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An alternative approach of Poisson and Geometric distributions having a more general form of sampling method are suggested in this paper and defined them as generalized Poisson and generalized Geometric distributions respectively. It is evident that the traditional forms of both the distributions are the special cases of the proposed generalized distributions. Moreover, some distributional properties of the suggested distributions are derived here. The study also fixes the assumptions under which generalized Binomial distribution reduces to generalized Poisson distribution along with proof.