IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Midterm Solutions 25th October 2006 Page 1 of ?
Midterm Solutions
1. (a) The relevant computations for part (a) are summarized in the following table: Customer 1 2 3 4 5 6 7 Arrival 1 5 8 9 11 14 16 Joins
Columbia University
IEOR 4404: Simulation Fall 2009
Solution to Assignment 2, due on Sep. 24th 1. Use simulation to approximate the following integrals. Compare your estimate with the exact answer if known.
1 2 1 0 terms go from 0 to .]
=
0
(2x x2 )(1 +
IEOR E4404
Solution to Assignment 6
2014 Spring
Problem 1.
(a). If we only consider the behavior of the rst server, its a FCFS single server queue. We have seen
the result for single server queue in class. Actually we consider two cases:
(1)
(1)
(1). An +
IEOR E4404
Solution to Assignment 2
2014 Spring
Prepared by Yanan Pei
1. Recall that a N B(r, p) r.v. has p.m.f given by
p(k) =
k1 r
p (1 p)kr , k = r, r + 1, .
r1
(a) Use the relationship between N B(r, p) and Geom(p), and the relationship between Geom(p
IEOR E4404 Simulation, Fall 2014
September 24, 2014
Assignment 2
Due date: October 8, 2014
Problem 1. Recall that for two random variables X and Y , Cov(X, Y ) = E[XY ] E[X]E[Y ]. Use
Monte Carlo simulation to estimate Cov(U, eU ) where U U nif (0, 1). Co
IEOR 4404
Simulation
Prof. Mariana Olvera-Cravioto
Assignment #1 Solutions
September 16, 2012
Page 1 of 5
Assignment #1 Solutions
1. (a) Dn Vn denotes the time at which the nth customer enters service. The nth may enter
service in one of two ways:
1. The
IEOR E4404 Simulation, Spring 2014
February 10, 2014
Assignment 3
Due date: February 17, 2014
Problem 1.
Use Monte Carlo simulation to numerically approximate the integral
x2
e(x+y) sin(xy)dydx.
0
0
You should attach your codes and the numerical estimates
IEOR E4404 Simulation, Spring 2015
January 21, 2015
Assignment 1
Due date: February 3, 2015
Problem 1. Suppose that there are 20 dierent types of coupons and you wish to collect all of them.
You collect one coupon every day, and it is equally likely for y
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #6 Solutions 24th October 2006 Page 1 of ?
Assignment #6 Solutions
1. The following code asks the user to input the probability mass vector and then generates a value of a random variable havin
IEOR E4404 Simulation, Spring 2014
February 24, 2014
Assignment 5
Due date: March 3, 2014
Problem 1. Let (X, Y ) be uniformly distributed in a circular region of radius 1. Prove that if R is the
distance from the center of the circle to (X, Y ), then R2 i
IEOR E4404 Simulation, Spring 2014
January 27, 2014
Assignment 1
Due date: February 3, 2014
Problem 1.
You have a friend who is a probability enthusiast and who never lies about anything.
(a) She performed two independent ips of a fair coin and told you t
IEOR E4404
Solution to Assignment 5
2014 Spring
Prepared by Yanan Pei
1. Let (X, Y ) be uniformly distributed in a circular region of radius 1. Prove that if R is the distance
from the center of the circle to (X, Y ), then R2 is uniformly distributed on t
IEOR 4404
Simulation
Prof. Mariana Olvera-Cravioto
Assignment #3
September 26, 2012
Page 1 of 2
Assignment #3 due October 3rd, 2012
1. Let X1 , X2 , . . . , Xn denote a sample from a population whose mean value is unknown. Let
1 , 2 , . . . , n be any num
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #8 Solutions 16th November 2006 Page 1 of ?
Assignment #8 Solutions
1. Here, all you have to do is use the algorithm given in class and modify it very slightly: using the notation in Ross (pp 9
IEOR E4404 Simulation, Fall 2014
Start: 8:40am, October 15
Midterm Part II Solutions
Due: 9:55am, October 15
Problem 6 (30 points).
(a) (15 points) On a particular day, buses arrive at the Staples center to drop o sports fans at regular
time intervals of
Simulation Assignment 4 Solutions
Problem 1
The task is to simulate f (x, y, z) = K exp x2
the acceptance/rejection method.
Consider g(x, y, z) = g1 (x)g2 (y)g3 (z), where
z 2 + sin (xy) z and estimate E [X]. We can do this using
y2
g1 (x)
g2 (y)
=
g3 (z)
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Solutions to the Practice Midterm Exam March 3, 2012 Page 1 of 11
Solutions to the Practice Midterm Exam
Place all answers on the question sheet provided. The exam is open book/notes/handouts/homework. Yo
IEOR E4404
Solution to Assignment 3
2014 Spring
1. Observe that e(x+y) is the joint density function of 2 independent Exp(1) random variables.
x2
e(x+y) sin(xy)dxdy =
0
0
e(x+y) sin(xy)1cfw_y x2 dxdy
0
0
= E[sin(XY )1cfw_Y X 2 ]
While X, Y are 2 independe
IEOR 4404: The Inverse Transform Method
1
Introduction
Previously we saw how to generate pseudorandom numbers between 0 and 1. Although the sequence of
numbers generated by any procedure is deterministic, they look random enough for us. The theory of
pseu
IEOR E4404 Simulation, Spring 2014
March 12, 2014
Assignment 6
Due date: March 26, 2014
Problem 1. (Two queues in series) In this problem, we are interested in simulating a system with
two servers in series (sometimes called a tandem system).
The system c
IEOR 4404: Solutions to Homework 1
Do the problems at the end (Page 15) of posted Lecture Notes Review of probability theory
(BUT DO NOT HAND THAT IN, SOLUTIONS ARE POSTED) and also do the following
(AND HAND IN ON Monday, JANUARY 28 in class):
Let U be u
IEOR E4404 Simulation, Spring 2014
Midterm Solution
Due:
Problem 2. (a) We will suppose that n and m are nonnegative real numbers. (Some of you assumed
that n and m are positive integers; that is also okay.) Since the random variable X has density functio
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #2 September 17, 2008 Page 1 of 2
Assignment #2 due September 23rd, 2008
1. Suppose that X and Y are jointly discrete random variables with x + y , for x = 0, 1, 2 and y = 0, 1, 2, 3 30 p(x, y
IEOR 4404 (MS): Homework 8
1. Estimating using Antithetic-Variates Recall that one can estimate by observing that
1
= the area of a disk of radius 1 (cfw_(x, y) : x2 + y 2 1); /4 = 0 1 x2 dx =
E( 1 U 2 ). So Monte Carlo can be used to estimate by generat
IEOR 4404 (MS): Homework 10
1. Control Variates: Consider a GI/GI/1 FIFO queue in which interarrival times cfw_Tn are
iid distributed as U nif (2, 6), and service times cfw_Sn are iid distributed as P (S x) =
1 e x , x 0. ( Thus S has a Weibull distribu
IEOR E4404 Simulation, Spring 2014
March 26, 2014
Assignment 7
Due date: April 2, 2014
Problem 1. (Another Insurance Risk Model) We saw a fairly simple insurance risk model in
class. In this problem, we will explore a more complicated (and perhaps more re
Simulation Assignment 7 Solutions
a. Ik is a random variable representing whether policy holder k stays with the company or leaves. Since the time
that a policy holder stays with the company is geometric with parameter , the probability of them leaving
at
IEOR 4404: Homework 2
U1 , U2 , . . . denotes a sequence of independent identically distributed (iid) uniform (0, 1) rvs U .
1/3
1. Suppose F (x) = 1 ex , x 0, an example of a Weibull distribution. Find F 1 (y )
and give the inverse transform algorithm fo