1. Spiros D. Dafnis
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  2. Markos V. Koutras
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  3. Frosso S. Makri
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Correspondence to Markos V. Koutras.

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Spiros D. Dafnis and Frosso S. Makri have contributed equally to this work.

This article is part of the topical collection “Advances in Probability and Statistics: an Issue in Memory of Theophilos Cacoullos” guest edited by Narayanaswamy Balakrishnan, Charalambos A. Charalambides, Tasos Christofides, Markos Koutras, and Simos Meintanis.

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Dafnis, S.D., Koutras, M.V. & Makri, F.S. Binomial Distribution of Order k in a Modified Binary Sequence. J Stat Theory Pract 16, 42 (2022). https://doi.org/10.1007/s42519-022-00267-7

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