1
ISyE 6644 Spring 2012
Homework #5 Solutions
1. Do some of the following random number generation problems from Law, pp. 419
421: #7.1, 7.2(a), 7.3(d), 7.11, 7.12, 7.13.
Problem 7.1: If Xi+1 = (5Xi + 3)mod(16) nd X500 .
Solution: X0 = 7. Then X1 = 38 mod
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ISyE 6644 Spring 2012
Homework #6 Solutions
1. Suppose the c.d.f. of X is F (x) = x4 /16, 0 x 2. Develop a generator for X .
Demonstrate with U = 0.54.
Solution: Set F (X ) = U U (0, 1). Then U = X 4 /16, and so X = 2U 1/4 . 2
For U = 0.54, we get X = 1
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ISyE 6644 Fall 2015
Homework #3 Due Tuesday 9/22/15
In what follows, well work with a couple of the Arena models we introduced during lecture.
The models referenced below are from my webpage,
www.isye.gatech.edu/sman/courses/6644/.
1. Arena Baby Model 3
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ISyE 6644 Spring 2012
Homework #2 Solutions
1. Suppose that X is a discrete random variable having probability function
Pr(X = k ) = ck 2 for k = 1, 2, 3. Find c, Pr(X 2), E[X ], and Var(X ).
Solution. Since
1=
3
Pr(X = k ) =
3
ck 2 = 14c,
k=1
k=1
2
2
2
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ISyE 6644 Simulation Spring 2010
Professor: Dave Goldsman; Oce: Groseclose 433; Phone: 404-894-2365; email:
[email protected]; website: www.isye.gatech.edu/sman/courses/6644.
Class Times and Location: This is a Distance Learning course.
Oce Hours: Wheneve
1
ISyE 6644 Fall 2015
Homework #2 Solutions
1. Use the basic Monte Carlo technique from class to integrate
1
exp(x2 /2) dx.
I =
0
(a) Use n = 100 Unif(0,1) random variates to produce your answer. Repeat this
1000 times and make a histogram of the results.
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ISyE 6644 Spring 2012
Homework #3 Solutions
1. Use the basic Monte Carlo technique from class to integrate
I=
1
1
1
expcfw_x2 /2 dx.
2
(a) Use n = 100 Unif(0,1) random variates to produce your answer. Repeat this
50 times and make a histogram of the re
ISyE 6414: Regression Analysis
HW#3 (due in class on Friday, June 07, 2013)
There are 3 questions, and please look at both sides. Total points = 20 + 30 + 30 = 80 pts.
(It is OK to use R or other statistical softwares for this homework.)
1. Employee Salar
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ISyE 6644 Fall 2017 Homework #4
Due 9/26/17
These problems primarily cover Modules 3 and 4.
1. Consider the dierential equation f 0 (x) = (x + 1)f (x) with f (0) = 2. Solve this
approximately via Eulers method using increment h = 0.01 for x 2 [0, 0.20].
A Whirlwind Tour of Computer
Simulation
Dave Goldsman
Georgia Tech
Atlanta, GA, USA
[email protected]
www.isye.gatech.edu/~sman
5/31/16
1
Outline
1.
2.
3.
4.
5.
6.
Intro to Simulation
Some Easy Examples
Generating Randomness
Analyzing Randomness
Some Bigger
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ISyE 6644 Fall 2016
Homework 7 Due Tuesday, 11/22/16
1. Suppose that X1 , X2 , . . . , Xn are i.i.d. Exp(). What is the expected value of the
What is E[1/X]?
Is it unbiased for ?
sample mean, X?
2. Do the following MLE questions from Law, where we ass
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ISyE 6644 Fall 2016
Homework #6 Solutions (revised 12/2/16)
1. Generate and plot 10000 bivariate normal random variates (X, Y ), where both
X and Y are standard normal but such that the correlation between X and Y is 0.9.
2. Lets look at a first-order a
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ISyE 6739 Summer 2017
Homework #1 (Modules 1.11.5) Solutions
1. Suppose we define the following line segments: U = [0, 2], A = [0.5, 1], and B =
[0.5, 1.5]. What are A, A B, A B, A B, and A B?
Solution: First of all, A = A = [0.5, 1].
2
Now, note that A
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ISyE 6644 Fall 2016
Homework 7 Solutions
1. Suppose that X1 , X2 , . . . , Xn are i.i.d. Exp(). What is the expected value of the
What is E[1/X]?
Is it unbiased for ?
sample mean, X?
= E[Xi ] = 1/.
Solution: Obviously, E[X]
2
Now, let f (y) y 1 ey /(
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ISyE 3044 Fall 2016
Homework 7 Due Tuesday, 11/22/16
1. Generate and plot 10000 bivariate normal random variates (X, Y ), where both
X and Y are standard normal but such that the correlation between X and Y is 0.9.
2. (BCNN 8-35.) Suppose that the arriv
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ISyE 6644 Fall 2017 Homework #1
Due 9/5/17
1. (Deterministic Model.) Suppose you throw a rock o a cli having height h0 = 500
feet. Youre a strong bloke, so the initial downward velocity is v0 = 100 feet/sec
(slightly under 70 miles/hr). Further, in this
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ISyE 6644 Fall 2017 Homework #2
Due 9/12/17
These are probability / statistics review problems + a couple of simulation problems.
1. If Pr(A) = Pr(B) = Pr(C) = 0.7, and A, B, and C are independent, find the
probability that exactly one of A, B, and C oc
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ISyE 6644 Fall 2017 Homework #1
Due 9/5/17
1. (Deterministic Model.) Suppose you throw a rock off a cliff having height h0 = 500
feet. Youre a strong bloke, so the initial downward velocity is v0 = 100 feet/sec
(slightly under 70 miles/hr). Further, in
Since I find that I cannot type in the text box given. I just write my comments here.
My name is Haotian Wu. GTID # 903330569.
I wrote my homework by hand, took a picture of it and converted it to a PDF version. Please see the
attached documents.
Thanks f
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ISyE 6644 Fall 2017
Homework #4 Solutions
These problems primarily cover Modules 3 and 4.
1. Consider the differential equation f 0 (x) = (x + 1)f (x) with f (0) = 2. Solve this
approximately via Eulers method using increment h = 0.01 for x [0, 0.20].
S
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ISyE 6644 Fall 2017 Homework #3
Due 9/19/17
These are more probability / statistics review problems.
1. Some TRUE/FALSE questions. The RVs X and Y must be independent if. . .
(a) f (y|x) = fY (y) for all x, y.
(b) Corr(X, Y ) = 0.
(c) f (x, y) = cx, for
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ISyE 6739 Summer 2017
Homework #2 (Covers Modules 1.61.9) Solutions
1. Suppose that 12 clowns are trying to get into a small car that will only accommodate a maximum of 7 clowns. How many possible choices of 7 clowns are there?
!
Solution:
12
= 120.
7
2
9/28/2017
Special Generation Methods for Specic Distributions
Last updated Thursday 28 September 2017 00:20 Eastern
Efcient Sampling Techniques for Specic
Distributions
As we have seen in the rvg-by-inversion section, many common distributions like the No
9/21/2017
Non-Uniform Random Variate Generation
Last updated Tuesday 19 September 2017 15:42 Eastern
Non-uniform Random Variate Generation:
Inversion
Uniform random number generation via a computer is a mature eld with a long history, and methods to
gener
9/21/2017
Needed Probability and Staticstics Tools
Last updated Monday 18 September 2017 23:45 Eastern
Necessary Probability and Statistics Tools
This is a collection of tools we will need as we go through the course. This is NOT an exhaustive list, but i
9/28/2017
Non-Uniform Random Variate Generation
Last updated Tuesday 26 September 2017 07:32 Eastern
Non-uniform Random Variate Generation:
Inversion
Uniform random number generation via a computer is a mature eld with a long history, and methods to
gener
9/21/2017
Preliminary Output Analysis
Last updated Monday 18 September 2017 10:02 Eastern
How to Run a Simulation Analysis.
As we saw in the last lecture, in simulation analyses of stochastic systems, we are interested in estimating some
deterministic pro
10/5/2017
Dependent Models
Last updated Thursday 5 October 2017 14:07 Eastern
Dependent Input Models
We have studied various methods to generate samples from independent one-dimensional random variables.
While these form the bedrock of all simulation mode
Last updated Friday 29 September 2017 01:43 Eastern
Home Work 4: given out Sept 28, due (beginning of class) Oct 5
1. Minion Mania. Lego toys / Disney have added an interesting new twist to the sales strategy of the popular set of
toys they bring out with
10/5/2017
Acceptance / Rejection
Last updated Tuesday 3 October 2017 15:21 Eastern
Non-uniform r.v. Generation: Acceptance /
Rejection Approach
Earlier in the course, we saw the inverse transform method, a general simulation technique that, given the
comp
9/21/2017
Introduction to Simulation
Last updated Friday 15 September 2017 16:20 Eastern
An Introduction to Simulation
What is it?
We build stochastic models to represent real-world systems of interest, for example a service queue, a trafc
network, a port
10/5/2017
TEMPLATE
Last updated Monday 2 October 2017 07:54 Eastern
The Poisson Family of Processes
We have studied many techniques for generating random variables from the simple U nif (0, 1). The next
natural question is: how can we use these techniques
Last updated Monday 18 September 2017 16:54 Eastern
Home Work 2: given out Sept 14, due (by 5.45PM in class) Sept 21
of variance is
1. Variance estimation and bias. In class, we have discussed that the straightforward estimator
biased because variance i
1
NAME
ISyE 6644 Fall 2016 Test #1 Solutions
This test is 85 minutes. Youre allowed one cheat sheet. Good luck!
1. Suppose X has p.d.f. f (x) = 3x2 , 0 < x < 1. Find E[3X + 2].
Solution: E[X] =
R1
0
x 3x2 dx = 3/4. Thus, E[3X + 2] = 17/4.
2
2. Suppose X