IEOR E4404 Simulation, Fall 2013
Assignment 1
Due on September 18th (Wednesday) in class
1. If X and Y are independent Binomial random variables with respective parameters (n, p) and
(m, p), argue, without any calculation, that X + Y is Binomial with para
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #4 Solutions March 4, 2010 Page 1 of 4
Assignment #4 Solutions
1. First note that an immediate algorithm is to generate an exponential random variable X with rate 1, and return X whenever X < 0
IEOR E4404.001 SIMULATION Prof. Mariana Olvera-Cravioto
Assignment #3 Solutions
1. The following MATLAB code computes 95% approximate condence intervals for the expected number dice rolls that are needed: N = 1000; counts = []; for n=1:N outcomes = zeros(
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #5 February 24, 2010 Page 1 of 1
Assignment #5 due March 5th, 2010
1. Let Y1 , Y2 , . . . be a sequence of iid exponential random variables with parameter . Dene the random variable N = supcfw_
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #12 April 21, 2010 Page 1 of 3
Assignment #12 due April 30th, 2010
1. Download the text le Data1.txt and import it into MATLAB by typing > Data = csvread(Data1.txt) The vector Data is a sample
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #6 March 3, 2010 Page 1 of 1
Assignment #6 due March 12th, 2010
1. Suppose in the insurance risk model presented in Lecture 11 that, conditional on the event that the rms capital goes negative
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #10 April 7, 2010 Page 1 of 2
Assignment #10 due April 16th, 2010
1. In certain situations a random variable X , whose mean is known, is simulated so as to obtain an estimate of P (X a) for a g
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #3 February 14, 2010 Page 1 of 2
Assignment #3 due February 19th, 2010
1. A pair of dice are to be continually rolled until all the possible outcomes 2, 3,. . . , 12 have occurred at least once
IEOR E4404.001 SIMULATION Prof. Mariana Olvera-Cravioto
Assignment #6 Solutions
1. Here, all you have to do is use the algorithm given in class and modify it very slightly: using the same notations as the lecture notes , we need to output the value tE whe
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #11 April 14, 2010 Page 1 of 1
Assignment #11 due April 23th, 2010
1. Consider a single server queue where customers arrive according to a Poisson process with rate 2 per minute and the service
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #10 Solutions April 16, 2010 Page 1 of 5
Assignment #10 Solutions
1. (a) Let X be a uniform random variable, over [0, 1]. Then, it is well known that E [X ] = 1/2 and V ar[X ] = 1/12. Instead o
IEOR E4404
Solution to Assignment 8
2013 Fall
1. Suppose that you wish to compute = E(X ) via simulation. You can simulate random variables
Y and Z such jointly with X (that is you can simulate (X, Y, Z ). Suppose that all of these are correlated.
Moreove
IEOR 4404 Simulation
Practice Final Exam Solution
Prepared by: Zhen Qiu
1. (40 points) Given the goal of computing
= Ef (X )
for some function f (), formulate and re-derive the selection of the optimal control variate
parameter, , for the estimator of th
IEOR E4404
Solution to Assignment 5
2013 Fall
1. Starting with N = 50 to obtain an approximate copy of the Asian payo; denote this by Y1 .
Repeat this (independently) n = 1000 times and average to get the approximation for th eoption price
as
n
1
C0 =
Yj
1
IEOR 4404: Assignment 8
Instructor: Jose Blanchet
Fall 2013
Due date: 12/09/13
1.- Suppose that you wish to compute = E (X ) via simulation. You can simulate random variables
Y and Z such jointly with X (that is you can simulate (X; Y; Z ). Suppose that
1
IEOR 4404: Assignment 5
Instructor: Jose Blanchet
Fall 2013
Due date: 11/20/13
1.- Suppose that P (X > 0) = 1 and consider the problem of estimating = E (X ) via simulation. For
simplicity assume that V ar (X ) = 2. Use the CLT (Central Limit Theorem) t
IEOR E4404 Simulation, Fall 2013
Assignment 4
Due on October 16th (Wednesday) in class
1. Provide a procedure that clearly explains how to sample uniformly from the region
A=
1 x 1 : |y |
1
|x|1/4
as depicted in the following picture
Notice that the pict
IEOR E4404.001 SIMULATION Prof. Mariana Olvera-Cravioto
Assignment #11 Solutions
1. (a) Let X = T1 + . + T10 (the raw estimator for ). For n = 1000 simulation runs the following values were obtained X (x) = 35.2562 and S 2 (n) = 318.1027 (b) For the antit
IEOR E4404.001 SIMULATION Prof. Mariana Olvera-Cravioto
Assignment #12 Solutions
1. We run the K-S test on several distributions: Gamma: a = 3.26, b = 0.40, p = 0.73. There is not enough evidence to reject the null hypothesis. Lognormal: = 0.11, = 0.61, p
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #2 February 4, 2010 Page 1 of 2
Assignment #2 due February 12th, 2010
1. For any random variables X1 , X2 and any numbers c1 , c2 , show that var(c1 X1 + c2 X2 ) = c2 var(X1 ) + 2c1 c2 cov(X1 ,
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #7 March 11, 2010 Page 1 of 2
Assignment #7 due March 26th, 2010
1. Messages arrive at a communication facility in accordance with a Poisson process having a rate of 2/hour. The facility consis
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #7 Solutions March 28, 2010 Page 1 of 7
Assignment #7 Solutions
1. Solution: (a) events: Arrivals to the facility Departures from channels variables: time variable: t counter variable: NL = num
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #8 March 24, 2010 Page 1 of 2
Assignment #8 due April 2nd, 2010
1. To estimate a certain parameter , we generated 20 independent vales of a random variable having mean . If the successive value
IEOR E4404.001 SIMULATION Prof. Mariana Olvera-Cravioto
Assignment #8 Solutions March 24, 2010
Assignment #8 Solutions
1. We are trying to reach an absolute error of 0.5 with 99
S 2 (20) = 282.24 z10.99/2 = 2.5758 m z10.99/2 S 2 (n)
2
7490 2. Just subst
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #9 March 31, 2010 Page 1 of 2
Assignment #9 due April 9th, 2010
1. For the system in Problem 5 from Assignment #8, make one replication of length 200 days and let Mi be as previously dened. Use
IEOR E4404.001 SIMULATION Prof. Mariana Olvera-Cravioto
Assignment #5 Solutions February 19, 2010
Assignment #5 Solutions
1.
P (N = n) = P (Sn 1, Sn+1 > 1)
=
0 1
P (SN 1, Sn+1 > 1|Sn = y )Gamma(n, )dy P (Xn+1 > 1 y )Gamma(n, )dy
0 1
= = =
0 n e
e(1y) y n
2
IEOR 4404, Assignment #5 Solutions
3. The variables needed are: (1) the entering time of each customer, denoted by E (i) for the ith customer. (2) the tolerance time length of each customer, denoted by R(i) for the ith customer. (3) the leaving time of
IEOR 4404 Simulation Prof. Mariana Olvera-Cravioto
Assignment #4 February 20, 2010 Page 1 of 2
Assignment #4 due February 26th, 2010
1. Let X be an exponential random variable with mean 1. Give an ecient algorithm for simulating a random variable whose di
IEOR E4404.001 SIMULATION Prof. Mariana Olvera-Cravioto
Assignment #9 Solutions
1. We modify the above code as follows: T=4800; %time length (hours), changed to 200 days SS=1; 0umber of replications, changed to 1 Then we get a new Y whose dimension is T=4