Simulation and the Monte Carlo Method: Solutions Manual to Accompany, Second Edition

186 Pages·2008·7.321 MB·English

by  Dirk P. Kroese, Thomas Taimre, Zdravko I. Botev, Rueven Y. Rubinstein(auth.)

Solutions Manual to Accompany
Simulation and the
Monte Carlo Method
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Solutions Manual to Accompany
Simulation and the
Monte Carlo Method
Second Edition
Dirk P. Kroese
University of Queensland
Department of Mathematics
Brisbane, Australia
Thomas Taimre
University of Queensland
Department of Mathematics
Brisbane, Australia
Zdravko I. Botev
University of Queensland
Department of Mathematics
Brisbane, Australia
Reuven Y. Rubinstein
Technion-Israel Institute of Technology
Faculty of Industrial Engineering and Management
Haifa, Israel
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CONTENTS
Preface vii
Acknowledgments ix
I Problems 1
1 Preliminaries 1
2 Random Number, Random Variable, and Stochastic Process Generation 9
3 Simulation of Discrete-Event Systems 15
4 Statistical Analysis of Discrete-Event Systems 19
5 Controlling the Variance 25
6 Markov Chain Monte Carlo 31
7 Sensitivity Analysis and Monte Carlo Optimization 37
8 The Cross-Entropy Method 41
9 Counting via Monte Carlo 47
10 Appendix 51
11 Solutions 53
11 Preliminaries 55
12 Random Number, Random Variable, and Stochastic Process Generation 69
13 Simulation of Discrete-Event Systems 85
14 Statistical Analysis of Discrete-Event Systems 95
15 Controlling the Variance 105
16 Markov Chain Monte Carlo 123
17 Sensitivity Analysis and Monte Carlo Optimization 145
18 The Cross-Entropy Method 151
19 Counting via Monte Carlo 177
20 Appendix 187
v
PREFACE
The only effective way to master the theory and practice of Monte Carlo simulation is
through exercises and experiments. For this reason, we (RYR and DPK) included many
exercises and algorithms in Simulation and the Monte Carlo Method, 2nd Edition (SMCM2),
Wiley & Sons, New York, 2007.
This companion volume to SMCM2 is written in the same style and contains a wealth of
additional material: worked solutions for the over 180 problems in SMCM2, lots of Matlab
algorithms and many illustrations.
Like SMCM2, this solution manual is aimed at a broad audience of students, instructors
and researchers in engineering, physical and life sciences, statistics, computer science and
mathematics, as well as anyone interested in using Monte Carlo simulation in his or her study
or work. One of the main goals of the book is to provide a comprehensive solutions guide to
instructors, which will aid student assessment and stimulate further student development.
In addition, the book offers a unique complement to SMCM2 for self-study. All too often a
stumbling block for learning is the unavailability of worked solutions and actual algorithms.
The problem set includes the major topics in Monte Carlo simulation. Starting with
exercises that review various important concepts in probability and optimization, the book
covers a wide range of exercises in random variable/process generation, discrete-event
simulation, statistical analysis of output data, variance reduction techniques, Markov chain
Monte Carlo, simulated annealing, sensitivity analysis, cross-entropy methods, and Monte
Carlo counting techniques. The original questions from SMCM2 have been added for easy
reference. More difficult sections and exercises are marked with an asterisk (*) sign. Our
choice of using Matlab was motivated by its ease of use and clarity of syntax.
Reference numbers to SMCM2 are indicated in boldface font. For example, Defini-
tion 1.1.1 refers to the corresponding definition in SMCM2, and (1.7) refers to equation
(1.7) in SMCM2, whereas Figure 1.1 refers to the first numbered figure in the present book.
DIRK KROESE, THOMAS TAIMRE, ZDRAVKO BOTEV AND REUVEN RUBINSTEIN
Brisbane and Haifa
July, 2007
vii
ACKNOWLEDGMENTS
We thank Gareth Evans for providing the code for Problem 3.8.
This book was supported by the Australian Research Council, under Grants DP056631
and DP055895. Thomas Taimre acknowledges the financial support of the Australian
Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems.
DPK, TT, Z1B, RYR
ix
PARTI
PROBLEMS
Solutions Manual for SMCM, 2nd Edition. By D.P. Kroese, T. Taimre, Z.I. Botev, and R.Y. Rubinstein 1
Copyright © 2007 John Wiley & Sons, Inc.
CHAPTER 1
PRELIMINARIES
Probability Theory
1.1 Using the properties of the probability measure in Definition 1.1.1:
A probability P is a rule that assigns a number 0 < Ψ(Α) < 1 to each event A, such
that Ρ(Ω) = l and such that for any sequence Ai, A,... of disjoint events
t 2
prove the following results.
(a) F(Ac) = l-F(A).
(b) Ψ(Α U B) = P(i4) + P(JB) - P(i4 Π B).
1.2 Prove the product rule (1.4):
For any sequence of events A\, A,..., A ,
2 n
Ψ(Αι · · · Λ„) = ?(Αι) Ψ(Α | Αι) Ψ(Α | Λι A) · · · Ρ(Λ \A · · · Λ -ι) ,
2 3 7 η X η
using the abbreviation Λ1Λ2 · · - Ak = Ai Π Λ2 Π · · · O Λ*.
for the case of three events.
Solutions Manual for SMCM, 2nd Edition. By D.P. Kroese, T. Taimre, Z.I. Botev, and R.Y. Rubinstein 1
Copyright © 2007 John Wiley & Sons, Inc.
2 PRELIMINARIES
1.3 We draw three balls consecutively from a bowl containing exactly five white and five
black balls, without putting them back. What is the probability that all drawn balls will be
black?
1.4 Consider the random experiment where we toss a biased coin until heads comes up.
Suppose the probability of heads on any one toss is p. Let X be the number of tosses
required. Show that X ~ G(p).
1.5 In a room with many people, we ask each person his/her birthday, for example, May
5. Let N be the number of people queried until we get a "duplicate" birthday.
(a) Calculate F(N > π), π = 0,1,2,....
(b) For which n do we have F(N ζ η) 3* 1/2?
(c) Use a computer to calculate E[N].
1.6 Let X and Y be independent standard normal random variables, and let U and V be
random variables that are derived from X and Y via the linear transformation
/£7\ _ /sin a —cosa\ fX\
\VJ ~~ \cosa sin a ) \Y)
(a) Derive the joint pdf of U and V.
(b) Show that U and V are independent and standard normally distributed.
1.7 Let X ~ Εχρ(λ). Show that the memoryless property holds:
¥{X>t + s\X>t) = ¥(X>s) foralls,t>0.
1.8 Let X\, Xi, X$ be independent Bernoulli random variables with success probabilities
1/2,1/3, and 1/4, respectively. Give their conditional joint pdf, given that X\ +X2 +Xz =
2.
1.9 Verify the expectations and variances in Table 1.1 below.
Table 1.1 Expectations and variances for some well-known distributions.
Dist. E[X] Var(X) Dist. E[X] Var(X)
Βίη(π,ρ) ηρ np(l - p) Gamma(α, λ) - ^
Χ-ψ- Ν(μ,σ2) μ σ*
Ροί(λ) λ X Beta(a,/3) ^ (o+wt+o+ti
i^f- Weib(a,A) m£l Sßfsl - (Eü^)2
Εχρ(λ) I