Certain mathematical formulas for moments of Geometric distribution by means of the Apostol-Bernoulli polynomials

Even though the first and second order moments of Geometric distribution are easy to compute, the higher order moments of this distribution are very complicated to derive and calculate. There is an immense need for statisticians to calculate these moments manually or programmatically using statistical softwares. The aim of this paper is to identify certain mathematical formulas. For derivation of the moments, factorial moments, and moment generating function of Geometric distribution, we apply the generating functions for the Apostol-Bernoulli number and polynomials. These newly derived moment formulas are linked to Stirling numbers and other special functions. In addition, when we apply the z-transform to the probability distribution of Geometric distribution, we obtain some new computational formulas that can help calculating the higher order factorial moments of Geometric distribution. These new formulas give alternative way of calculating them without dealing with the cumbersome derivations moments and moment generating function of this distribution.