We introduce the Type II binomial distribution of order k with gap g and derive its double pgf in closed form. Expressions for the moments, the pgf, and the pmf as nested binomial sums are also derived. For the case \(g=2\), closed from expressions for the mean and variance are obtained. For \(g>2\), we derive closed form asymptotic expressions for all the moments (in addition to the full expressions as nested binomial sums mentioned above). We also correct some published results for \(g=1\).
Type II binomial distribution Bernoulli trials Gaps between success runs
Mathematics Subject Classification
primary 62E15 secondary 60E05
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S. J. Dilworth was supported by the National Science Foundation under Grant Number DMS–1361461.
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