Success Run Waiting Times and Fuss-Catalan Numbers
We present power series expressions for all the roots of the auxiliary equation of the recurrence relation for the distribution of the waiting time for the first run of k consecutive successes in a sequence of independent Bernoulli trials, that is, the geometric distribution of order k. We show that...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | Journal of Probability and Statistics |
| Online Access: | http://dx.doi.org/10.1155/2015/482462 |
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| _version_ | 1832558134056452096 |
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| author | S. J. Dilworth S. R. Mane |
| author_facet | S. J. Dilworth S. R. Mane |
| author_sort | S. J. Dilworth |
| collection | DOAJ |
| description | We present power series expressions for all the roots of the auxiliary equation of the recurrence relation for the distribution of the waiting time for the first run of k consecutive successes in a sequence of independent Bernoulli trials, that is, the geometric distribution of order k. We show that the series coefficients are Fuss-Catalan numbers and write the roots in terms of the generating function of the Fuss-Catalan numbers. Our main result is a new exact expression for the distribution, which is more concise than previously published formulas. Our work extends the analysis by Feller, who gave asymptotic results. We obtain quantitative improvements of the error estimates obtained by Feller. |
| format | Article |
| id | doaj-art-d1f21f718cdf4707b180352e62e5d2e3 |
| institution | Kabale University |
| issn | 1687-952X 1687-9538 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Probability and Statistics |
| spelling | doaj-art-d1f21f718cdf4707b180352e62e5d2e32025-02-03T01:33:09ZengWileyJournal of Probability and Statistics1687-952X1687-95382015-01-01201510.1155/2015/482462482462Success Run Waiting Times and Fuss-Catalan NumbersS. J. Dilworth0S. R. Mane1Department of Mathematics, University of South Carolina, Columbia, SC 29208, USAConvergent Computing Inc., P.O. Box 561, Shoreham, NY 11786, USAWe present power series expressions for all the roots of the auxiliary equation of the recurrence relation for the distribution of the waiting time for the first run of k consecutive successes in a sequence of independent Bernoulli trials, that is, the geometric distribution of order k. We show that the series coefficients are Fuss-Catalan numbers and write the roots in terms of the generating function of the Fuss-Catalan numbers. Our main result is a new exact expression for the distribution, which is more concise than previously published formulas. Our work extends the analysis by Feller, who gave asymptotic results. We obtain quantitative improvements of the error estimates obtained by Feller.http://dx.doi.org/10.1155/2015/482462 |
| spellingShingle | S. J. Dilworth S. R. Mane Success Run Waiting Times and Fuss-Catalan Numbers Journal of Probability and Statistics |
| title | Success Run Waiting Times and Fuss-Catalan Numbers |
| title_full | Success Run Waiting Times and Fuss-Catalan Numbers |
| title_fullStr | Success Run Waiting Times and Fuss-Catalan Numbers |
| title_full_unstemmed | Success Run Waiting Times and Fuss-Catalan Numbers |
| title_short | Success Run Waiting Times and Fuss-Catalan Numbers |
| title_sort | success run waiting times and fuss catalan numbers |
| url | http://dx.doi.org/10.1155/2015/482462 |
| work_keys_str_mv | AT sjdilworth successrunwaitingtimesandfusscatalannumbers AT srmane successrunwaitingtimesandfusscatalannumbers |