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International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 13, Pages 801-815

Generalized distributions of order k associated with success runs in Bernoulli trials

1Department of Medical Statistics, Faculty of Medicine, Demokritos University of Thrace, Alexandroupolis 68100, Greece
2Department of Mathematics, University of Patras, Patras, Greece

Received 16 July 2002

Copyright © 2003 Hindawi Publishing Corporation. 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.


In a sequence of independent Bernoulli trials, by counting multidimensional lattice paths in order to compute the probability of a first-passage event, we derive and study a generalized negative binomial distribution of order k, type I, which extends to distributions of order k, the generalized negative binomial distribution of Jain and Consul (1971), and includes as a special case the negative binomial distribution of order k, type I, of Philippou et al. (1983). This new distribution gives rise in the limit to generalized logarithmic and Borel-Tanner distributions and, by compounding, to the generalized Pólya distribution of the same order and type. Limiting cases are considered and an application to observed data is presented.