On a Distribution Associated with a Stochastic Process in Ecology

• Published in: Biometrical journal Vol. 44; no. 4; pp. 510 - 522
• Main Author: Janardan, K.G.
• Format: Journal Article
• Language: English
• Published: Berlin WILEY-VCH Verlag Berlin GmbH 01.06.2002; WILEY‐VCH Verlag Berlin GmbH; Wiley-VCH
• Subjects: Analysis. Health state; Biological and medical sciences; Birth process; Continuous time Markov chain; Epidemiology; General aspects; Integral representation; Medical sciences; Over- (under-) dispersion; Oviposition; Parasites; Poisson process; Poisson-binomial; Poisson-geometric; Public health. Hygiene; Public health. Hygiene-occupational medicine; Stochastic models; Stochastic model; Generating function; Parasite; Continuous time; Ecology; Poisson distribution; Stochastic process; Poisson process; Birth process; Poisson geometric distribution; Dispersion; Markov chain; Probability density; Poisson binomial distribution; Binomial distribution; Integral representation
• ISSN: 0323-3847; 1521-4036
• DOI: 10.1002/1521-4036(200206)44:4<510::AID-BIMJ510>3.0.CO;2-K

Poisson processes {X(t), t ≥ 0} are suitable models for a broad variety of counting processes in Ecology. For example, when analyzing data that apparently came from Poisson population, over‐dispersion [i.e. V(X(t)) > E(X(t))] or under‐dispersion [i.e. V(X(t)) < E(X(t))] is encountered. This led Consul and Jain (1973), and Janardan and Schaeffer (1977) to consider a generalization of the Poisson distribution called Lagrangian Poisson distribution. Janardan (1980) modified the Poisson process and derived a stochastic model for the number of eggs laid by a parasite on a host. This distribution is very suitable for fitting data with over‐ (or under‐) dispersion. Janardan et al. (1981) considered this stochastic model and applied it to study the variation of the distribution of chromosome aberrations in human and animal cells subject to radiation or chemical insults. Here, we present a new approach for the derivation of this distribution and provide some alternative chance mechanisms for the genesis of the distribution. Moments, moment properties, and some applications are also given.