Theory of Probability and Mathematical Statistics
Approximations for success run probabilities in Bernoulli trials
Stephen Kaczkowski
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Abstract: Concise and convenient bounds are obtained for the probability mass and cumulative distribution functions associated with the first success run of length k in a sequence of n Bernoulli trials. Results are compared to an approximation obtained by the Stein–Chen method as well as to bounds obtained from statistical reliability theory. These approximation formulas are used to obtain precise estimates of the expectation value associated with the occurrence of at least one success run of length k within N concurrent sequences of Bernoulli trials.
Keywords: Bernoulli trials, success runs, asymptotic approximations
Bibliography: Richard Arratia, Larry Goldstein, and Louis Gordon, Poisson approximation and the Chen-Stein method, Statist. Sci. 5 (1990), no. 4, 403–434. With comments and a rejoinder by the authors. MR 1092983
N. Balakrishnan and Markos V. Koutras, Runs and scans with applications, Wiley Series in Probability and Statistics, Wiley-Interscience [John Wiley & Sons], New York, 2002. MR 1882476
Ayman Baklizi, Approximating the tail probabilities of the longest run in a sequence of Bernoulli trials, J. Stat. Comput. Simul. 88 (2018), no. 14, 2751–2760. MR 3827406, DOI 10.1080/00949655.2018.1486841
A. D. Barbour, Ourania Chryssaphinou, and Małgorzata Roos, Compound Poisson approximation in systems reliability, Naval Res. Logist. 43 (1996), no. 2, 251–264. MR 1375386, DOI 10.1002/(SICI)1520-6750(199603)43:2<251::AID-NAV6>3.0.CO;2-9
É. Borel. Mécanique statistique et irréversibilité, Journal de Physique, 5e série 3 (1913), 189–196.
M. T. Chao, J. C. Fu and M. V. Koutras, Survey of reliability studies of consecutive-k-out-of-n:F and related systems, IEEE Trans. Reliab. 44(1) (1995), 120–127.
Ourania Chryssaphinou and Stavros G. Papastavridis, Limit distribution for a consecutive-k-out-of-n:F system, Adv. in Appl. Probab. 22 (1990), no. 2, 491–493. MR 1061213, DOI 10.2307/1427550
Paul Deheuvels, Luc Devroye, and James Lynch, Exact convergence rate in the limit theorems of Erdős-Rényi and Shepp, Ann. Probab. 14 (1986), no. 1, 209–223. MR 815966
Paul Erdős and Alfréd Rényi, On a new law of large numbers, J. Analyse Math. 23 (1970), 103–111. MR 272026, DOI 10.1007/BF02795493
William Feller, An introduction to probability theory and its applications. Vol. I, 3rd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0228020
J. C. Fu, Reliability of a large consecutive-k-out-of-n:F system, IEEE Trans. Reliab. vol R-34 (1985), 127–130.
Anant P. Godbole, Specific formulae for some success run distributions, Statist. Probab. Lett. 10 (1990), no. 2, 119–124. MR 1072498, DOI 10.1016/0167-7152(90)90006-S
Stephen Kaczkowski, Concurrent sequences of Bernoulli trials, Math. Gaz. 104 (2020), no. 561, 435–448. MR 4163925, DOI 10.1017/mag.2020.98
Marco Muselli, Simple expressions for success run distributions in Bernoulli trials, Statist. Probab. Lett. 31 (1996), no. 2, 121–128. MR 1421563, DOI 10.1016/S0167-7152(96)00022-3
Marco Muselli, New improved bounds for reliability of consecutive-k-out-of-n:F systems, J. Appl. Probab. 37 (2000), no. 4, 1164–1170. MR 1808882, DOI 10.1239/jap/1014843097
Andreas N. Philippou and Frosso S. Makri, Successes, runs and longest runs, Statist. Probab. Lett. 4 (1986), no. 2, 101–105. MR 829441, DOI 10.1016/0167-7152(86)90025-8
A. D. Solov′ev, A combinatorial identity and its application to the problem on the first occurrence of a rare event, Teor. Verojatnost. i Primenen. 11 (1966), 313–320 (Russian, with English summary). MR 0202217
