MATH 5652. Introduction to Stochastic Processes. Fall 2015.
Homework Assignment 10.
Solution.
Assigned 12/03/2015. Due 12/10/2015.
Total is 10 points.
1. (2 points) Two people are working in a small oce selling shares in a mutual
fund. Each is either on t
MATH 5652. Introduction to Stochastic Processes. Fall 2015.
Homework Assignment 10.
Assigned 12/03/2015. Due 12/10/2015.
Total is 10 points.
1. (2 points) Two people are working in a small oce selling shares in a mutual
fund. Each is either on the phone o
MATH 5652. Introduction to Stochastic Processes. Fall 2015.
Homework Assignment 9.
Solution.
Assigned 11/12/2015. Due 11/19/2015.
Total is 10 points.
1. (5 points) (Epidemic model.)
Suppose there is a population consisting of N individuals. Each pair of i
MATH 5652. Introduction to Stochastic Processes. Fall 2015.
Homework Assignment 11.
Assigned 12/08/2015. Due 12/15/2015.
Total is 10 points.
1. (4 points) Let (Bt )t0 be a standard Brownian motion. Show that the following
processes are standard Brownian m
MATH 5652. Introduction to Stochastic Processes. Fall 2015.
Homework Assignment 11.
Assigned 12/08/2015. Due 12/15/2015.
Total is 10 points.
1. (4 points) Let (Bt )t0 be a standard Brownian motion. Show that the following
processes are standard Brownian m
MATH 5652. Introduction to Stochastic Processes. Fall 2015.
Homework Assignment 8.
Assigned 11/05/2015. Due 11/12/2015.
Total is 10 points.
1. (5 points)
(1) (1 point) Let exp() and exp() be independent. Show that
exp( + )
and
P( < ) =
.
+
(2) (1 point)
MATH 5652. Introduction to Stochastic Processes. Fall 2015.
Homework Assignment 9.
Assigned 11/12/2015. Due 11/19/2015.
Total is 10 points.
1. (5 points) (Epidemic model.)
Suppose there is a population consisting of N individuals. Each pair of individuals
Notes on Stochastic Processes
Kiyoshi Igusa December 17, 2006
ii
These are lecture notes from a an undergraduate course given at Brandeis University in Fall 2006 using the second edition of Gregory Lawlers book Introduction to Stochastic Processes.
Conten
1. (a) Let N(t) be a Poisson process with rate > 0 and (Yn )n0 be a discrete time Markov
chain with transition matrix u. Compute the Q matrix of Xt = YN(t) for t 0. (b) Consider
the jump rate of a continuous time MC on the state space S = cfw_0, 1 dened b
1. Consider the Gamblers ruin chain in which p(i, i + 1) = p(i, i 1) = 1/2 for 0 < i < N
and p(0, 0) = p(N, N) = 1. Let T = infcfw_n 0 : Xn = 0, N. Find the exit time Ex T for all
0 x N.
2. In a large metropolitian area, communters either deive along (A),
MATH 5652. Introduction to Stochastic Processes. Fall 2015.
Homework Assignment 7.
Solution.
Assigned 10/29/2015. Due 11/05/2015.
Total is 10 points.
1. (1 point) Let 1 , ., k , . be a sequence of i.i.d. exponential random variables
with rate . Show that
MATH 5652. Introduction to Stochastic Processes. Fall 2015.
Homework Assignment 6.
Solution.
Assigned 10/22/2015. Due 10/29/2015.
Total is 10 points.
1. (1 point) Suppose the transition matrix P is periodic with period d > 1 and
1
stationary distribution
Practice problems for midterm 1
MATH 5652 Spring 2016
February 20, 2016
No calculator is needed unless explicitly mentioned in the problem.
1. Identify the recurrent and transient classes for the following Markov chain that has transition matrix
0.3 0.4 0
MATH 5652 - Spring 2017 - Homework 1
Due Tues. Jan. 31
1. Write out the transition
molecules.
0.1 0.4 0.2
0.5 0.1 0.1
1
0
2. Let M =
0
0
0 0.6
0.1 0.2 0.3
matrix for the Ehrenfest Chain with n = 5
0.2 0.1
0.2 0.1
0
0
be the transition matrix for a
MATH 5652 - Spring 2017 - Homeworks 4 and 5
Due Tues. Feb. 21
1. Do Exercise 1.56
2. Do Exercise 1.57. Here is the full set-up of the problem
There is a barber shop with two barbers. There are also two chairs for
customers to wait. When one barber finishe
MATH 5652 - Spring 2017 - Homework 1
Due Tues. Feb. 7
1. Do exercise 1.38. Also answer the following question: If the professor buys
a fourth umbrella, what percent of the time will the professor not have an
umbrella at the current location.
Note: The cur
MATH 5652 - Spring 2017 - Homework 2 Solutions
1. Do exercise 1.38. Also answer the following question: If the professor buys
a fourth umbrella, what percent of the time will the professor not have an
umbrella at the current location.
Note: The current lo
Practice problems for midterm 2
MATH 5652 Spring 2016
April 3, 2016
1. Customers arrive at a service kiosk in a bank as a Poisson process of rate 2. Being impatient, the
customers leave immediately unless the kiosk is free. Customers are served independen
1. Consider the weather chain given in Example 1.3 (page 3) in Durretts book.
(a) Suppose today is sunny. What is the expected number of days to wait for the next sunny day?
(b) In the long run, what is the fraction of days when it is rainy, but the next
Math 5652 (See 001) Spring 2016 Midterm 1 Page 2 of 6 October 13, 2016
1. (26 points) Consider a Markov chain on state space cfw_1, 2, 3, 4, 5, 6 with the following transition
matrix.
0 0 1 0 0 O
0 0 0 0
1 0 0 0 0 0
P * o a o o
0 0 0 0 0.6 0.4
0 0 0
W / wq w
x2;
_ 2 5
+ t/xpk/ i i:
Sh 5+ C:>( X A 3:
(2 + 33x % 55 J E ML
WWWQQ,
A (1Q g
m
M4 7! .5 W
X -;:
X; ; Cum 1/5
"WW ,7 rrrr
fI-lrilbhIvlli'lZ'Iim ;,;,;4;,v ; i ,
M ,
_
L L
%_
L cfw_:1
L
in 1% Lmk an MLL 25;?
MW Cm,
.mm m'nhylywmmamm
aix .
Yr:
Practice problems for midterm 2
MATH 5652 Fall 2016
November 11, 2016
1. Customers arrive at a service kiosk in a bank as a Poisson process of rate . Being impatient,
the customers leave immediately unless the kiosk is free. Customers are served independe