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Journal of Probability and Statistics
Volume 2015, Article ID 482462, 11 pages
http://dx.doi.org/10.1155/2015/482462
Research Article

Success Run Waiting Times and Fuss-Catalan Numbers

1Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
2Convergent Computing Inc., P.O. Box 561, Shoreham, NY 11786, USA

Received 10 June 2015; Accepted 3 September 2015

Academic Editor: Steve Su

Copyright © 2015 S. J. Dilworth and S. R. Mane. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 consecutive successes in a sequence of independent Bernoulli trials, that is, the geometric distribution of order . 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.