Moments through Tail Probabilities of Nonnegative Integer-Valued Random Variables

We present and prove some moments and factorial moments equalities for nonnegative integer-valued random variables. These equalities give the moments in terms of the tail probabilities of the nonnegative integer-valued random variables. We also derive the moment generating functions and the probability generating functions of these random variables when the distributions of the random variables are unknown, but their tail probabilities are known. This enables us to derive the probability density functions of some random variables when we only know their tail probabilities. This is especially good for stopping times whose distribution functions are unknown.

Keywords: Moments, tail probabilities, factorial moments, moment generating function, probability generating function, stopping times.