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
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
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
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
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
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, 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
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, 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
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
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
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
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
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
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
IEOR4106 Intro to OR  Stoch Models
David D. Yao
Sample Final Examination (150 minutes)
All problems are equally weighted. 1. (i) X is a random variable, with the first two moments given; Y is another random variables, with E(Y X) = a + bX, where a and b
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: 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 Assignment 9
Michael Hamilton
April 18, 2015
Problem 1
Qu Consider a singleserver bank for which customers arrive in accordance with a Poisson process
with rate . If a customer will enter the bank only if the server is free when he arrives, and
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 E4106: Intro to OR: Stochastic Model
Recitation 12
April 24th,2015
1. A supermarket has two exponential checkout counters, each operating at rate . Arrivals are
Poisson at rate . The counters operate in the following way:
(i) One queue feeds both cou
IEOR E4106: Intro to OR: Stochastic Model
Recitation 6
March 27th,2015
1. There are two machines, one of which is used as a spare. A working machine will function for
an exponential time with rate and will then fail. Upon failure, it is immediately replac
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana OlveraCravioto
Assignment #8
April 2, 2015
Page 1 of 1
Assignment #8  due Wednesday, April 8th, 2015
1. Suppose that the inter arrival distribution for a renewal process is Poisson with mean . That
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana OlveraCravioto
Assignment #7
March 17, 2015
Page 1 of 2
Assignment #7  due Wednesday, April 1st, 2015
1. Potential customers arrive at a singleserver station in accordance with a Poisson process wi
IEOR 4106, HMWK 6 Solutions, Professor Sigman
1. Suppose that subways leave West 116th Street (to go downtown) according to a renewal
process with iid interarrival times that have the continuous uniform distribution over
the interval (0, 1) hour. Meanwhil
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana OlveraCravioto
Assignment #4
February 17, 2015
Page 1 of 1
Assignment #4  due Wednesday, February 25th, 2015
1. Let Yn be the sum of n independent rolls of a fair die. Find
lim P (Yn is a multiple o
IEOR 4106
Intro to OR: Stochastic Models
Prof. David Yao
Assignment #2 Solutions
September 30, 2016
Page 1 of 5
Assignment #2 Solutions
2.43 (a) X =
Pn
i=1 Xi .
(b) We have that
E [Xi ] = P (Xi = 1) = P (red ball i is chosen before all m black balls) =
1
IEOR 4106
Intro to OR: Stochastic Models
Prof. David Yao
Assignment #5 Solutions
October 11, 2016
Page 1 of 3
Assignment #5 Solutions
4.2 We need 8 states: cfw_(RRR),(RRD),(RDR),(RDD),(DRR),(DRD),(DDR),(DDD). Each state
representing the weather condition
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 cfw_
IEOR 4106
Intro to OR: Stochastic Models
Prof. David Yao
Assignment #3 Solutions
October 11, 2016
Page 1 of 2
Assignment #3 Solutions
5.4 Let Sj and Tj be the service time and departure time, respectively, for j = A, B, C.
(a) 0, since SA = SB = SC = 10,
IEOR 4106
Intro to OR: Stochastic Models
Prof. David Yao
Assignment #1 Solutions
September 23, 2016
Page 1 of 2
Assignment #1 Solutions
1.36 Let B = event marble is black; Bi = event that box i is chosen. Now,
B = (B B1 ) (B B2 )
P (B) = P (B B1 ) + P (B