  A novel multinomial theorem for commutative idempotents is shown to lead to
new results about the moments, central moments, factorial moments, and their
generating functions for any random variable $X = \sum_{i} Y_i $ expressible as
a sum of Bernoulli indicator random variables $Y_i$. The resulting expressions
are functions of the expectation of products of the Bernoulli indicator random
variables. These results are used to derive novel expressions for the various
moments in a large number of familiar examples and classic problems including:
the binomial, hypergeometric, Poisson limit of the binomial, Poisson,
Conway-Maxwell-Poisson binomial, Ideal Soliton, and Benford distributions as
well as the empty urns and the matching problems. Other general results include
expressions for the moments of an arbitrary count random variable in terms of
its upper tail probabilities.
