International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 13, Pages 801-815
doi:10.1155/S0161171203207250
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 Gregory A. Tripsiannis et al. 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
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.