Expectation Identity of the Discrete Uniform Distribution and Its Application in the Calculations of Higher-Order Origin Moments

We provide a novel method to analytically calculate the high-order origin moments of a Discrete Uniform (DU) random variable, that is, the expectation identity method. First, the expectation identity of the DU distribution is discovered and summarized in a theorem. After that, we analytically calculate the first four origin moments and the general kth (k=1,2,…) origin moment of the DU distribution by the expectation identity method. After comparing the corresponding coefficients on both sides of an equation, we obtain a nonhomogeneous linear equations of first degree in k+1 variables. Furthermore, we have provided two ways to solve the nonhomogeneous linear equations. The first way is by matrix inversion, and the second way is by iterative solving. Moreover, the coefficients of the first ten origin moments of the DU distribution are summarized in a table. Finally, we have a proposition for special summations.