On the number of trials needed to obtain k consecutive successes

Steve Drekic and Michael Z. Spivey

Statistics & Probability Letters, 2021, vol. 176, issue C

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.

Keywords: Bernoulli trials; Consecutive successes; Factorial moments; Generating function; Polynomial coefficients; Bell polynomials (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1016/j.spl.2021.109132

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