Let be a random variable taking values in the non-negative integers, and let be the probability generating function of . Assuming is everywhere finite, show that
where is the mean of and is its variance. [You may interchange differentiation and expectation without justification.]
Consider a branching process where individuals produce independent random numbers of offspring with the same distribution as . Let be the number of individuals in the -th generation, and let be the probability generating function of . Explain carefully why
Assuming , compute the mean of . Show that
Suppose and . Compute the probability that the population will eventually become extinct. You may use standard results on branching processes as long as they are clearly stated.