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lain what is meant by the term branching process . (Uxford 1974) e nth generation of a branching process in which eac 6. (b) Let Xn be the size of th has probability generating function G, and assume generating function Gn of Xn satisfies Gni(s) - 1. Show fXn satisfies Gn+1 (s) = Gn(G(s)) forn>that the probabili m variable when ar, B (O, 1), and find GIn explicitly wh ß is the probability generating function of a non-ne (c) Show, that G (s) = I-α (1-s) integer-valued random variable is α n Gis thus ity that Xn = 0, and show that it converges as n → oo to I-α1/(1-p) given. 0 (d) Find the probability Explain why this implies that the probability of ultimate extinction equals 1-uni-n (Cambridge 2001)

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(a) Consider a population consisting of individuals able to produce offspring of the same kind. Suppose that each individual will, by the end of its lifetime, have produced j new offspring with probability Pj.J 0, independently of the numbers produced by other individuals. We suppose thatJ The number of individuals initially present, denoted by Xo, is called the size of the zeroth generation. All offspring of the zeroth generation constitute the first generation and their number is denoted by 1. In general, let Andenote the size of the nth generation. It follows thatt^n,nU, is a Markov chain having as its state space the set of nonnegative integers. Note that state o is a recurrent state, since clearly Poo 1. Also, if Poall other states are transient. This follows since Po Po, which implies that starting other states are transient. This follows since 0 = with i individuals there is a positive probability of at least 0 that no later generation will ever consist of i individuals. Moreover, since any finite set of transient states {1, 2, . . . , n} will be visited only finitely often, this leads to the important then the population will either die out or its size will converge to infinity. Such a process is called a Branching Process.(b) Let Gn be the PGF of Xn. Then Pj(n) = P(X,+,-JIX, = i); i,J = 0, 1,2, , n:s > 0 G(a) -m)s let Since individual particles behave independently,The formula means that the population X(n: i) involving in time n from i initial parents is same, in the sense of probability, as the combined sum of i population each having one initial parents. Also, this formula characterizes and distinguishes branching processes from other continuous time Markov chains. By the time homogeneity, the Chapman - Kolmogorov equations take the form Pi(n +1)-Σ Pk (n)Ag( 1 ) Pik n)Pkj -Gn (Gi(s)) at i-1, our proof is ready.Using above given proof, we have Gnt1(s) Gn(G(s)) This implies that G2(s) G(G(s)) G3 (s) = G2(G(s)), and so on Here, G(s)- 1-a(1-s) implies that G2(8)-1-a (-a( )+1) implies that (72 (s Iterating n times, we shall have, (1 - s)9

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