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

Gregory A. Tripsiannis,1 Afroditi A. Papathanasiou,1 and Andreas N. Philippou2

 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

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.