Homework 7, MATH 4410/6410
Due on Friday, April 25 (and will be counted as extra credit)
Problem 1 Ross, page 357 Problem 21.
Problem 2 Ross, page 357 Problem 22.
Problem 3 Ross, page 361 Problem 49.
Problem 4 Ross, page 361 Problem 50.
Problem 5 Ross, pa

Homework 5, MATH 4410/6410
Due on Friday, March 28
Problem 1 Suppose cfw_Xn n0 is a DTMC on the
probability matrix P is given by
0.7 0
0
0.1 0.1 0.1
0.3 0.2 0.4
P =
0
0
0
0.2 0
0
0.1 0
0
state space E = cfw_0, 1, 2, 3, 4, 5, whose transition
0 0.3 0
0

Homework 2, MATH 4410/6410
Due on Wednesday, January 29
Problem 1 Ross, pg. 173 Problem 7
Problem 2 Ross, pg. 174 Problem 15
Problem 3 Ross, pg. 175 Problem 20
Problem 4 Ross, pg. 175 Problem 21
Problem 5 Ross, pg. 177 Problem 27
Problem 6 Ross, pg. 177 P

Homework 4, MATH 4410/6410
Due on Wednesday, March 12
Problem 1 Consider a DTMC cfw_Xn n0 on a state space E = cfw_0, 1, 2, having a transition probability matrix P given by
0 0.2 0.8
P = 0.6 0 0.4
0.5 0.5 0
(a) Write down all communicating classes of th

Homework 1, MATH 4410/6410
Problem 1 Ross, pg. 16, Problem 14
Problem 2 Ross, pg. 17, Problem 21
Problem 3 Ross, pg. 18, Problem 35
Problem 4 Ross, pg. 87, Problem 8
Problem 5 Ross, pg. 87, Problem 12
Problem 6 Ross, pg. 88, Problem 25
Problem 7 Ross, pg.

Simulation Project
Due on the last week of class
This assignment is centered around learning how to simulate random variables and processes on
the computer. You are allowed to make use of whatever programming language/software you wish
to do this assignme

MATH 4410: Take-Home portion of Test 2
Instructions: This portion of the test is open-book, open notes, but you are not allowed to discuss
these problems with one another. These are due by test-time on Friday, April 11, and is worth 50
percent of Test 2.

Review of Probability Theory
We begin our review of probability theory by solving problems, whose solutions involve the use
of various concepts you should have encountered in MATH 4000.
This problem is Problem 22 of Chapter 1 of Introduction to Probabilit

MATH 4410: Take-Home portion of Test 1
Instructions: This portion of the test is open-book, open notes, but you are not allowed to discuss
these problems with one another. These are due by test-time on Wednesday, February 26, and is
worth 50 percent of Te

Comments on the Take-Home
I have received a number of questions from people in the class, so I have decided to make a
number of remarks on each Problem from the take-home, to ensure that everyones on the same
page.
Problem 1 All Im looking for here is for

Discrete-time Markov Chains
1
Introduction
We are now ready to begin our discussion of stochastic processes. The rst type of stochastic process
we will encounter is known as a Discrete-time Markov Chain, which is dened as follows.
Denition 1.1 Suppose cfw

MTHSC 4410 Simulation Project
Spring 2014
Zhongyi Zhou
April 24, 2014
1 Introduction
A discrete random variable can be simulated using a PC by first generating a uniform random
variable U on (0, 1). Since uniform random variable is evenly distributed on i

Comments on a problem in Homework 2
One of the assigned problems in Homework 2 is Problem 20 on pg. 175 of Rosss text, which
reads as follows.
Problem: An individual whose level of exposure to a certain pathogen is x will contract the disease
caused by th

Solutions to Homework 7, MATH 4410
Problem 1 The key to doing this problem is to property dene all exponential random variables
used. Let Xi,j represent the time spent by customer i at server j. All random variables are independent of one another, and all

Solutions to Homework 6, MATH 4410/6410
Problem 1
Throughout this problem, cfw_N (t); t 0 is a Poisson process with rate = 10, where N (t) = the
number of arrivals in [0, t], t being in units of hours. Here time t = 0 corresponds to 6am.
(a) The number of

Homework 6, MATH 4410/6410
Due on Monday, April 7
Problem 1 Suppose vehicles arrive to one of the I-185 toll booths outside of Greenville in accordance
to a Poisson process having rate = 10 cars per hour. Assume we arrive one morning at 6AM, and
we begin

Solutions to Homework 2, MATH 4410/6410
Problem 1
(a) We rst nd E[X | Y = 2]. First, note that since the range of X is the set cfw_1, 2, we have
E[X | Y = 2]
=
(1)P(X = 1 | Y = 2) + (2)P(X = 2 | Y = 2).
However,
P(X = 1 | Y = 2) =
P(X = 1, Y = 2)
P(Y = 2)

Homework 3, MATH 4410/6410
Due on Friday, February 14
Problem 1 Suppose cfw_Xn n0 is a time-homogeneous DTMC on the state space cfw_0, 1, 2, having as
its transition matrix
0.5 0.5 0
P = 0.2 0.3 0.5 .
0.4 0.1 0.5
Moreover, suppose X0 has as its pmf P(X0 =

Solutions to Homework 1, MATH 4410/6410
Problem 1
A person is said to have won if they are the rst to win on a single toss of a die. Players A and
B alternate rolling the die, with A rolling rst. Letting X represent the number of rolls needed to
record a

Copyright c 2007 by Karl Sigman
1
Simulating Markov chains
Many stochastic processes used for the modeling of nancial assets and other systems in engineering are Markovian, and this makes it relatively easy to simulate from them.
Here we present a brief i

Some Examples of LaTeX Commands
Brian Fralix
April 8, 2014
1
Introduction
This le is meant to show you some basic LaTeX commands, to help you write up your project
report based o of Problems 4 and 5. I dont expect you all to be experts in LaTeX, but at so

Solutions to Homework 4, MATH 4410/6410
Problem 1
(a) This chain is irreducible, meaning all states communicate with one another. Hence, the only
communicating class is C0 = cfw_0, 1, 2.
(b) Because the chain is irreducible, and the state space is nite, a