Lagrangian Probability Distribution

Lagrangian probability distributions form large class of important probabilitydistributions such as the generalized poisson, negative binomial and logarithmicseries and also power series distributions. The family of Lagrangian distributionsof the first kind are obtained from the first Lagrangian expansion and the familyof Lagrangian distributions of the second kind are obtained from the second Lagrangianexpansion. In this article, we shall first study convolution properties ofthe distributions in question and then show that the two families of Lagrangiandistributions are equivalent. Finally it is proved that some weighted distributionsof the family of the first kind Lagrangian distributions are members of the familyLagrangian distributions of the second kind.