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In this article, we discuss a bivariate geometric distribution whose conditionals are geometric distributions and the marginals are not geometric and exhibits negative correlation. Several useful structural properties of the bivariate geometric distribution namely marginals, moments, generating functions, stochastic ordering are investigated. Simple proofs of negative correlation, marginal over-dispersion, distribution of sum, and distribution of the conditional given the sum are also derived. The distribution is shown to be a member of the multi-parameter exponential family and some natural but useful consequences are also outlined. A characterization via conditional failure rates is provided. An extensive simulation study is also carried out to explore the flexibility of the bivariate geometric distribution with various choices of the model parameters. Finally, the distribution is fitted to one bivariate count data set with an inherent negative correlation to illustrate its suitability.

A bivariate geometric distribution via conditional specification: properties and applications