On the Number of Trials Needed to Obtain k Consecutive Successes
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Date
2021-04-30
Authors
Drekic, Steve
Spivey, Michael Z.
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Abstract
A sequence of independent Bernoulli trials, each of which is a success with probability p, is conducted. For k ∈ Z+, let Xk be the number of trials required to obtain k consecutive successes. Using techniques from elementary probability theory, we present a derivation which ultimately yields an elegant expression for the probability mass function of Xk, and is simpler in comparison to what is found in the literature. Following this, we use our derived formula to obtain explicit closed-form expressions for the complementary cumulative distribution function and the nth factorial moment of Xk.
Description
The final publication is available at Elsevier via http://dx.doi.org/https://doi.org/10.1016/j.spl.2021.109132.© (2021). This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Keywords
Bernoulli trials, Consecutive successes, Factorial moments, Generating function, Polynomial coefficients, Bell polynomials