Computational Probability: Algorithms and Applications in the Mathematical Sciences

216 Pages·2007·1.58 MB·English

by  John H. Drew

Computational Probability
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John H. Drew
Diane L. Evans
Andrew G. Glen
Lawrence M. Leemis
Computational Probability
Algorithms and Applications in the
Mathematical Sciences
ABC
JohnH.Drew DianeL.Evans
CollegeofWilliamandMary Rose-HulmanInstituteofTechnology
Williamsburg,VA,USA TerreHaute,IN,USA
AndrewG.Glen LawrenceM.Leemis
UnitedStatesMilitaryAcademy CollegeofWilliamandMary
WestPoint,NY,USA Williamsburg,VA,USA
SeriesEditor:
FredHillier
StanfordUniversity
Stanford,CA,USA
ISBN978-0-387-74675-3 e-ISBN978-0-387-74676-0
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Preface
Fordecades,statisticianshaveenjoyedtheuseof“statisticalpackages”which
read in a (potentially) large data set, process the observations, and print
out anything from histograms to sample variances, to p-values, to multi-
dimensional plots. But pity the poor probabilist, who through all those
decades had only paper and pencil for symbolic calculations. The purpose
of this monograph is to address the plight of the probabilist by providing al-
gorithmstoperformcalculationsassociatedwithunivariaterandomvariables.
Werefertoacollectionofdatastructuresandalgorithmsthatautomateprob-
ability calculations as “computational probability.” The data structures and
algorithms introduced here have been implemented in a language known as
APPL (A Probability Programming Language). Several illustrations of prob-
lemsfromthemathematicalsciencesthatcanbesolvedbyimplementingthese
algorithms in a computer algebra system are presented in the final chapters
of this monograph.
The algorithms for manipulating random variables (e.g., adding, multi-
plying, transforming, ordering) symbolically result in an entire class of new
problemsthatcannowbeaddressed.Theimplementationofthesealgorithms
in Maple-based APPL is available without charge for non-commercial use at
www.applsoftware.com. APPL is able to perform exact probability calcula-
tions for problems that would otherwise be deemed intractable. The work is
quitedistinctfromtraditionalprobabilityanalysisinthatacomputeralgebra
system, in this case Maple, is used as a computing platform.
The use of a computer algebra system to solve problems in operations re-
search and probability is increasing. Other researchers also sense the benefits
of incorporating a computer algebra system into fields with probabilistic ap-
plications, for example, Parlar’s Interactive Operations Research with Maple
[69],KarianandTanis’s2ndeditionofProbabilityandStatistics:Explorations
with Maple [42], Rose and Smith’s Mathematical Statistics and Mathematica
[74], and Hasting’s 2nd edition of Introduction to the Mathematics of Opera-
tions Research with Mathematica [33].
VI Preface
Thismonographsignificantlydiffersfromthefourtitleslistedaboveintwo
ways.First,thefourtitleslistedabovearealltextbooks,ratherthanresearch
monographs. They contain exercises and examples geared toward students,
rather than researchers. Second, the emphasis in most of these texts is much
broader than the emphasis being proposed here. For example, Parlar and
Hasting consider all of OR/MS, rather than the probabilistic side of OR/MS
proposed herein much more depth. Also, Karian and Tanis emphasize Monte
Carlosolutionstoprobabilityandstatisticsproblems,asopposedtotheexact
solutions given in APPL.
Themonographbeginswithanintroductorychapter,theninChapter2re-
viewstheMapledatastructuresandfunctionsnecessarytoimplementAPPL.
Thisisfollowedbyadiscussionofthedevelopmentofthealgorithms(Chapters
3–5 for continuous random variables and Chapters 6–8 for discrete random
variables), and by a sampling of various applications in the mathematical
sciences (Chapters 9–11). The two most likely audiences for the monograph
are researchers in the mathematical sciences with an interest in applied prob-
ability and instructors using the monograph for a special topics course in
computational probability taught in a mathematics, statistics, operations re-
search, management science, or industrial engineering department. The in-
tended audience for this monograph includes researchers, MS students, PhD
students, and advanced practitioners in stochastic operations research, man-
agement science, and applied probability.
An indication of the proven utility of APPL is that the research efforts of
theauthorsandothercolleagueshaveproducedmanyrelatedrefereedjournal
publications, many conference presentations, the ICS Computing Prize with
INFORMS, a government patent, and multiple improvements to pedagogical
methods in numerous colleges and universities around the world. We believe
that the potential of this field of computational probability in research and
education is unlimited. It is our hope that this monograph encourages people
to join us in attaining future accomplishments in this field.
We are grateful to Camille Price and Gary Folven from Springer for their
supportofthisproject.WealsothankProfessorLudolfMeesterfromTUDelft
for class-testing and debugging portions of the APPL code. We thank our co-
authors Matt Duggan, Kerry Connell, Jeff Mallozzi, and Bruce Schmeiser
for their collaboration on the applications given in Chapter 11. We also
thank the editors, anonymous referees, and colleagues who have been help-
ful in the presentation of the material, including John Backes, Donald Barr,
Barbara Boyer, Don Campbell, Jacques Carette, Gianfranco Ciardo, Mike
Crawford, Mark Eaton, Jerry Ellis, Bob Foote, Greg Gruver, Matt Hanson,
Carl Harris, Billy Kaczynski, Rex Kincaid, Hank Krieger, Marina Kondra-
tovitch, Sid Lawrence, Lee McDaniel, Lauren Merrill, David Nicol, Raghu
Pasupathy, Steve Roberts, Evan Saltzman, Jim Scott, Bob Shumaker, Paul
Stockmeyer, Bill Treadwell, Michael Trosset, Erik Vargo, Mark Vinson, and
Marianna Williamson. We thank techie Robert Marmorstein for his LATEX
support. The authors gratefully acknowledge support from the Clare Boothe
Preface VII
Luce Foundation, the National Science Foundation (for providing funding for
anEducationalInnovationGrantCDA9712718“UndergraduateModelingand
Simulation Analysis” and for scholarship funds provided under the CSEMS
Grant0123022“EffectiveTransitionsThroughAcademetoIndustryforCom-
puter Scientists and Mathematicians”), and the College of William & Mary
for research leave to support this endeavor.
Williamsburg, VA John Drew
Terre Haute, IN Diane Evans
West Point, NY Andy Glen
Williamsburg, VA Larry Leemis
NOTE
The authors and publisher of this book have made their besteffortin prepar-
ing this book and the associated software. The authors and publisher of this
bookmakenowarrantyofanykind,expressedorimplied,withrespecttothe
software or the associated documentation described in this book. Neither the
authors nor the publisher shall be liable in any event for incidental or conse-
quential damages in connection with, or arising out of, the furnishing, perfor-
mance, or use of these programs or the associated documentation described
in this book.
Contents
Part I Introduction
1 Computational Probability ................................ 3
1.1 Four Simple Examples of the Use of APPL ................. 3
1.2 A Different Way of Thinking.............................. 8
1.3 Overview............................................... 10
2 Maple for APPL........................................... 13
2.1 Numerical Computations ................................. 13
2.2 Variables ............................................... 15
2.3 Symbolic Computations .................................. 16
2.4 Functions............................................... 17
2.5 Data Types............................................. 18
2.6 Solving Equations ....................................... 20
2.7 Graphing............................................... 22
2.8 Calculus................................................ 24
2.9 Loops and Conditions.................................... 26
2.10 Procedures ............................................. 27
Part II Algorithms for Continuous Random Variables
3 Data Structures and Simple Algorithms ................... 33
3.1 Data Structures ......................................... 33
3.2 Simple Algorithms....................................... 37
4 Transformations of Random Variables ..................... 45
4.1 Theorem ............................................... 46
4.2 Implementation in APPL................................. 48
4.3 Examples............................................... 50