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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