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Unformatted text preview: One Hundred 1 Solved 2 Exercises 3 for the subject: Stochastic Processes I 4 Takis Konstantopoulos 5 1. In the Dark Ages, Harvard, Dartmouth, and Yale admitted only male students. Assume that, at that time, 80 percent of the sons of Harvard men went to Harvard and the rest went to Yale, 40 percent of the sons of Yale men went to Yale, and the rest split evenly between Harvard and Dartmouth; and of the sons of Dartmouth men, 70 percent went to Dartmouth, 20 percent to Harvard, and 10 percent to Yale. (i) Find the probability that the grandson of a man from Harvard went to Harvard. (ii) Modify the above by assuming that the son of a Harvard man always went to Harvard. Again, ±nd the probability that the grandson of a man from Harvard went to Harvard. Solution. We ±rst form a Markov chain with state space S = { H,D,Y } and the following transition probability matrix : P = . 8 . 2 . 2 . 7 . 1 . 3 . 3 . 4 . Note that the columns and rows are ordered: ±rst H , then D , then Y . Recall: the ij th entry of the matrix P n gives the probability that the Markov chain starting in state i will be in state j after n steps. Thus, the probability that the grandson of a man from Harvard went to Harvard is the upperleft element of the matrix P 2 = . 7 . 06 . 24 . 33 . 52 . 15 . 42 . 33 . 25 . It is equal to . 7 = . 8 2 + . 2 × . 3 and, of course, one does not need to calculate all elements of P 2 to answer this question. If all sons of men from Harvard went to Harvard, this would give the following matrix for the new Markov chain with the same set of states: P = 1 . 2 . 7 . 1 . 3 . 3 . 4 . The upperleft element of P 2 is 1, which is not surprising, because the o²spring of Harvard men enter this very institution only. 2. Consider an experiment of mating rabbits. We watch the evolution of a particular 1 More or less 2 Most of them 3 Some of these exercises are taken verbatim from Grinstead and Snell; some from other standard sources; some are original; and some are mere repetitions of things explained in my lecture notes. 4 The subject covers the basic theory of Markov chains in discrete time and simple random walks on the integers 5 Thanks to Andrei Bejan for writing solutions for many of them 1 gene that appears in two types, G or g. A rabbit has a pair of genes, either GG (dominant), Gg (hybrid–the order is irrelevant, so gG is the same as Gg) or gg (recessive). In mating two rabbits, the oFspring inherits a gene from each of its parents with equal probability. Thus, if we mate a dominant (GG) with a hybrid (Gg), the oFspring is dominant with probability 1 / 2 or hybrid with probability 1 / 2. Start with a rabbit of given character (GG, Gg, or gg) and mate it with a hybrid. The oFspring produced is again mated with a hybrid, and the process is repeated through a number of generations, always mating with a hybrid....
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 Spring '09
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