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MTHE/STAT455, STAT855
Fall 2015
MTHE/STAT455, STAT855, Stochastic Processes
Midterm Exam
Instructions:
(a) The exam is closed book. No books are allowed. You may use one 8.5 × 11 inch sheet
of notes and a calculator.
(b) There are 3 questions. Stat 855 students must do all of the problems. Mthe/Stat 455
students must do 2 of questions 1, 2 and 3, and if you answer all three questions, you
must specify which two you want graded (the default if you do not specify is questions
1 and 2).
(c) Each question is worth 15 marks for a total of 30 marks (Mthe/Stat455) or 45 marks
(Stat855)
(d) Show all your work. Partial credit is given.
(e) Read the hints!
(f) You have 120 minutes. Good luck!
MTHE/STAT455, STAT855 -- Midterm Exam, 2015
Page 2 of 2
1. (15 marks) A fair die is rolled repeatedly. Let Xk denote the number of rolls required
until k consecutive rolls of the die all turn up the same number.
(a) (8 marks) In this problem we want to find E[X3 ]. By conditioning on the first,
then second roll of the die, write down a set of equations for m1 and m2 , where
mi is the expected further number of rolls required if the previous i rolls were all
the same number (and prior to that the roll, if any, was different), and solve for
m1 . Note that E[X3 ] = 1 + m1 .
(b) (7 marks) By conditioning on Xk−1 , show that E[Xk ] = 6E[Xk−1 ] + 1.
2 (15 marks) Let X = {Xn : n ≥ 0} be a time-homogeneous Markov chain and let
Y = {Yn : n ≥ 0}, where Yn = X2n , for n ≥ 0. Then Y is also a time-homogeneous
Markov chain (which you don’t need to show).
(a) (7 marks) If state i has period 2 in the X chain then
(i) (3 marks) show by contradiction that state i has period 1 in the Y chain.
(ii) (4 marks) if state i is also recurrent in the X chain then show that state i is
recurrent in the Y chain.
(b) (8 marks) If state i has period 1 in the X chain then
(i) (4 marks) removed from exam.
(ii) (4 marks) if the X chain is also irreducible then show that the Y chain is
irreducible. Hint: For the X chain, pij (m + n) ≥ pii (m)pij (n). Choose m and
n appropriately. If n is odd choose m odd. If n is even choose m even. Why
can you do this?
3. (15 marks) Consider a Galton-Watson branching process for which the family size
distribution has generating function
G(s) = 1 − α(1 − s)p ,
where 0 < α < 1 and 0 < p < 1 are both fixed numbers.
(a) (3 marks) Find the mean family size.
(b) (12 marks) Let Gn (s) be the generating function of Xn , the population size in
generation n. Find Gn (s) (recall that Gn (s) is the n-fold composition of G(s)).
Then use Gn (s) to find the probability of ultimate extinction in terms of α and p.