Some properties and a characterization of trivariate and multivariate binomial distributions

By considering a trivariate binomial distribution, the regression equations are obtained and a set of necessary and sufficient conditions are given for the regression to be linear. Under the limiting conditions, the expressions approach those of trivariate Poisson distribution derived by Mahamunulu (1967). A characterization of the trivariate binomial distribution is also established based on the distribution of the sum of two trivariate random vectors. The results are shown to extend to the multidimensional case in a natural way.