IEOR 4102, HMWK 6, Professor Sigman
Use (on our course Website Lecture Notes 9 (Review of the Exponential distribution) to do
the HMWK here. Pay particular attention to Section 1.4 which introduces the Poisson process.
1. You arrive at the West 96th Stree
IEOR 4102, HMWK 5, Professor Sigman
1. Recall that for a finite state Markov chain with transient states 1, . . . , b we can
compute Si,j = the expected total number of visits of the chain into transient state
j given that X0 = i (transient state) via sol
IEOR 3106 Solutions to Midterm Exam I
1. (40 points) A small business sells professional digital cameras one by one. On any given day, either a camera is delivered to them from the factory (causing the inventory level to go up by 1) with probability
IEOR 4106: Introduction to Operations Research: Stochastic Models SOLUTIONS to Second Midterm Exam, April 2, 2007 Open Book: but only the Ross textbook plus the CTMC lecture notes and one extra 8 11 page of notes
Justify your answers; show your work. Hono
IEOR 3106: Introduction to Operations Research: Stochastic Models
Fall 2014, Professor Whitt
Homework Assignment 3: Tuesday, September 16
Introduction to Discrete-Time Markov Chains
Due on Tuesday, September 23 before class; to be discussed at the
recitat
IEOR 3106: Introduction to Operations Research: Stochastic Models
Fall 2014, Professor Whitt
Numerical Part of Homework Assignment 3: Tuesday, September 16
Markov Chains
Due on Tuesday, Septmber 30, an extra week is given, To be discussed at
the recitatio
IEOR 3106: Introduction to Operations Research: Stochastic Models
Homework Assignment 2: Tuesday, September 9, 2014
Due on Tuesday, September 16 before class; to be discussed at the recitation on Sunday, September 14, 4:00-6:00pm, in 303 Mudd.
More Probab
IEOR 3106, HMWK 7, Professor Sigman
1. You arrive at the West 96th Street Subway station to go Downtown. Suppose that Express
trains arrive according to a Poisson process at rate 3 (per hour), and independent of this,
the Local trains arrive according to
IEOR 4102, HMWK 10, Professor Sigman
1. Program up the Polar Method Algorithm for generating iid pairs of N (0, 1) rvs:
1. Generate U1 , U2 .
2. Set R2 = 2 ln (U1 ), = 2U2 and set R =
3. Set
R2 .
X = R cos
Y = R sin .
4. Stop. Output X, Y .
SOLUTION: Sam
IEOR 4102, HMWK 4, Professor Sigman
1. Each of the following transition matrices is for a Markov chain. For each, find the
communication classes for breaking down the state space, S = C1 C2 and
for each class Ck tell if it is recurrent or transient.
(a)
1
IEOR 4102, HMWK 9, Professor Sigman
1. Consider the CTMC inventory model (Problem 1) from HMWK 8 (repeated here for
completeness):
Inventory model I: A retailer sells headphones one at a time according to demand which
forms a Poisson process at rate : At
IEOR 4102, HMWK 7, Professor Sigman
1. Suppose that the arrival of buy and sell orders in a financial markets firm occur (together)
according to a Poisson processes at rate = 200 (per hour). Suppose further that
independently, each order is a buy order wi
IEOR 4102, HMWK 8, Professor Sigman
1. Inventory model I: A retailer sells headphones one at a time according to demand which
forms a Poisson process at rate : At Poisson arrival time tn (nth demand request), the
inventory drops by 1 if the inventory is n
IEOR 3106: First Midterm Exam, Chapters 1-4, October 3, 2013
SOLUTIONS
1. A Five-Room Maze for Markov Mouse (35 points, 5 points for each part)
Markov Mouse is placed in room 1 of the 5-room maze below and then moves randomly
from room to room through one
IEOR 3106: Introduction to Operations Research: Stochastic Models
Fall 2014, Professor Whitt
Homework Assignment 1: Tuesday, September 2, 2014
Due on Tuesday, September 9 before class; to be discussed at the recitation on Sunday, September 7, 4:00-6:00pm,
IEOR 3106: Introduction to Operations Research: Stochastic Models
Professor Whitt
SOLUTIONS to Homework Assignment 2
More Probability Review: In the Ross textbook, Introduction to Probability Models,
read Sections 3.1-3.5 up to (not including) Example 3.2
IEOR 3106: Second Midterm Exam, Chapters 5-6, November 7, 2013
SOLUTIONS
Honor Code: Students are expected to behave honorably, following the accepted code of
academic honesty. You may keep the exam itself. Solutions will eventually be posted on line.
1.
IEOR 3106, HMWK 1, Professor Sigman
1. An asset price starts off initially at price $7.00 at the end of a day (day 0), and at
the end of each consecutive day, independent of the past, the price goes up by one
dollar (with probability p = 0.6) or down by o
IEOR 3106, HMWK 2, Professor Sigman
1. An asset (stock) starts off at $4.00 per share, and moves day by day as a simple
random walk with p = 0.4 (negative drift). What is the probability that the price
will ever reach a value $7.00 (before drifting off to
IEOR 3106, HMWK 3, Professor Sigman
1. Consider from class lecture, the weather Markov chain Xn = (Wn1 , Wn ) where
Wn cfw_0, 1 (1 = rain, 0 = no rain), and the labeling is given by
0
1
2
3
=
=
=
=
(0, 0)
(0, 1)
(1, 0)
(1, 1);
the state space is thus S =
c 2016 by Karl Sigman
Copyright
1
Gamblers Ruin Problem
Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with
an initial fortune of $i and then on each successive gamble either wins $1 or loses $1
independent of the past with probabil
IEOR 3106: Introduction to Operations Research: Stochastic Models
Fall 2014, Professor Whitt
Homework Assignment 4: Tuesday, September 23.
Due on Tuesday, September 30. To be discussed at the recitation on
Sunday, September 28.
Note that the rst midterm e
IEOR 3106: Introduction to Operations Research: Stochastic Models
Professor Whitt
SOLUTIONS to Homework Assignment 3
Introduction to Discrete-Time Markov Chains
Read Sections 4.1-4.5 in the Ross text, up to, but not including Section 4.5.2, i.e., pages
18
IEOR 3106: Introduction to Operations Research: Stochastic Models
Professor Whitt
Solutions to Homework Assignment 4
Even More Markov Chains
Read Sections 4.5.1, 4.5.2, 4.6 and part of 4.8. In Section 4.8 read pages 237-243 (249-255
in the 10th edition),
IEOR 3106: Introduction to Operations Research: Stochastic Models
Professor Whitt
SOLUTIONS to Homework Assignment 1
Probability Review: Read Chapters 1 and 2 in the textbook, Introduction to Probability Models, tenth edition, by Sheldon Ross. Please do t
IEOR 3106
Intro to OR: Stochastic Models
Prof. Mariana Olvera-Cravioto
Assignment #5
October 14, 2015
Page 1 of 1
Assignment #5 - due Wednesday, October 21st, 2015
1. Random walk on a clock. Consider the numbers 1, 2, . . . , 12 written around a ring as t
Lecture 18
Mariana Olvera-Cravioto
Columbia University
[email protected]
November 16th, 2015
IEOR 3106, Intro to OR: Stochastic Models
Lecture 18
1/17
Transition probabilities
I
I
Let cfw_X(t) : t
0 be a general continuous-time Markov chain.
Dene