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MTHE/STAT455, STAT855 -- Final Exam, 2013
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QUEEN’S UNIVERSITY
DEPARTMENT OF MATHEMATICS AND STATISTICS
FACULTY OF ARTS AND SCIENCE
MTHE/STAT455, STAT855 FALL 2013, FINAL EXAM
7:00PM, DECEMBER 4, 2013
GLEN TAKAHARA
Instructions:
• “Proctors are unable to respond to queries about the interpretation of exam questions.
Do your best to answer exam questions as written.”
• This material is copyrighted and is for the sole use of students registered in MTHE/STAT455,
STAT855 and writing this examination. This material shall not be distributed or disseminated. Failure to abide by these conditions is a breach of copyright and may
also constitute a breach of academic integrity under the University Senates Academic
Integrity Policy Statement.
• This examination is THREE HOURS in length. It is closed-book – no books, notes,
or any other resource material is allowed except as indicated in the next item.
• Simple non-communicating calculators without text storage capabilities (Casio 991,
blue or gold sticker) and any notes on a single 8 12 × 11 inch sheet of paper (both sides)
are allowed.
• Please answer all questions in the booklets provided. Put your student number on the
front of all answer booklets and number the answer booklets if more than one answer
booklet is used.
• There are 5 questions. STAT855 students must do all 5 questions. MTHE/STAT455
students must to 4 of the 5 questions. For MTHE/STAT455 students, if you answer
all 5 questions please indicate which 4 questions you want marked. By default, unless
otherwise specified, the first 4 questions will be marked.
• Each question is worth 15 marks for a possible total of 60 for MTHE/STAT455 students
and 75 for STAT855 students.
• Show all your work. You may receive partial credit if you get an answer wrong but
show your work, whereas you will receive no credit if you get an answer wrong and do
not show your work.
MTHE/STAT455, STAT855 -- Final Exam, 2013
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1. (Total 15 marks) Consider a Galton-Watson branching process with family size probability mass function f (that is, P (family size is k) = f (k), for k ≥ 0) which has
generating function G. Now modify this process such that each individual in any generation (independently across individuals and generations) is afflicted with a pathogen
with probability r, (0 ≤ r ≤ 1), that kills that individual, resulting in no offspring for
that individual. The case r = 0 is the original branching process.
(a) (6 marks) Suppose that in the original branching process (r = 0) the probability
of ultimate extinction, η, is less than 1. Prove that in the modified branching
process with r > 0 the probability of ultimate extinction is greater than η. (This
is intuitive, but you should be able to prove it mathematically by considering the
family size generating function for the modified process).
(b) (9 marks) Now suppose that f (k) = ( 41 )( 34 )k for k = 0, 1, . . . (that is, the family size
distribution is Geometric(1/4) on {0, 1, 2, . . .}). Find the probability of ultimate
extinction for the modified process with parameter r, for any r ∈ [0, 1].
2. (Total 15 marks) Let {Xn : n = 0, 1, 2, . . .} be a discrete-time Markov chain with M
states (M < ∞) and transition probability matrix P . All of the questions below have
simple, one to three line answers. Don’t overthink them.
(a) (4 marks) Show if all the column sums of P are equal to c, then c must equal 1.
(b) (4 marks) If all the column sums of P are the same, find the stationary distribution
of the Markov chain.
(c) (5 marks) Suppose P has the block diagonal form
"
P =
A 0
0T B
#
,
where A and B are (finite) square matrices representing irreducible, positive recurrent classes (and 0 is a matrix of all zeroes of dimension |A| × |B|). Show that
this Markov chain has infinitely many stationary distributions. Hint: Let π 1 and
π 2 denote the (unique) stationary distributions corresponding to the transition
matrices A and B, respectively. Go from there.
(d) (2 marks) If a Markov chain is irreducible and positive recurrent it has a unique
stationary distribution, but it need not be aperiodic. Give a simple example.
MTHE/STAT455, STAT855 -- Final Exam, 2013
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3. (Total 15 marks) Let N = {N (t) : t ≥ 0} be a Poisson process with rate λ. Suppose
an event that occurs at time s is a type 1 event with probability p1 (s) = e−as , for
some fixed a > 0, and is a type 2 event with probability 1 − p1 (s). For a fixed t, find
the probability that the last event before time t was a type 1 event. (You’ll want to
first condition on N (t); then, conditioned on N (t) = n condition on the time of the
last event before time t, which has the distribution of the maximum of n Uniform(0, t)
random variables. You may assume that conditioned on N (t) = 0 the event in question
has probability 0).
4. (Total 15 marks) Suppose a particle moves among 4 states (states 0, 1, 2, and 3) as
follows. When it enters state 0 is stays there for an Exponential(λ0 ) amount of time
then moves to state i with probability pi , i = 1, 2, 3, where pi > 0 for i = 1, 2, 3
and p1 + p2 + p3 = 1. For i = 1, 2, 3, when it enters state i it stays there for an
Exponential(λi ) amount of time and then moves to state 0. Let X(t) by the state of
the particle at time t, t ≥ 0. The process {X(t) : t ≥ 0} is a continuous-time Markov
chain.
(a) (7 marks) Write the transition probability matrix for the embedded discrete-time
jump chain, and find the stationary distribution of this embedded discrete-time
chain.
(b) (8 marks) Write the generator matrix for the continuous-time Markov chain
{X(t) : t ≥ 0} and find the stationary distribution for this chain.
5. (Total 15 marks)
For h > 0 partition the interval [0, ∞) into the subintervals [0, h), [h, 2h), [2h, 3h), . . ..
Let λ > 0 be given (and fixed). Suppose in each subinterval, independently from
interval to interval, an event occurs with probability λh and does not occur with
probability 1 − λh (for definiteness assume that if an event occurs in an interval it
occurs at the start of the interval). Let Nh (t) denote the number of events that occur
in the interval [0, t) for any real, nonnegative t.
(a) (10 marks) Compute limh→0 P (Nh (t) = k) for k ≥ 0 and state the limiting distribution of Nh (t). (Note that as h decreases the partition gets finer and the
probability of an event in any given subinterval decreases). Hint: Note that Nh (t)
has a Binomial distribution.
(b) (5 marks) Let Th denote the first time an event occurs in the process {Nh (t) : t ≥
0}. Compute limh→0 P (Th > t) and state the limiting distribution of Th .
MTHE/STAT455, STAT855 -- Final Exam, 2013
Some distributions (for reference):
• Poisson(λ) pmf: f (k) =
λk −λ
e ,
k!
for k = 0, 1, 2, . . .
λ ∈ (0, ∞)
(E[X] = λ; Var(X) = λ).
• Exponential(λ) pdf: f (x) = λe−λx , for x ≥ 0
λ ∈ (0, ∞)
(E[X] = 1/λ; Var(X) = 1/λ2 ).
• Uniform(a, b) pdf: f (x) =
1
,
b−a
for a ≤ x ≤ b
−∞ < a < b < ∞
(E[X] = (a + b)/2; Var(X) = (b − a)2 /12).
HAPPY HOLIDAYS!!
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