MTHE/STAT455, STAT855
Fall 2013
Stochastic Processes
Assignment #1, Solutions
Total Marks: 37 for 455 and 45 for 855.
1. From Sheet. (10 marks)
(a) (4 marks) First note that if n is even then Xn must be even while if n is odd then
Xn must be odd. Also, wh
MTHE/STAT455, STAT855
Fall 2014
Stochastic Processes
Assignment 4
Due Friday, Nov.28
1. Consider a continuous time Markov chain with state space S = cfw_1, 2 and generator
matrix
"
#
G=
,
where > 0 and > 0.
(a) Show by induction that
"
n
G =
(1)n ( +
MTHE/STAT455, STAT855
Fall 2014
Stochastic Processes
Assignment #2
Due Tuesday, Oct.14
Starred questions are for 855 students only.
1. Let cfw_Xn : n 0 be a time-homogeneous Markov chain with state space S.
(a) Show that, for 1 < r < n and states k and xi
MTHE/STAT455, STAT855 Midterm Solutions
Fall, 2013
1. (15 marks)
(a) (7 marks) Dene M1 , M2 and M3 , where Mi is the expected number of steps to
get to the target vertex starting at a vertex that is i edges away from the target
vertex. By symmetry, the ex
STAT455/855
Fall 2010
Stochastic Processes
Final Exam, Solutions
1. (15 marks)
(a) (8 marks) Let Sm be the number of steps until the walk rst reaches state m
starting in state 0, and let Si,i+1 be the number of steps until the walk rst
reaches state i + 1
STAT455/855
Fall 2007
Applied Stochastic Processes
Final Exam, Solutions
1. (15 marks)
(a) (11 marks) Let q = P (Xi > m). This is the same for all Xi , and is equal to
j1
p(1 p)
q=
m
(1 p)j =
= (1 p) p
j=m+1
j=0
(1 p)m p
= (1 p)m .
1 (1 p)
(i) (2 marks) T
MTHE/STAT455, STAT855
Fall 2012
Stochastic Processes
Final Exam, Solutions
1. (15 marks)
(a) (10 marks) Condition on the rst pair to bond; each of the n 1 adjacent pairs
is equally likely to bond. Given that (mr , mr+1 ) is the rst pair to bond the
probab
STAT 455/855 Midterm Solutions
Fall, 2010
1. (15 marks)
(a) (4 marks) Each individual in the initial population is the root of a Galton-Watson
branching process starting at a single individual. If the initial population is of
size k then we have k indepen
MTHE/STAT455, STAT855 Midterm Solutions
Fall, 2012
1. (15 marks)
(a) (5 marks) Following the hint let mk denote the expected additional number of
rolls required if the rst roll is k. Let X denote the number of rolls required until
two consecutive rolls ar
MTHE/STAT455, STAT855
Fall 2013
Stochastic Processes
Assignment #2, Solutions
Total Marks: for 455 and for 855.
1. From Sheet. (8 marks)
(a) (4 marks) The example should be such that knowing the information X0 = 2 in
addition to X1 cfw_3, 4 tells us somet
MTHE/STAT455, STAT855
Fall 2013
Stochastic Processes
Assignment #3, Solutions
Total Marks: 33 for 455 and 43 for 855.
1. From Sheet. (6 marks)
(a) (3 marks) If we sum gij (n) over n we get the probability that the Y chain ever
visits state j starting in s
MTHE/STAT455, STAT855
Fall 2013
Stochastic Processes
Assignment #4, Solutions
Total Marks: 38 for both 455 and 855.
1. From Sheet. (10 marks)
(a) (4 marks) The problem is to compute the probability that there are r arrivals
in the N2 process during an Exp
MTHE/STAT455, STAT855
Fall 2013
Stochastic Processes
Final Exam, Solutions
1. (15 marks)
(a) (6 marks) If fr denotes the probability mass function of the family size in the
modied branching process with parameter r then
fr (0) = r + (1 r)f (0)
and
fr (k)
MTHE/STAT455, STAT855
Fall 2013
Stochastic Processes
Assignment #5, Solutions
Total Marks: 20 for both 455 and 855.
1. From Sheet. (10 marks)
(a) (3 marks) The given expression for Gn is clearly correct for n = 1. Assume it is
true for n. Direct matrix mu
MTHE/STAT455, STAT855
Fall 2014
Stochastic Processes
Assignment #1
Due Friday, Sep.26
Starred questions are for 855 students only.
1. Consider the simple random walk, cfw_Xn : n 0, starting at 0 (X0 = 0), where the
probability of moving up at each step is