Expectation identity of the negative binomial distribution and its application in the calculations of high-order origin moments

Analytically calculating high-order origin moments of a negative binomial distribution has long been a persistent and challenging problem. In this study, we employ an expectation identity method to analytically compute these moments. First, we restate the expectation identity theorem for the negative binomial distribution. Subsequently, using this method, we derive analytical expressions for the first four origin moments as well as the general <i>k</i>th (k=1,2,…) moment of the negative binomial distribution. Finally, we present a table displaying polynomial coefficients for the first 10 origin moments and highlight that their leading coefficients correspond to Stirling numbers of the second kind.