Laurence T. Maloney

English

Research Papers in Economics

JSS Journal of Statistical Software
April 2009, Volume 30, Book Review 1. http://www.jstatsoft.org/
Reviewer: Laurence T. Maloney
New York University
Introduction to Probability with R
Kenneth Baclawski
Chapman & Hall/CRC, Boca Raton, FL, 2008.
ISBN 978-1-4200-6521-3. 363 pp. USD 89.95 (P).
http://www.ccs.neu.edu/home/kenb/stochas/
Introduction to Probability with R by Kenneth Baclawksi is a new addition to Chapman &
Hall's series of books featuring R and it is a welcome addition. It is intended to be used
by undergraduate students in economics, engineering, computer science, and biology, among
other disciplines. It covers core topics in probability theory and its prerequisites and also
more advanced topics such as Markov chains. Course slides in Powerpoint and a solutions
manual are available to qualied instructors from the publisher.
The book is clearly written and very well-organized and it stems in part from a popular course
at MIT taught by the late Gian-Carlo Rota which was originally designed in conjunction with
the author of this book. The book goes well beyond the MIT course in making extensive use
of computation and R. Teaching simulation methods is of value in itself, of course, but they
also allow the student to develop a better understanding of the central results of probability
theory. There is a large number of problems some of which are solved in detail. The author
also includes historical notes throughout the text giving brief summaries of the history of
probability theory and early researchers in probability theory (Pascal, Fermat, Huyghens,
...). These also serve to capture the students' interest and convey the fascinating history of
probability since the 1600s.
While the text makes extensive use of R it does not attempt to teach students how to program.
It is likely that a course based on this text would have to be supplemented by a separate text
on programming in R and some class time devoted to teaching programming concepts as well
as R. This book is a course on probability theory and while a few topics in statistical estimation
and hypothesis testing are touched on, this book does not cover introductory statistics. It
would serve as an exemplary test for the rst semester of a two-semester course on probability
and statistics.
The book consists of ten chapters and two appendices. The rst chapter develops the set
algebra needed to present axiomatic probability theory; this material is hard to present in
a way that captures the students' interest. The author complements the presentation of
set theory and probability theory with several intriguing problems that go beyond the usual2 Introduction to Probability with R
regurgitation of results in the main text. For example, a problem from Chapter 1 (credited
to Prendergast):
Two technicians are discussing the relative merits of two rockets. One rocket has
two engines, the other four. The engines used are all identical. To ensure success
the engines are somewhat redundant: the rocket will achieve its mission even if
half its engines fail. The rst technician argues that the four-engine rocket ought
to be the better one. The second technician replies,\Although I cannot reveal the
failure probability of an engine, because it is classied top secret, I can assure you
that either rocket is as likely to succeed as the other."
The rst technician replies, \Thank you. What you just told me allows me to
compute the failure probability both for an engine and for a rocket."
Can you do this computation also? (Exercise 1.20, p. 21)
Chapter 2 introduces combinatoric problems and R rst by using built-in functions to compute
factorials and then to study the birthday problem in some detail by writing a function that
computes the probability that the birth dates of two or more people in a group of n people
have the same month and day. The presentation is clear but compressed (students should
have a good idea of what a logarithm is and why it is useful in working with large factorials)
and this particular example deserves supplementary treatment in class. The result is a plot
of the probability of one or more pairs of coincident birthdays as a function of group size,
illustrating how to use R's graphical facilities and also illustrating that probability grows to
near-certainty between group sizes of 10 to 40.
Discrete random variables are introduced in Chapter 3 and general random variables in Chap-
ter 4 and the author introduces and makes extensive use of R's built-in functions for each
distributional family considered. One slightly unusual and welcome topic in Chapter 4 is
consideration of the distribution of order statistics, introduced before discussion of mean, me-
dian and mode and their distributions also in Chapter 4. The student who masters these two
chapters has learned a useful vocabulary of random variables that could be used in developing
simulations in biology, economics, and other areas of the natural and social sciences.
Chapter 5 is devoted to the normal distribution with concomitant introduction of variance
and the central limit theorem as well as a rudimentary discussion of hypothesis testing and
signicance levels. These topics are followed by presentation of the strong law of large num-
bers and several topics usually reserved to the graduate level: Chebyshev's inequality and
Kolmogorov's 0-1 Law. These results are not proven but they are clearly presented and ex-
plained. The chapter ends with a discussion of the Cauchy distribution that is remarkably
well motivated and { fun! By presenting a distribution for which the law of large numbers
does not apply, the author nicely emphasizes the importance of the law of large numbers.
The extensive use of R code and graphics allows students to develop real insight into the
odd behavior of Cauchy random variables. Where is the weak law of large numbers? In the
exercises to the chapter where the student is challenged to prove one version of WLLN from
the central limit theorem.
Chapter 6 continues with conditional probability, Bayes' theorem and topics in sampling.
Neyman-Pearson hypothesis testing is introduced in one of the problems. Chapter 7 contin-
ues with the Poisson distribution and the exponential distribution accompanied by relevantJournal of Statistical Software { Book Reviews 3
physical examples. One especially useful feature of the chapter is the manner in which the
author makes connections between uniform, Bernoulli, and Poison random variables.
The rst seven chapters form a remarkably clear presentation of the central topics of probabil-
ity theory. The remainder of the book consists of advanced topics that make less use of specic
features of R and which could be approached in any order. Chapter 8 introduces advanced
material that likely would be omitted from most undergraduate courses. These include the
Laplace transform and a proof of the central limit theorem as well as consideration of renewal
processes and a remarkably concise introduction to Bayes networks. Chapter 9 contains a
magnicent summary of entropy and information theory and Chapter 10 concludes the main
text of the book with a concise discussion of Markov chains. The two appendices summarize
material that complements earlier chapters (Appendix A, random walks and Appendix B,
memorylessness).
In summary, Introduction to Probability with R is a well-organized course in probability theory.
It presupposes that students are comfortable with programming and can pick up R from
examples. Alternatively, a second text introducing R could be assigned and covered early
in the semester. The text is most impressive in its clear treatment of advanced topics and
as such is comparable to Elementary Probability Theory by Chung and AitSahlia (2003) or
Strang's (1988) treatment of linear algebra. In comparison to Chung's text, Introduction
to Probability in R will appeal more to sophisticated students with no specic preparation
in mathematics beyond calculus but an interest in scientic research and a preference for
hands-on investigation of probability theory through simulation in R.
References
Chung KL, AitSahlia F (2003). Elementary Probability Theory: With Stochastic Processes
and an Introduction to Mathematical Finance. Springer-Verlag, New York, NY.
Strang G (1988). Linear Algebra and its Applications. 3rd edition. Brooks Cole, Florence,
KY.
Reviewer:
Laurence T. Maloney
New York University
Department of Psychology
Center for Neural Science
6 Washington Place, 2nd Floor
New York, NY 10003, United States of America
E-mail: ltm1@nyu.edu
URL: http://www.psych.nyu.edu/maloney/
Journal of Statistical Software http://www.jstatsoft.org/
published by the American Statistical Association http://www.amstat.org/
Volume 30, Book Review 1 Published: 2009-04-27
April 2009

Introduction to Probability with R

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