  In this paper we develop a very general class of bivariate discrete
distributions. The basic idea is very simple. The marginals are obtained by
taking the random geometric sum of a baseline distribution function. The
proposed class of distributions is a very flexible class of distributions in
the sense the marginals can take variety of shapes. It can be multimodal as
well as heavy tailed also. It can be both over dispersed as well as under
dispersed. Moreover, the correlation can be of a wide range. We discuss
different properties of the proposes class of bivariate distributions. The
proposed distribution has some interesting physical interpretations also.
Further, we consider two specific base line distributions namely; Poisson and
negative binomial distributions for illustrative purposes. Both of them are
infinitely divisible. The maximum likelihood estimators of the unknown
parameters cannot be obtained in closed form. They can be obtained by solving
three and five dimensional non-linear optimizations problems, respectively. To
avoid that we propose to use the method of moment estimators and they can be
obtained quite conveniently. The analyses of two real data sets have been
performed to show the effectiveness of the proposed class of models. Finally,
we discuss some open problems and conclude the paper.
