Compound Poisson law generalized by negative binomial distribution

For the compound Poisson law generalized by the negative binomial distribution we derive the explicit representation of the probability function, the finite-difference recurrence, and expressions for the derivatives with respect to all parameters. Explicit representations of ordinary and factorial cumulants to sixth order are given and asymptotic normality of the distribution is proved. The distribution functions are constructed and analyzed for the most typical parameter values. Used to solve direct and inverse problems by moment methods. © 2006 Springer Science+Business Media, Inc.