Branching with immigration. Each generation of a branching process (with a single progenitor) is augmented by a random number of immigrants who are indistinguishable from the other members of the population. Suppose that the numbers of immigrants in different generations are independent of each other and of the past history of the branching process, each such number having probability generating function H(s). Show that the probability generating function Gn of the size of the nth generation satisfies Gn+1(s) = Gn (G(s))H(s), where G is the probability generating function of a typical family of offspring.
Branching with immigration. Each generation of a branching process (with a single progenitor) is augmented by a random number of immigrants who are indistinguishable from the other members of the population. Suppose that the numbers of immigrants in different generations are independent of each other and of the past history of the branching process, each such number having probability generating function H(s). Show that the probability generating function Gn of the size of the nth generation satisfies Gn+1(s) = Gn (G(s))H(s), where G is the probability generating function of a typical family of offspring.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 61CR
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