any help is appreiated. Thanks! Recall the Galton-Watson branching process discu

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any help is appreiated. Thanks!

Recall the Galton-Watson branching process discussed in class. Let Xn be the number of individuals at generation n; by default Xo 1. For this problem, suppose that the offspring distribution fpk) 0 is given by the binomial distribution Bin(2,p), that is 0 for all k 3 Let u be the mean offspring number. According to the general analysis mentioned in class (and also in Durrett, Example 1.52): If u 1, the process will go extinct with probability 1 and in finite time If 1, the process will go extinct with probability 1, but the time to extinction is infinite, If 1, there is positive probability that the process will survive. We call each of the three cases subcritical, critical, and supercritical, respectively. (a) For what values of the parameter p is the Galton-Watson process supercritical? For the following questions, you may answer them using a specific value of p (as long as it gives the supercritical case), or keep the parameter p general. (b) Find an explicit expression for the extinction probability (denoted by P(extinct)). Hint: You will need to compute the generating function of the offepring distribution. Recall that for a general offspring distribution pk)k, its generating function is d(z) .opkzk.] (c) (BONUS, 2 points) Compute the distribution of X1 conditional upon extinction: POX i and extinct) PIX1 ilextinctl P(extinct) Define X1 through PX lextinct] PIX, i. Show that EX] 1.

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