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 weather on day n,
Wn , be R for rain, or S for sun.
(a) I
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 0
0 1/6 0 5/6
2/5 0 3/5 0
Solution
P
To find the statio
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 at the airport, he will be at hotel B next with probabi
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 marked with * are of increased difficulty. You dont nee
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 the recent warm weather Jill and Kelly are doing spring c
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 S = cfw_1, 2, 3, 4 and transition matrix:
1
2
3
4
1
2
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 Poisson process with rate
10 per hour. Suppose that each
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):
(2pts) 1.57
Solution 1: The problem is the same as the gamb
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 pumps, at rate of 30 cars per hour. However,
customers
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 20 points, you
will still only be credited 20 points f
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 Branching process (discussed in the lectures), where at eac
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), where at each time step, each member of the
population
526 Stochastic Processes, Fall 2015.
Suggested problems on Stationary Distributions of Markov Chains. This is for practicing ONLY, it is NOT a
part of your assignment and will not be graded. Hwks and quizzes may contain similar problems.
From the Essentia
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 algorithm for computing the probability that a MC reaches one
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 been paid, the next quarter dividend will be paid with pr
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 has failed once before. If the machine fails the first
ti
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 squares that results if we pick one of bishops legal move
526 Stochastic Processes, Winter 2017.
Suggested Problems on the Construction of Markov chains and transition probabilities.
This is for practicing ONLY, it is NOT a part of your assignment and will not be graded. Hwks, quizzes, and
exams may contain simi
Solution to Quiz 2
526-W17,
Section 1
1. Consider a Markov chain with the state space cfw_0, . . . , 5, and with the transition probabilities given by:
p(0, 0) = 1/4,
p(1, 1) = 1/4,
p(m, m + 1) = 3/4,
m = 0, 1, 2, 3,
p(m, m 1) = 1/4,
m = 2, 3, 4, 5,
p(5,
Chapter 12
Multiple Regression
Learn.
To use Multiple Regression
Analysis to predict a response
variable using more than one
explanatory variable.
Agresti/Franklin Statistics, 1 of 141
Section 12.1
How Can We Use Several
Variables to Predict a
Response?
Fall 2016 MATH 526
Hw 5
Show all your work for full credit. Due: December 8, in class. All four questions are of the same value.
1. In discrete time, assuming the interest rate is zero (which is what we currently observe), a very general model for
the sto
526 Stochastic Processes, Winter 2017.
Lecturer: Alon Nishry.
Lecture 2. Markov chains (cont.). Markov chains: multistep transition probabilities
1
Markov chains (cont.)
Ex 1. (Inventory chain) Suppose an electronics store sells a video game system. The d
526 Stochastic Processes, Winter 2017.
Lecturer: Alon Nishry.
Lecture 5. Markov chains: limit behavior.
1
Limit behavior
This section is concerned with the limiting behavior of the distribution of a MC over long time horizons: (pn (x, y)x,yS ,
as n .
The
526 Stochastic Processes, Winter 2017.
Lecturer: Alon Nishry.
Lecture 4. Markov chains: classification of states (continued).
1
Classification of states
It turns out that the type of each state y is determined by yy = Py (Ty < ). There are two possibiliti
526 Stochastic Processes, Winter 2017.
Lecturer: Alon Nishry.
Lecture 5. Markov chains: stationary distributions.
1
Stationary distributions
Recall that we already have some understanding of the limiting behavior of MCs: at least, if the state space is fi
526 Stochastic Processes, Winter 2017.
Solution to some suggested problems on Limit Behavior of Markov Chains and Detailed Balance Condition.
From the Essentials of Stochastic Processes, 2nd ed. (Durrett):
1.15
1.17
1.19
Solution T=transit and C=car
T
526 Stochastic Processes, Winter 2017.
Solution to suggested problems on Markov Chains with Infinite State Spaces. This is for practicing ONLY, it is
NOT a part of your assignment and will not be graded. Hwks, quizzes, and exams may contain similar
proble
526 Stochastic Processes, Winter 2017.
Solution to some suggested problems on Stationary Distributions, Limit Behavior of Markov Chains, Detailed
Balance Condition, Exit Probabilities and Expected Exit Times.
From the Essentials of Stochastic Processes, 2
Solution to Practice Quiz 3
Math 526 - Winter 2017
1. Two people are working in a small office selling shares in a mutual fund. Each is either
on the phone or not. Suppose that salesmen i is on the phone for an exponential amount
of time with rate i , and
Math 526 - 003 Winter 2017 - Quiz #2
Time limit: 15 Minutes
Name: Solution
Date: 2/9/2017
Show an appropriate amount of work for each exercise. Good luck!
Problem 1 (5 points). Consider a Markov chain with state space t0, 1, 2, 3u, and transition probabil
Math 526 - 003 Winter 2017 - Quiz #1
Time limit: 15 Minutes
Name: Solution
Date: 1/26/2017
Show an appropriate amount of work for each exercise. Good luck!
Problem 1 (8 points). A cab driver moves between the airport, two hotels, A and B, and his home,
ac
Math 286 Homework 10
Due: 4/10
Complete the following problems. Be sure to show your work. Correct answers with no explanations will
receive little to no credit.
Book problems
Problem 1. B+D 7.4.8
Problem 2. B+D 7.4.9
Problem 3. B+D 7.5.17
Problem 4. B+D
Math 286 Homework 9
Due: 3/24
Complete the following problems. Be sure to show your work. Correct answers with no explanations will
receive little to no credit.
B+D refers to Boyce and DiPrima
M refers to K.R. Matthews online linear algebra notes <http: