Question

9. Consider the Branching Process {X_n, n = 0, 1, 2, 3, ...} where X_n is the population size at the nth generation. Assume P(X_0 = 1) = 1 and that the probability generating function of the common offspring distribution A(z) = 1 / (5 - 4z), which is defined for 0 ? z < 5/4. (a) Express A(z) as a power series and hence find the probability that an individual has four offspring. (b) Find the expected value and the variance of the number of offspring that an individual has. (c) If q_n = P(X_n = 0) for n = 0, 1, ..., write down an equation relating q_{n+1} and q_n. Hence, or otherwise, evaluate q_n for n = 0, 1, 2. (d) Find the extinction probability q = lim_{n ? ?} q_n.

          9. Consider the Branching Process {X_n, n = 0, 1, 2, 3, ...} where X_n is the population size at the nth generation. Assume P(X_0 = 1) = 1 and that the probability generating function of the common offspring distribution
A(z) = 1 / (5 - 4z),
which is defined for 0 ? z < 5/4.
(a) Express A(z) as a power series and hence find the probability that an individual has four offspring.
(b) Find the expected value and the variance of the number of offspring that an individual has.
(c) If q_n = P(X_n = 0) for n = 0, 1, ..., write down an equation relating q_{n+1} and q_n. Hence, or otherwise, evaluate q_n for n = 0, 1, 2.
(d) Find the extinction probability q = lim_{n ? ?} q_n.

9. Consider the Branching Process Xn, n = 0, 1, 2, 3, ... where Xn is the population size at the nth generation. Assume P(X0 = 1) = 1 and that the probability generating function of the common offspring distribution
A(z) = 1 / (5 - 4z),
which is defined for 0 ? z < 5/4.
(a) Express A(z) as a power series and hence find the probability that an individual has four offspring.
(b) Find the expected value and the variance of the number of offspring that an individual has.
(c) If qn = P(Xn = 0) for n = 0, 1, ..., write down an equation relating qn+1 and qn. Hence, or otherwise, evaluate qn for n = 0, 1, 2.
(d) Find the extinction probability q = limn ? ? qn.

Added by Jose Angel M.

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