Applications of Fuss-Catalan Numbers to Success Runs of Bernoulli Trials

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 Fu...

Full description

Saved in:
Bibliographic Details
Main Authors: S. J. Dilworth, S. R. Mane
Format: Article
Language:English
Published: Wiley 2016-01-01
Series:Journal of Probability and Statistics
Online Access:http://dx.doi.org/10.1155/2016/2071582
Tags: Add Tag
No Tags, Be the first to tag this record!
_version_ 1832561031508918272
author S. J. Dilworth
S. R. Mane
author_facet S. J. Dilworth
S. R. Mane
author_sort S. J. Dilworth
collection DOAJ
description 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≥1 occurrences of success runs of length k with l-overlapping. The case l<0, where there must be a gap of at least l trials between success runs, is also studied. Next we treat the distribution of the waiting time for the rth nonoverlapping appearance of a pair of successes separated by at most k-2 failures (k≥2).
format Article
id doaj-art-91f801b76c4d4010a5258b299a9a052f
institution Kabale University
issn 1687-952X
1687-9538
language English
publishDate 2016-01-01
publisher Wiley
record_format Article
series Journal of Probability and Statistics
spelling doaj-art-91f801b76c4d4010a5258b299a9a052f2025-02-03T01:26:08ZengWileyJournal of Probability and Statistics1687-952X1687-95382016-01-01201610.1155/2016/20715822071582Applications of Fuss-Catalan Numbers to Success Runs of Bernoulli TrialsS. J. Dilworth0S. R. Mane1Department of Mathematics, University of South Carolina, Columbia, SC 29208, USAConvergent Computing Inc., P.O. Box 561, Shoreham, NY 11786, USAIn 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≥1 occurrences of success runs of length k with l-overlapping. The case l<0, where there must be a gap of at least l trials between success runs, is also studied. Next we treat the distribution of the waiting time for the rth nonoverlapping appearance of a pair of successes separated by at most k-2 failures (k≥2).http://dx.doi.org/10.1155/2016/2071582
spellingShingle S. J. Dilworth
S. R. Mane
Applications of Fuss-Catalan Numbers to Success Runs of Bernoulli Trials
Journal of Probability and Statistics
title Applications of Fuss-Catalan Numbers to Success Runs of Bernoulli Trials
title_full Applications of Fuss-Catalan Numbers to Success Runs of Bernoulli Trials
title_fullStr Applications of Fuss-Catalan Numbers to Success Runs of Bernoulli Trials
title_full_unstemmed Applications of Fuss-Catalan Numbers to Success Runs of Bernoulli Trials
title_short Applications of Fuss-Catalan Numbers to Success Runs of Bernoulli Trials
title_sort applications of fuss catalan numbers to success runs of bernoulli trials
url http://dx.doi.org/10.1155/2016/2071582
work_keys_str_mv AT sjdilworth applicationsoffusscatalannumberstosuccessrunsofbernoullitrials
AT srmane applicationsoffusscatalannumberstosuccessrunsofbernoullitrials