STAT455/855
Fall 2009
Applied Stochastic Processes
Assignment #3, Solutions
Total Marks: 44 for 455 and 50 for 855.
1. From Sheet. (7 marks)
(a) (3 marks) Conditioning on X0 we have
P (Xn = j X0 = i)P (X0 = i) =
P (Xn = j) =
i∈S
pij (n)P (X0 = i).
i∈S
The
STAT455/855
Fall 2009
Applied Stochastic Processes
Assignment #2, Solutions
Total Marks: 55 for 455 and 65 for 855.
1. From Sheet. (10 marks)
(a) (4 marks)
(i) (4 marks) Let Dk and Nk denote, respectively, the period and the number of
equivalence classes
STAT455/855
Fall 2009
Applied Stochastic Processes
Assignment #2 Due Friday, Oct.16
Starred questions are for 855 students only. 1. (a) Let cfw_Xn : n 0 be an irreducible Markov chain with period d > 1 and transition matrix P. Consider the Markov chain wi
STAT 455/855 Midterm Solutions
Fall, 2008
1. (15 marks) (a) (5 marks) The equivalence classes are cfw_1, 3, 6, cfw_2, 5, and cfw_4. Class cfw_1, 3, 6 is transient and has period 1. Class cfw_2, 5 is recurrent and has period 2. Class cfw_4 is recurrent and
STAT455/855
Fall 2008
Applied 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 p
STAT 455/855 Midterm Solutions
Fall, 2007
1. (15 marks)
(a) (6 marks) Let p1 be the probability that player 1 wins when he rolls rst and
let p1 be the probability that player 1 wins if he doesnt roll rst. By symmetry,
p1 doesnt depend on which of players
STAT455/855
Fall 2007
Applied 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 p
STAT 455/855 Solutions: Midterm
Fall, 2006 1. (a) Yes. Examples are a random walk on the vertices of a square (or any polygon with an even number of sides), or the random walk with reflecting barrier at 0 described in problem 2, both of which have period
STAT455/855
Fall 2006
Applied 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 p
STAT455/855
Fall 2005
Applied Stochastic Processes
Midterm, Brief Solutions
1. (15 marks) (a) (9 marks) (i) (1 mark) M0 = 1, since the rst step of the random walk will go to state 1 or state -1 with probability 1. (ii) (2 marks) Conditioning on the rst mo
STAT455/855
Fall 2009
Applied 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 p
STAT 455/855 Midterm Solutions
Fall, 2009
1. (15 marks)
(a) (6 marks) The chain is irreducible (so only 1 equivalence class). It is recurrent
and the period is 2. The cyclic classes are C0 = cfw_0, 5 and C1 = cfw_1, 2, 3, 4, 6.
(b) (4 marks) Let M01 denot
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6
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STAT455/855
Fall 2005
Applied 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 (both sides) and a calculator.
(b) Stat 855 students must do all of the problems. S
STAT455/855
Fall 2003
Applied Stochastic Processes
Midterm, Brief Solutions
1. (15 marks)
(a) (2 marks) When N = 1, each move from the line x = 0 will go to one of the lines
x = 1 or x = 1 with probability 1/2. Thus, the number of moves to go from the lin
STAT455/855
Fall 2003
Applied 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) Stat 855 students must do all of the problems. Stat 455 stude
STAT455/855
Fall 2003
Applied Stochastic Processes
Final Exam, Brief Solutions
1. (15 marks)
(a) (7 marks) The distribution of Y is given by
P (Y = y ) =
1
6
5
6
y 2
for y = 2, 3, . . .
The above follows because each of the rolls 2, . . . , y 1 should be
STAT 455/855 - Final Exam, 2003
Page 1 of 4
QUEENS UNIVERSITY
DEPARTMENT OF MATHEMATICS AND STATISTICS
STAT 455/855 FALL 2003
FINAL EXAMINATION
2:00PM, DECEMBER 8, 2003
GLEN TAKAHARA
Instructions:
This examination is THREE HOURS in length. It is closed-b
STAT455/855
Fall 2009
Applied Stochastic Processes
Assignment #5, Solutions
Total Marks: 32. 1. Ross, Chapter 6 #1. (4 marks) The state space of the process (N1 (t), N2 (t) is S = cfw_(n, m) : n 0, m 0. If in state i = (n, m), we leave state i as soon as
STAT455/855
Fall 2009
Applied Stochastic Processes
Assignment #5
Due Friday, Dec.4
1. Ross, Chapter 6, #1 (8th Edition: Chapter 6, #1).
2. PASTA: Poisson Arrivals See Time Averages. Consider a continuous time Markov
chain observed at the times of a Poisso
STAT455/855
Fall 2009
Applied Stochastic Processes
Assignment #4, Solutions
Total Marks: 30 for 455 and 35 for 855.
1. Ross, Chapter 5, #44. (6 marks)
(a) (2 marks) The probability that the woman’s waiting time is 0 is equal to the
probability that when s
STAT455/855
Fall 2009
Applied Stochastic Processes
Assignment #3 Due Monday, Nov.9
Starred questions are for 855 students only. 1. Let cfw_Xn : n 0 be an irreducible, aperiodic, positive recurrent Markov chain, with stationary distribution . (a) Show that
STAT455/855
Fall 2009
Applied Stochastic Processes
Assignment #1, Solutions
Total Marks: 35 for 455 and 40 for 855. 1. Ross, Chapter 3 #28. (12 marks) Let us define Yi = Then Xk = Y1 + . . . + Yk . (a) (1 mark) The expected value of X1 is the expected val