For example we might ask q if we are going to operate

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Unformatted text preview: about 5 years), then how many parts of types 1, 2, and 3 will we use? Example 1.8. Branching processes. These processes arose from Francis Galton’s statistical investigation of the extinction of family names. Consider a population in which each individual in the nth generation independently gives birth, producing k children (who are members of generation n + 1) with probability pk . In Galton’s application only male children count since only they carry on the family name. To define the Markov chain, note that the number of individuals in generation n, Xn , can be any nonnegative integer, so the state space is {0, 1, 2, . . .}. If we let Y1 , Y2 , . . . be independent random variables with P (Ym = k ) = pk , then we can write the transition probability as p(i, j ) = P (Y1 + · · · + Yi = j ) for i > 0 and j 0 When there are no living members of the population, no new ones can be born, so p(0, 0) = 1. Galton’s question, originally posed in the Educational Times of 1873, is Q. What is the probability that the line of a man becomes extinct?, i...
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