Answer Key to Quiz 1
526-W15
1. (2 pts) Suppose that the probability it rains today is 0.3 if neither of the last two days
was rainy, but 0.6 if at least one of the last two days was rainy. Let the we
526 Stochastic Processes, Winter 2015.
Solution to Homework 1
1. (2 pts) Find the stationary distribution of a Markov chain with the following transition matrix:
1
2
3
4
1
2
3
4
0 2/3 0 1/3
1/3 0 2/3
526 Stochastic Processes, Fall 2015, Sections 1 and 2.
Homework 1
Due: Thursday, Oct 1, 2015, NO LATER than 3:30pm.
The maximum number of points you can receive for this homework is 12.
The exercises
526 Stochastic Processes, Fall 2016
Homework 3
Due: Wednesday, Nov 9, 2016 in class
1. (15 pts) Assume that a company stock price (Xn ) is a random walk reflected at 1 and
at 4: i.e. it is an MC, with
526 Stochastic Processes, Winter 2015.
Problem Set 3
Due: Thursday, March 12, 2015, NO LATER than 1:00pm.
The maximum number of points you can receive for this homework is 12.
1. (3 pts) Excited by th
526 Stochastic Processes, Fall 2016
Homework 1
Due: Monday, Oct 3, 2016 in class
1. (20 pts) A taxicab driver moves between airport A and two hotels B and C according
to the following rules. If he is
526 Stochastic Processes, Winter 2015.
Homework 2
The maximum number of points you can receive for this homework is 12.
Solutions
From the Essentials of Stochastic Processes, 2nd ed. (Durrett):
(2pt
Z
=6
e5v
0
Thus, we conclude
ET =
e2v
1
2
4
dv = 1
3
7
=
10
10
59
1 1 1 1 7
+ +
=
hours
3 2 5 2 10
60
(3 pts)
2. (3 pts) Suppose that the number of calls per hour to an answering service follows a Po
526 Stochastic Processes, Winter 2015.
Solution to Homework 4
The maximum number of points you can receive for this homework is 14.
1. (5 pts) Customers arrive at a full-service gas station, with two
Solution to Midterm Exam
526-W15
To achieve the total contribution of the midterm exam to the final grade, i.e. 20%, it is
sufficient to get 20, out of 26, points for this exam. If you score more than
526 Stochastic Processes, Winter 2015.
Problem Set 5
Due: Tuesday, April 14, 2015, NO LATER than 1:00pm.
The maximum number of points you can receive for this homework is 12.
1. (1 pt) Consider a Bran
526 Stochastic Processes, Winter 2015.
Solution to Homework 5
The maximum number of points you can receive for this homework is 12.
1. (1 pt) Consider a Branching process (discussed in the lectures),
Midterm-Math 526, Fall 2016
Name:
This is a closed book exam.
Please write all your solutions in this exam booklet (front and back of the page if necessary).
Keep your explanations concise (as time is
526 Stochastic Processes, Winter 2017, Sections 1 and 2.
Solution to Homework 2
1. (3 pts) Consider 8 8 chess board. A bishop can move any number of squares diagonally. Let (Xn ) be the
sequence of sq
526 Stochastic Processes, Fall 2015.
Suggested problems on Limit Behavior of Markov Chains and Detailed Balance Condition. This is for
practicing ONLY, it is NOT a part of your assignment and will not
Practice Quiz 1
Math 526 - Winter 2017
1. In a retail shop, they check the warehouse each morning at 8am. If they are out of
a specific product, e.g. product A, they order three and receive them next
Quiz 2 Solution
Math 526 Sec 1 - Fall 17
Name:
1. [3 pts] Let (Xn ) be an irreducible Markov chain with a finite state space. Show that
if there exists a state with the period strictly greater than 1,
Midterm Exam
Name
526-W17
uniquename
You are obliged to comply with the Honor Code of LSA. After you have completed the examination,
please sign the Honor Pledge below. A test where the signed honor p
Solution to Practice Final exam
526-W17
1. A bus station has taxis and buses waiting for passengers. Groups of passengers arrive according to a
Poisson process with rate 100 groups per hour. The size
Solution to Quiz 1
526.2-W17
1. Let Xn be the number of days since David last shaved, calculated on the morning of
day n, when he is deciding whether to shave or not (i.e. Xn 1). For example, if David
526 Stochastic Processes, Winter 2017, Sections 1 and 2.
Solution to Homework 4
The maximum amount of points you can receive for this homework is 16.
1. (2 pts) The following problem arises in molecul
526 Stochastic Processes, Winter 2017.
Solution to Homework 5
The maximum number of points you can receive for this homework is 12.
1. (Branching process) (1 pt) Consider a discrete-time Branching pro
526 Stochastic Processes, Winter 2017, Sections 1 and 2.
Homework 4, due by 3:30pm on Apr 5
The maximum amount of points you can receive for this homework is 16.
1. (2 pts) The following problem arise
526 Stochastic Processes, Winter 2017.
Lecturer: Sergey Nadtochiy.
Lecture 7. Markov chains: exit probabilities and expected exit times.
1
Exit probabilities
In this section, we will develop an algori
526 Stochastic Processes, Winter 2017, Sections 1 and 2.
Solution to Homework 1
1. (2 pts) In unprofitable times corporations sometimes suspend dividend payments. Suppose that, after a dividend
has be
Solution to Midterm Exam
526-W17
1. Consider a machine installed in a factory. On any given day, the machine fails with probability 0.4 if it
has never failed before, and with probability 0.7 if it ha
Solution to Quiz 4.
526-W17
1. Consider a sequence of i.i.d. r.v.s cfw_i
i=0 , taking values 1 and 1 with probability 1/2
each. Consider a process (Xn )n=0 , defined as follows:
X0 = 0,
Xn =
n
X
i i1
Midterm Exam
Name
526-F16
uniquename
Good luck
1. A car rental company has one Toyota in its office, which can be rented at the beginning of each day
for 1 or 2 days. The Toyota is always returned imm
526 Stochastic Processes Fall 2017
Homework 2 Solution
Due on Thursday, October 12, at the beginning of class. 10 points for each problem.
1. Let X be a Markov chain with state space S = cfw_a, b and
Math 526 - HW 1 Solution
Fall 2017
HW 1 Solution
Question 1
Consider a Markov chain cfw_Xn with state space S = cfw_0, 1, 2 and transition matrix
1 1
0 2 2
P = 12 21 0 .
1 0 0
Let the mapping f : S
526 Stochastic Processes Fall 2017
Homework 1
Due on Thursday, September 28, at the beginning of class. 10 points for each problem.
1. Consider a Markov chain cfw_Xn with state space S = cfw_0, 1, 2
526 Stochastic Processes Fall 2017
Homework 2
Due on Thursday, October 12, at the beginning of class. 10 points for each problem.
1. Let X be a Markov chain with state space S = cfw_a, b and transitio