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We meet in the literature the bivariate Poisson distribution put in evidence by Berkhout and Plug. From this distribution, Elion et al. put in evidence the bivariate weighted Poisson distribution like crossing of two univariate weighted Poisson distributions. The structure of the covariance of this bivariate weighted Poisson distribution has been put not again in evidence in the literature. Thus, in this paper, we remedy this hiatus. The overdispersion, underdispersion and the equidispersion will be valued with the help of the Fisher dispersion index for multivariate count distributions introduces by Kokonendji et al. An illustrative example based on the Aleurodicus data is presented.

Kimberly et al. had proposed in 2016 a bivariate function as a bivariate Conway-Maxwell-Poisson distribution (COM-Poisson) using the generalized bivariate Poisson distribution and the probability generating functions of the follow distributions: bivariate bernoulli, bivariate Poisson, bivariate geometric and bivariate binomial. By examining this paper we have shown that this bivariate function is constant and it double series is divergent, when it should have been 1. To overcome this deadlock, we propose a new bivariate Conway-Maxwell-Poisson distribution which is definetely a probability distribution based on the crossing method, method highlighted by Elion et al. in 2016 and revisited by Batsindila et al. and Mandangui et al. in 2019. And this is the purpose of this paper. To this bivariate distribution is attached two generalized linear models (GLM) whose resolution allows us to highlight, not only the independence between the variables forming the couple, but also the effect of the factors (or predictors) on these variables. The resulting correlation is negative, zero or positive depending on the values of a parameter; in particular for the bivariate Poisson distribution according to Berkhout and Plug. A simulation of data will be given at the end of the article to illustrate the model.

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