A form of bivariate binomial conditionals distributions.

In this article, we discuss a bivariate distribution whose conditionals are univariate binomial distributions, but the marginals are not binomial that exhibits both positive and negative correlation. Some useful structural properties of this distribution namely marginals, moments, generating functions, stochastic ordering are investigated. Simple proofs of both positive and negative correlation, marginal over-dispersion, the distribution of the maximum and minimum order statistics for a random sample of size 2 are also derived. The distribution is shown to be a member of the multi-parameter exponential family, and some natural and useful consequences are also outlined. The proposed distribution reduces to another bivariate discrete distribution, namely the bivariate Poisson conditionals recently studied by Ghosh, Marques, and Chakraborty (2021) under certain parametric conditions. Finally, the distribution is fitted to two bivariate count data sets having positive and negative correlation separately to illustrate its' suitability. [ABSTRACT FROM AUTHOR]