PYTHON

for reference:

(7) (4.24) Let Tn = Zo + 2 1 + . . . + Zn be the total number of individuals up through generation n. Let T := limnoo Tn be the total progeny of the branching process. Find E(T) for the subcritical ( 1 ) cases. In this problem it is not necessarily Zo 1; We may answer the question in terms of E[Zo (8) (4.25) Let (s) = E[s be the probability generating function of Tz as in the previous problem. Assume Zo 1 for simplicity. (a) Show that Pn satisfies the recurrence relation n (s) = sG(Pn-1(s)) for n = 1, 2, , where G(-) is the probability generating function of the offspring distribution. (b) From (a), argue that (s) = sG(d(s)) , where (s) is the probability generating function of the total progeny T. (c) Use (b) to find the mean of T in the subcritical case (7) (4.24) Let Tn = Zo + 2 1 + . . . + Zn be the total number of individuals up through generation n. Let T := limnoo Tn be the total progeny of the branching process. Find E(T) for the subcritical ( 1 ) cases. In this problem it is not necessarily Zo 1; We may answer the question in terms of E[Zo (8) (4.25) Let (s) = E[s be the probability generating function of Tz as in the previous problem. Assume Zo 1 for simplicity. (a) Show that Pn satisfies the recurrence relation n (s) = sG(Pn-1(s)) for n = 1, 2, , where G(-) is the probability generating function of the offspring distribution. (b) From (a), argue that (s) = sG(d(s)) , where (s) is the probability generating function of the total progeny T. (c) Use (b) to find the mean of T in the subcritical case