Question 2 i 20 marks This question is intended to assess your understanding of the Galton-Watson branching process. You should be able to answer this question after working through Part] of Book 3. (a) Suggest a random phenomenon, different from any mentioned in Part I of Book 3, for which the Galton-Watson branching process might provide a reasonable model. Be clear about the random variable being counted (Zn) and the interpretation of the notion of a 'generation'. [6] (b) The offspring random variable of a Galton-Watson branching process with a single ancestor in generation 0 has probability generating function no) = 855 (i) Find the probability generating function of Z2, the number of individuals in the second generation, and by reference to the Handbook, identify the probability distribution of 22. [4] (ii) Calculate each of the following probabilities correct to four decimal places. (1) The probability of extinction by the second generation. (2) The probability of extinction at the fourth generation. [5] (iii) Find the probability that the process will eventually become extinct. Do not use an iterative method to nd this probability. [5]