Find the mean and variance of the compound Poisson distribution (Example H in Section 4....

Find the mean and variance of the compound Poisson distribution (Example H in Section 4.5).

Reference

Compound Poisson Distribution

This example presents a model that occurs for certain chain reactions, or “cascade” processes. When a single primary electron, having been accelerated in an electrical field, hits a plate, several secondary electrons are produced. In a multistage multiplying tube, each of these secondary electrons hits another plate and thereby produces a number of tertiary electrons. The process can continue through several stages in this manner. Woodward (1948) considered models of this type in which the number of electrons produced by the impact of a single electron on the plate is random and, in particular, in which the number of secondary electrons follows a Poisson distribution. The number of electrons produced at the third stage is described by a random sum of the type just described, where N is the number of secondary electrons and Xi is the number of electrons produced by the ith secondary electron. Suppose that the Xi are independent Poisson random variables with parameter λ and that N is a Poisson random variable with parameter μ. According to the preceding result, the mgf of S,