Applications of Fuss-Catalan numbers to success runs of Bernoulli trials
type: Item
instance of: scholarly article
title: Applications of Fuss-Catalan numbers to success runs of Bernoulli trials
zbMATH Open document ID: 1426.60008
DOI: 10.1155/2016/2071582
author: S. R. Mane
author: S. J. Dilworth
published in: Journal of Probability and Statistics
publication date: 2018-08-14T00:00:00Z
review text: Summary: In a recent paper, the authors derived the exact solution for the probability mass function of the geometric distribution of order \(k\), expressing the roots of the associated auxiliary equation in terms of generating functions for Fuss-Catalan numbers. This paper applies the above formalism for the Fuss-Catalan numbers to treat additional problems pertaining to occurrences of success runs. New exact analytical expressions for the probability mass function and probability generating function and so forth are derived. First, we treat sequences of Bernoulli trials with \(r \geq 1\) occurrences of success runs of length \(k\) with \(\ell\)-overlapping. The case \(\ell < 0\), where there must be a gap of at least \(\left|\ell\right|\) trials between success runs, is also studied. Next we treat the distribution of the waiting time for the \(r\)th nonoverlapping appearance of a pair of successes separated by at most \(k - 2\) failures (\(k \geq 2\)).
Mathematics Subject Classification ID: 60C05
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 62E15
zbMATH DE Number: 6917503
Wikidata QID: Q59129337
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2016/2071582
OpenAlex ID: W2263124322
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description: scientific article; zbMATH DE number 6917503