Approximations for success run probabilities in Bernoulli trials

Kaczkowski, Stephen

doi:10.1090/tpms/1186

Author: Stephen Kaczkowski
Journal: Theor. Probability and Math. Statist. 108 (2023), 45-57
MSC (2020): Primary 60E05; Secondary 41A60
DOI: https://doi.org/10.1090/tpms/1186
Published electronically: May 2, 2023
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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.

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