Property of bivariate poisson distribution and its application to stochastic processes

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.