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LING 201 (6)
Final

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School
Department
Linguistics
Course
LING 201
Professor
Brian Mc Gill
Semester
Summer

Description
FACULTY OF SCIENCE SAMPLE FINAL EXAMINATION Mathematics and Statistics 447 Elementary Stochastic Processes Examiner: Professor W.J. Anderson Answer all questions. No aids other than pocket calculators are allowed. This examination paper must be returned with your answer booklet. 1. (a) Pet {Zn,n ≥ 0} be a branching process with basic probabilitiks p ,k = 0,1,2,..., and let P(s) = k=0 pks . If 0 = 1, prove that the probability ζ of extinction of the process is the smallest non-negative root of the equation P(s) = s. (b) The population growth of a certain species of trout is approximated by a branching process with probabilities 1 3 k pk= 4 · 4 ) , k ≥ 0. Assuming that Z0= 3, calculate the expected size of the fourth generation and the probability of extinction. 2. Consider a Markov chain {Xn,n ≥ 0} having state space E = {1,2,3,4,5,6} and transition matrix  .1 .2 .1 0 .3 .3  .3 .2 .2 .3 0 0   .3 .3 .1 .1 .1 .1  0 0 0 .7 .3 0  .   0 0 0 .5 .5 0 .2 .2 .1 .1 .1 .3 (a) Find Pr{X 3 5|X =05} and Pr{X = 16X = 5}0 (b) Find all closed sets and classify all states. (c) If the initial probability distribution vector for the chain is (0,0,0,0,1,0), what is the expected number of transitions until the chain returns to state 5 for the ﬁrst time? 3. (a) With respect to a Markov chain with transition matrix P, a set C of states is closed if no state
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