The aim of this paper is to derive many novel formulas involving the sum of powers of consecutive integers, the Bernoulli polynomials, the Stirling numbers and moments arise from conditional probability, moment generating functions and arithmetic functions by using the methods and techniques, which are used in discrete distributions in statistics such as uniform distribution, moment generating functions, and other probability distributions. Moreover, relations among the generalized Euler totient function, finite distributions containing special numbers and polynomials, discrete probability formula, and other special function are given. By using the Riemann zeta function and the Liouville's function, we derive a novel moment formula probability distribution on the set positive integers. Finally, by using approximation formulas for certain family of finite sums, approximation formulas for the sum of powers of consecutive integers involving the Bernoulli polynomials,and certain classes conditional probability involving the Laplace's rule of succession are derived.

Although it is very easy to calculate the 1st moment and 2nd moment values of the geometric distribution with the methods available in existing books and other articles, it is quite difficult to calculate moment values larger than the 3rd order. Because in order to find these moment values, many higher order derivatives of the geometric series and convergence properties of the series are needed. The aim of this article is to find new formulas for characteristic function of the geometric random variable (with parameter p) in terms of the Apostol-Bernoulli polynomials and numbers, and the Stirling numbers. This characteristic function characterizes the geometric distribution. Using the Euler’s identity, we give relations among theis characteristic function, the Apostol-Bernoulli polynomials and numbers, and also trigonometric functions including sin w and cos w . A relations between the characteristic function and the moment generating function is also given. By using these relations, we derive new moments formulas in terms of the Apostol-Bernoulli polynomials and numbers. Moreover, we give some applications of our new formulas.