This thesis presents an alternative way of generating first moments of the selected discrete probability distributions through the use of the recurrence formulas. The Polya-Eggenberger distribution, which constitutes a hierarchy of family composed of special discrete distributions is introduced in this paper. The recurrence formulas of these distributions are derived using the Polya-Eggenberger recurrence formula and also the relationships that hold with the members of the family. Also, the first moments of these distributions are also obtained using the relationships of the members of the Polya-Eggenberger family. All the equations and formulas in this thesis are results given by J. Wanzer Drane, Suhua Cao, Lixia Wang, and T. Postelnicu in their article Limiting Forms of Probability Mass Functions via Recurrence Formulas . The researchers provided the proofs and derivations of the equations
