IEOR E4106: Intro to OR: Stochastic Model
Homework 6
Solution
1. (a) By the property of memoryless, given that Y is greater than X, no matter where Y falls
in the remaining life of X is exponential with rate . Hence W and Z are independent.
(b)
E[min(X, Y
IEOR 4106 Midterm Exam. Open text book and class notes; 1.5 hours. 100 Points total 1. (35 points) Voice messages are made from a cell phone according to a Poisson process at rate 8 per hour, and independent of this, text messages are sent from the p
Copyright c 2013 by Karl Sigman
1
Inventory Models
1.1
Classic (s, S ) policy model
A company sells a product (items) which it keeps in inventory (a warehouse for example).
Customers make requests one at a time according to Poisson process at rate . Each
Introduction to Operations Research: Stochastic Modeling
IEOR 4106

Fall 2010
HMWK 6 Solutions
1. A stock has an initial price of S0 = 40. Sn denotes the price at time t = n, where we assume the binomial lattice model with parameters u = 1.25 d = 0.8 p = 0.60. The interest rate is r = 0.05. (Note that ud = du = 1.) (a) Compute E (S
IEOR4106 Intro to OR  Stoch Models
David D. Yao
Practice Midterm Examination 150 minutes All problems are equally weighted.
1. The joint density of X and Y is f (x, y) = 1 x/y y e e , y 0 < x < , 0 < y < .
Derive E(X), Var(X), and Cov(X, Y ). 2. Let cf
IEOR4106 Intro to OR  Stoch Models
David D. Yao
Practice Final Examination (150 minutes) All problems are equally weighted.
1. Let Sn be the number of points after rolling an unbiased die n times. Let Xn = Sn mod k where k = 6 (i.e., Xn is the remainder
IEOR 4106, Solutions to HMWK 11, Professor Sigman
1. Let cfw_B (t) : t 0 denote standard BM.
(a) Show that cfw_B (t) : t 0 is also a standard BM.
SOLUTION:
We simply must check that the sample paths are continuous (true since if a function
g (t) is contin
1. Let X and Y be continuous random variables with joint probability dens1ty Iunctlon
_ 3(x2+y2)/2 if0<:1:<1,0<y<1;
f ($:y){ 0 otherwise.
Find E(XY).
c 2.
why»
3 : ______,________
lexlH ) + 33;
\ aww)
ww so « a
I)
r\
+\*
+
w
L:
p 2. Taxi dri
IEOR4106 Intro to OR  Stoch Models
David D. Yao
Practice Midterm Examination 150 minutes All problems are equally weighted.
1. The joint density of X and Y is f (x, y) = 1 x/y y e e , y 0 < x < , 0 < y < .
Derive E(X), Var(X), and Cov(X, Y ). 2. Let cf
IEOR 4106, Midterm Exam Fall 2013SOLUTIONS. Professor
Sigman: 1 hour and 15 minutes
1. (60 points) The amount of money in reserve of an insurance risk business (in billions of
dollars) moves year by year according to a Markov chain cfw_Xn with state spa
Introduction to Operations Research: Stochastic Models
IEOR 4106

Spring 2011
IEOR 4106: Introduction to Operations Research: Stochastic Models
Spring 2011, Professor Whitt
Homework Assignment 1: Tuesday, January 18, 2011
Due on Tuesday, January 25 at (before) class.
Probability Review: Read Chapters 1 and 2 in the textbook, Introd
IEOR 4106, HMWK 2 SOLUTIONS, Professor Sigman
1. Consider the Rat in open Maze with 4 rooms (outside = state 0) From HMWK
1, Problem 1; the transition matrix P = (Pij ) is given below. S = cfw_0, 1, 2, 3, 4.
Given that X0 = 1, compute the probability that
IEOR 4106, HMWK 4, Professor Sigman
1. Martingale MC: Consider the MC with state space S = cfw_0, 1, 2, 3, 4 and transition matrix
1
0
0
0
0
1/2
0
1 /2
0
0
0
3 /5
0
1/5 1/5 .
P =
1/10 1/10 1/10 1/10 6/10
0
0
0
0
1
(a) Show that E (Xn+1 Xn = i) = i, i
Theoretical Economics 8 (2013), 365403
15557561/20130365
Achievable outcomes of dynamic contribution games
S teven A. M atthews
Department of Economics, University of Pennsylvania
This paper concerns multistage games, with and without discounting, in whi
IEOR E4106: Intro to OR: Stochastic Model
Recitation 4
Feb 20,2015
1. A company requires N employees to function properly. If an employee becomes sick then he
or she is replaced by a new one. It takes a week for a new employee to be recruited and to
start
IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2004, Professor Whitt First Midterm Exam: Thursday, February 19 Chapters 14 in Ross, SOLUTIONS
Justify your answers; show your work. 1. Satisfaction Survey (25 points) In its n
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana OlveraCravioto
Assignment #1
January 27, 2015
Page 1 of 1
Assignment #1  due Wednesday, February 4th, 2015
1. There are three coins in a box. One is a twoheaded coin, another is a fair coin, and th
Introduction to Operations Research: Stochastic Models
IEOR 4106

Spring 2009
IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2005, Professor Whitt, Second Midterm Exam Chapters 56 in Ross, Thursday, March 31, 11:00am1:00pm Open Book: but only the Ross textbook plus one 8 11 page of notes
Justify your ans
IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2005, Professor Whitt, Second Midterm Exam Chapters 56 in Ross, Thursday, March 31, 11:00am1:00pm Open Book: but only the Ross textbook plus one 8 11 page of notes
Justify yo
IEOR 4106, SOLUTIONS to HMWK 8, Professor Sigman
1. Consider 5 iPhones, each independently having a battery lifetime that is exponentially
distributed with mean 1 year. Once a battery breaks down, the iPhone immediately goes
to a facility to have the batt
IEOR 4106, HMWK 1, Professor Sigman
1. Consider the Rat in the Open Maze; 4 rooms, and the outside (state 0), but now the
probabilities are dierent whenever the rat leaves room 2: P2,1 = 3/4, P2,4 = 1/4;
all the other probabilities are equally likely as b
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana OlveraCravioto
Assignment #11
April 24, 2015
Page 1 of 2
Assignment #11  due Monday, May 4th, 2015
1. A group of m customers frequents a singleserver station in the following manner. When
a custome
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana OlveraCravioto
Assignment #2
February 3, 2015
Page 1 of 1
Assignment #2  due Wednesday, February 11th, 2015
1. A manuscript is sent to a typing rm consisting of typists A, B and C. If it is typed by
IEOR 4106, HMWK 1, Professor Sigman
1. Consider the Rat in the Open Maze; 4 rooms, and the outside (state 0), but now the
probabilities are dierent whenever the rat leaves room 2: P2,1 = 3/4, P2,4 = 1/4;
all the other probabilities are equally likely as b
IEOR 4106, HMWK 4, Professor Sigman
1. Martingale MC: Consider the MC with state space S = cfw_0, 1, 2, 3, 4 and transition matrix
1
0
0
0
0
1/2
0
1 /2
0
0
0
3 /5
0
1/5 1/5 .
P =
1/10 1/10 1/10 1/10 6/10
0
0
0
0
1
(a) Show that E (Xn+1 Xn = i) = i, i
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana OlveraCravioto
Assignment #9
April 9, 2015
Page 1 of 2
Assignment #9  due Wednesday, April 15th, 2015
1. Consider a singleserver bank for which customers arrive in accordance with a Poisson process
IEOR 4106, HMWK 5, Professor Sigman
1. A stock has an initial price of S0 = 40. Sn denotes the price at time t = n, where we
assume the binomial lattice model with parameters
u = 1.25
d = 0 .8
p = 0.60.
(a) Compute E (S1 ) and E (S2 ).
SOLUTION: Note that
Copyright c 2013 by Karl Sigman
1
Rare event simulation and importance sampling
Suppose we wish to use Monte Carlo simulation to estimate a probability p = P (A) when the
event A is rare (e.g., when p is very small). An example would be p = P (Dk > b) wit
c 2017 by Karl Sigman
Copyright
1
Some basic renewal theory: The Renewal Reward Theorem
Here, we will present some basic results in renewal theory such as the elementary renewal
theorem, and then the very useful Renewal Reward Theorem (RRT). As we shall
IEOR 4102, HMWK 7, 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 4102, HMWK 6, Professor Sigman
1. Cars are rented at times that form a Poisson process at rate = 150 per day. Each rental
keeps the car independently for an amount of time (days) that are iid with a continuous
uniform distribution over the interval (
IEOR 4102, HMWK 5, 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 4 (per hour), and independent of this,
the Local trains arrive according to
IEOR 4102, Midterm Exam, Spring 2016. 75 Minutes.
110 Points Total. Professor K. Sigman
Open Notes (anything on the course website plus your notes from class), but no books and
no electronic devices of any kind.
1. (40 points, 10 each) You arrive at the W