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 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
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 (MS): Homework 5
1. In the Acceptance-Rejection algorithm that we derived for generating from |Z|, where
Z N (0, 1), prove that when using g(x) = ex , x 0, the optimal value of is = 1
as we in fact used. In other words prove that the value of c
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)
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 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, 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 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 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 4404 (MS): Homework 2
U1 , U2 , . . . denotes a sequence of independent identically distributed (iid) uniform (0, 1) rvs U .
1. For > 0 xed, suppose P (X x) = F (x) = 1 e x , x 0, an example of a Weibull
distribution. Find F 1 (y) and give the invers
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 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
Columbia University
IEOR 4404: Simulation Spring 2009
Solution to Assignment 3 1. Exercise 3 Solution: function SimulateDiscrete(p) % SimulateDiscrete(p) generates a discrete random variable with % probability mass function given by the vector p p=[0.3,0.
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Practice Midterm Exam February 27, 2012 Page 1 of 11
Practice Midterm Exam
Place all answers on the question sheet provided. The exam is open book/notes/handouts/homework. You are allowed to use a calcula
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
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
c 2012 by Karl Sigman
Copyright
1
Review of Probability
Random variables are denoted by X, Y , Z, etc. The cumulative distribution function (c.d.f.)
of a random variable X is denoted by F (x) = P (X x), < x < , and if the random
variable is continuous th
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 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 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 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 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 E4404 Simulation, Spring 2014
April 9, 2014
Assignment 9
Due date: April 16, 2014
Problem 1. (Chebychevs Inequality)
We will prove Chebychevs inequality in this problem.
(a) (Markovs inequality) Let Y be a nonnegative random variable with nite mean.
IEOR 4404 (MS): Homework 4
1. Consider the binomial lattice model in which S0 = 50 (dollars per share), u = 1.2,
d = .90, and p = 0.75. Consider a barrier call option in which the payo at time T = 10
is (S10 60)+ as long as the price Sn never falls below
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #4 Solutions February 18, 2012 Page 1 of 6
Assignment #4 Solutions
1. The following MATLAB code computes 95% approximate confidence intervals for the expected number dice rolls that are needed:
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
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