The question regards Galton-Watson branching process 1. Suggest a random phenomenon 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'. 2. The offspring random variable in a Galton Watson branching proces: with a single ancestor in generation 0 has probability generating function II(s) = = 5 - 2s 10 - 7s 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 Z2. Calculate each of the following probabilities correct to four decimal places. (1) The probability of extinction by the third generation. (2) The probability of extinction at the fourth generation. Find the probability that the process will eventually become extinct. Do not use an iterative method to find this probability.

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