Property of bivariate poisson distribution and its application to stochastic processes

Abstract

Assuming that the random vectors X ₁ and X ₂ have independent bivariate Poisson distributions, the conditional distribution of X ₁ given X ₁ + X ₂ = n is obtained. The conditional distribution turns out to be a finite mixture of distributions involving univariate binomial distributions and the mixing proportions are based on a bivariate Poisson (BVP) distribution. The result is used to establish two properties of a bivariate Poisson stochastic process which are the bivariate extensions of the properties for a Poisson process given by Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, Academic Press, New York.

Keywords:

• Bivariate Poisson distribution
• Conditional distribution
• Stochastic process

Acknowledgements

This work was completed when the first author visited the Department of Mathematics and Statistics, McMaster University, Canada, during May–June 2002. The second author thanks the Natural Sciences and Engineering Research Council of Canada for funding this research. The authors also express their sincere thanks to the referee for making some suggestions which led to an improvement in the presentation of this article.