Negative Moments of Positive Random Variables

We investigate the problem of finding the expected value of functions of a random variable X of the form f(X) = (X+A)−n where X+A>0 a.s. and n is a non-negative integer. The technique is to successively integrate the probability generating function and is suggested by the well-known result that successive differentiation leads to the positive moments. The technique is applied to the problem of finding E[1/(X+A)] for the binomial and Poisson distributions.