ORIE 4580/5580/5581
Homework Assignment 1
Due by Sep. 12, 12:00 pm (noon), in the dropbox located on the second oor of Rhodes Hall.
1. The probability density function of the random variable X is given by
cfw_
3
3
3
x2 19 x + 76 if 2 x 2
19
fX (x) =
0
oth

ORIE 4580/5580/5581 Solutions for Homework Assignment 1
Question 1 a) Note that the largest value Y can take is 4 and the smallest value Y can take is 0. Therefore, 0 if y < 0 FY (y) = Pcfw_Y y = Pcfw_X 2 y = Pcfw_- y X y if 0 y 4 1 if y > 4 To compute th

ORIE 4580/5580/5581 Homework Assignment 2
Due by Sep. 16, 11:00 am, in the drop-box located on the second floor of Rhodes Hall. 1. a) A company selling oil operates a storage tank to serve 20 customer locations dispersed around the country. The monthly de

ORIE 4580/5580/5581 Homework Assignment 4
Due by Sep. 30, 11:00 am, in the drop-box located on the second floor of Rhodes Hall. If you are taking ORIE 5581, then you can turn in this homework assignment by Oct. 3, 11:00 am, in the drop-box located on the

Proof of the Fact that Box-Muller Algorithm Generates Samples
from the Standard Normal Distribution
1. Some Background
In this document, we show that the Box-Muller algorithm indeed generates two independent
samples from the standard normal distribution.

ORIE 4580/5580/5581
Homework Assignment 2
Due by Sep. 10, 2:40pm in class. Please show your work and clearly mark your answers.
Turn in the spreadsheet that you use for your simulation models, including an explanation of
how you used your spreadsheet to o

ORIE 4580/5580/5581 Homework Assignment 1
Due by Sep. 14, 11:00 am, in the drop-box located on the second floor of Rhodes Hall. Please note that this homework deviates from our announced schedule in the syllabus and it is due on a Wednesday. 1. The probab

ORIE 4580/5580/5581 Solutions for Homework Assignment 2
Question 1 a) Let D be the total customer demand in a month. Since sum of independent normally distributed random variables is normally distributed, D N (20 60, 20 202 ) N (1200, 8000) We want to fin

ORIE 4580/5580 Homework Assignment 8 Due by Nov. 4, 11:00 am, in the drop-box located on the second floor of Rhodes Hall.
1. Consider a two-server queueing system, where the customers arriving into the system join a single queue, wait for their turn, rece

ORIE 4580/5580/5581
Homework Assignment 2
Due by Sep. 20, 12:00 pm, in the drop-box located on the second oor of Rhodes Hall.
1. a) A company selling oil operates a storage tank to serve 10 customer locations dispersed
around the country. The monthly dema

Question 3 a) We have, EX 1 / 2 , since X in uniformly distributed over [0,1],
E 1 /(Y 1) 2
1
1 1 1 dy (1 y ) 1 1 / 2 2 0 0 ( y 1)
Then,
1 EZ E X /(1 Y ) 2 EX E 1 /(Y 1) 2 . 4 We also have,
EX
1 2
x3 x 1 dx 3 0
2
1 4
1
1/ 3 ,
0
1
1 E /(Y 1) 1 1 dy 4 3

1. Suppose that f(x) = sin x for 0 x /2.
(a) Sketch this probability density function.
(b) Give an acceptance rejection algorithm for generating samples of a random
variable with probability density function f.
(c) Use your algorithm and the random number

ORIE 4580/5580/5581 Homework Assignment 3
Due by Sep. 23, 11:00 am, in the drop-box located on the second floor of Rhodes Hall. Please show your work and clearly mark your answers. Make sure that your spreadsheet printouts are properly formatted and label

ORIE 4580 / 5580 Simulation Modeling and Analysis
ORIE 5581 Monte Carlo Simulation
Fall 2013
Aug. 27, 2013
Simulation deals with predicting certain aspects of the behavior of some system by creating a n
approximate model of it. Manufacturers use simulatio

ORIE 4580/5580/5581
Solutions for Homework Assignment 3
Question 2.
a) In order to ensure that the LCG has full period, we use Theorem 1 in the lecture notes
Generating Samples of Uniformly Distributed Random Variables. We pick m = 206 .
Since m = 206 = 2

ORIE 4580/5580/5581
Solutions for Homework Assignment 6
Question 1.
a) Choose m = 231 , a = 9 and c = 3. Following Theorem 1 in the course packet, m and c are
relatively prime; the only prime divisor of m is 2 and 2 divides a 1; since m is divisible by
4,

ORIE 4580/5580/5581
Solutions for Homework Assignment 5
Question 1.
a) See the attached spreadsheet.
b) We have to pick M such that
M max f (x).
x[1,3]
From the gure in the accompanying spreadsheet, it is easy to see that picking M = 0.8 would
do the job.

ORIE 4580/5580/5581
Solutions for Homework Assignment 4
Question 1.
a) The c.d.f of X is
x
F (x) =
ea
1
ln(x) a
du =
(b a)u
ba
for ea x eb . We now compute the inverse of F ():
u=
ln(x) a
= (b a) u + a = ln(x) = x = ea e(ba)u .
ba
Thus,
F 1 (u) = ea e(ba)

ORIE 4580/5580
Homework Assignment 7
Due by Oct. 31, 12:00 pm, in the drop-box located on the second oor of Rhodes Hall.
Please show your work and clearly mark your answers. Make sure that your spreadsheet
printouts are properly formatted and labeled. The

ORIE 4580/5580
Homework Assignment 8
Due by Nov. 7, 12:00 pm, in the drop-box located on the second floor of Rhodes Hall.
1. 1. Sit&Relax Co. manufactures customized chairs. The orders for chairs arrive
according to a Poisson process with rate 1/60 minute

ORIE 4580/5580/5581
Homework Assignment 1
Due by Thursday Sep. 1, 2:40 pm in class. Please show your work and clearly mark
your answers. We will separately deduct up to 20% of the potential grade for this
assignment if your work is messy or disorganized.