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...
Saved in:
| Main Authors: | , |
|---|---|
| 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 |