In this paper, we define a bivariate weighted Poisson distribution by the crossing of two univariate weighted Poisson distribution, in particular, the bivariate translated Poisson distribution and provide some of its properties. This distribution allows us to treat the statistical data of a 56 years old patient in 2008, not smoker, of male sex, suffering from arterial hypertension and diabetes in taking account of his weight. The object of this statistical study is to be able to detect the influence of the weight and the blood sugar level on the blood pressure, to possibly see the interaction between the weight and the blood sugar level. With this intention, we use the regression of Poisson model. We show that the estimators of the coefficients of regression of Poisson are the estimators of the maximum likelihood function of the bivariate translated Poisson distribution.

overdispersion, Poisson distribution, regression of Poisson model, underdispersion, univariate weighted Poisson distribution, test of adequacy of chi-2.