Type II success runs of Bernoulli trials separated by a gap
S. J. Dilworth, S. R. Mane
TL;DR
The paper extends the Type II binomial distribution of order k by introducing a gap g between success runs, deriving exact pmf expressions for the number of runs via double generating functions and inclusion-exclusion, and analyzing the distribution of the longest run with both single- and double-derivation approaches. It provides concise reformulations of factorial moments and connects these results to NBII(k,g,r), including recurrence and generating-function tools for P(L ≥ t). The work generalizes classic g=1 results (Muselli, de Moivre, Kopocinsky) to arbitrary gaps, yielding exact distributions, moments, and asymptotics that are valuable for reliability, quality control, and risk assessments where delays between success sequences occur. Overall, it delivers comprehensive, exact methods for analyzing runs with gaps, with practical formulas and clear connections to existing success-run theory.
Abstract
We treat success runs of independent identically distributed Bernoulli trials (with success parameter $p$) distributed according to the Type II binomial distribution of order $k$. However, the success runs are separated by a gap $g\ge1$ (a failure followed by $g-1$ arbitrary outcomes). Most of the literature treats the case $g=1$ only. Our main results are expressions for the probability mass function (we present two derivations) and the distribution of the longest success run. We also present more concise expressions for previously published results for the factorial moments. We present results for the mean, variance, probability mass function and factorial moments for $\textrm{NB}_{\rm II}(k,g,r)$, the Type II negative binomial distribution of order $k$, where the number of success runs $r$ is fixed and the number of trials $n$ is variable. Let $L$ denote the length of the longest success run. We present a recurrence and generating function for the distribution of $L$ and derive expressions for the mean, variance and factorial moments of $L$.