Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Distribution of Runs and Patterns in Four State Trials

  • Jungtaek Oh (Department of Biomedical Science, School of Medicine, Kyungpook National University, Clinical Omics Center, School of Medicine, Kyungpook National University, The Institute of Industrial Technology, Changwon National University)
  • Received : 2024.01.17
  • Accepted : 2024.03.11
  • Published : 2024.06.30
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Abstract

From the mathematical and statistical point of view, a segment of a DNA strand can be viewed as a sequence of four-state (A, C, G, T) trials. Herein, we consider the distributions of runs and patterns related to the run lengths of multi-state sequences, especially for four states (A, B, C, D). Let X1, X2, . . . be a sequence of four state independent and identically distributed trials taking values in the set 𝒢 = {A, B, C, D}. In this study, we obtain exact formulas for the probability distribution function for the discrete distribution of runs of B's of order k. We obtain longest run statistics, shortest run statistics, and determine the distributions of waiting times and run lengths.

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Acknowledgement

The author would like to thank Dr. habil. Tommy Rene Jensen whose comments led to significant improvements in this manuscript. This research was partially supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2019R1A6A3A01090663, NRF-2020R1I1A1A01057479) and partially supported under the framework of international cooperation program managed by the National Research Foundation of Korea(NRF-2022K2A9A1A01098016).

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