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 wi
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 inde
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
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
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
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 a
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. Le
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
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 = 0.6
IEOR 4106, HMWK 7, Professor Sigman
1. Cars arrive to a beach parking area (huge, assume of unlimited size) according to a Poisson
process at rate = 20 per hour. The length of time a car spends parked
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
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
IEOR 4106, HMWK 3, Professor Sigman
1. Each of the following transition matrices is for an irreducible Markov chain (from Problem
1 of HMWK 2). For each one, since the state space is nite, the chain m
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
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
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.
Probabilit
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
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
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. O
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 Ros
IEOR 4106, HMWK 3, Professor Sigman
1. Each of the following transition matrices is for an irreducible Markov chain (from Problem
1 of HMWK 2). For each one, since the state space is nite, the chain m
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
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
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 Ros
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
IEOR4106 Intro to OR  Stoch Models
David D. Yao
Practice Final Examination
(180 minutes)
All problems are equally weighted.
1. Consider the M/M/1 in steady state; let and denote the arrival and servi
Assignment #9 Solutions
April 11, 2017
Page 1 of 2
IEOR 4106
Intro to OR: Stochastic Models
Prof. David Yao
Assignment #9 Solutions
10.1 B(s) + B(t) = 2B(s) + [B(t) B(s)].
Now 2B(s) is normal with mea
HMWK 7 Solutions
1. Xn = Y1 Y2 Yn where X0 = 1 and the Yi are iid with P (Y = 0.5) = P (Y =
1) = P (Y = 1.5) = 1/3. Argue that Xn converges and find the limit.
SOLUTION: Since E(Y ) = 1, Xn is a nonn