Intuitive Probability and Random Processes using MATLAB. Steven KAY. New York: Springer, 2006, xviii+833 pp., $59.95 (H), ISBN:0-387-24157-4.

Intuitive Probability and Random Processes

using Matlab departs from the standard definition/theorem/proof set-up found in many textbooks. Through the use of motivating examples, or “particular instances,” followed by “generalizations,” the emphasis is mainly on intuition and conceptualization. As the author observes, this approach promotes better understanding but is not as clean and compact as the one found in standard texts. In regard to the latter point, at 833 pages you will not take Intuitive Probability to the beach, but at $60 (U.S.) you will get a lot of book for the buck. The graphics are quite nice, and the figures (all of them generated by MATLAB) are in refreshing black and white. The MATLAB programs that the author uses to generate realizations of random variables and processes are provided to the student, so that he or she can do some simulating on their own. Computer simulation is an effective aid to the understanding of established results and concepts and is also a valuable skill in its own right. Among the programs provided are those for probability density function (PDF) estimation of transformed variables, estimation of expectations, generation of scatterplots for bivariate random variables, verification of the central limit theorem (CLT), computation of repeated convolutions, and computation of normal curve areas. All are available online. The book is composed of 22 chapters. From Chapter 2 on, each begins with a one- or two-page introduction and summary and ends with a computer simulation and “Real-World” example. The introductions are short and to the point. I am not too keen on the summaries, however, as these have the habit of referring to figures, expressions, and terminologies that are yet to come and are thus rather hard to follow. The book contains a multitude of examples, some involving that ubiquitous urn of colored marbles but also many more that are both interesting and illuminating. The author states in the introduction that his book was written mainly for a first-year, graduate-level course in probability and random processes; the reader is assumed to have some exposure to basic probability theory. The book

strikes me as sophomore- or junior-level undergraduate through the first 15 chapters (I really do not think much background is needed for these), and advanced undergraduate to beginning graduate in the remaining seven chapters (the last third of the book) when the focus shifts to basic limit theory and random processes. Both groups can benefit by using this book. In a two-semester course, the former can skip the harder topics at the end, while the latter can breeze through the opening chapters and really dig into the final third. This is a very readable book. One could assign a chapter for the weekend and on Monday be reasonably sure that they have done it. For the professor, some of the discussions will seem a bit wordy. But for the students, especially those attempting to learn the material on their own, these same discussions will make for some nice reading. The inclusion of the word MATLAB in the title suggests that Intuitive Probability might skimp on the mathematics. On the contrary: all the important statements are proved in detail. MATLAB is used solely as an aid to understand the concepts.

Chapter 1 provides the usual “what is probability?” discussion. Chapter 2 contains an introduction to computer simulation of random experiments. Through several motivating examples, the author provides an answer to the oftasked question (“why use computer simulation?”). From Chapter 3 onward, the ordering of topics is very natural, though the development is a bit slow to unfold: basic probability, conditional probability, discrete random variables and their expectation, bivariate discrete random variables, conditional PDFs, discrete ndimensional random variables, continuous random variables and their expectation, conditional PDFs, n-dimensional continuous random variables, limit theory (CLT, Weak Law of Large Numbers), stationary random processes, “widesense” stationary processes (WSSRP), linear systems and WSSRP, Gaussian and Poisson random processes, and Markov chains. The entire treatment of continuous variables takes advantage of the student’s earlier exposure to the discrete case by pointing out the various analogies with regard to cumulative distribution functions, transformations, expectations, conditional PDFs, etc. The author makes fairly frequent use of matrix notation (e.g., when

writing the variance of a linear combination, or the PDF of a transformed multivariate normal vector). Included in Chapter 10 is a very welcome discussion on mixed random variables. Many undergraduates remain blissfully unaware of mixed random variables until their first attempt to extract payment from their car insurance company. The treatment of bivariate continuous random variables is (I am happy to say) very geometric (marginal PDF values as cross-sectional areas and conditional PDFs as scaled up cross-sectional shapes, etc.) with lots of nice pictures and is quite intuitive. The author has chosen to work with characteristic functions rather than moment-generating functions (MGFs). I like that generating functions are introduced early, so they can be put to immediate use for increased efficiency of presentation. This also gives the student more exposure to what many will find a tricky topic. However, most (if not all) of the applications could have been pulled off with MGFs. In the limit theory section (Chapter 15), I was a bit put off by the persistent use of the word “approaches” when referring to the movement of a sequence of random variables to a limit, without it being made clear that this can occur in a variety of senses. But overall, this is a nice chapter that I think will succeed in getting the student to understand a not-so-easy to understand idea. The coverage of random processes is impressive. Chapters 16–19 discuss stationarity, the autocorrelation sequence, ergodicity, the power spectral density, Wiener filtering, and transformations of multiple processes. Chapter 20 provides a nice introduction to Gaussian processes (GPs) (including transformed GPs as well as several special GPs). Chapters 21 and 22 offer a fairly standard introduction to Poisson processes and Markov chains. Intuitive Probability is a very well-written book with an informal style of presentation, containing many discussions that I found to be both clear and interesting. And the author has a nice sense of humor occasionally; for example, in Chapter 10, the image of a wedge-shaped piece of Jarlsburg cheese is used to make a point about linear density. Throughout the book, Kay makes frequent use of the foreshadowing technique, by which I mean that he often seizes the opportunity to introduce a concept informally, long before it appears formally. For example, the notion of a Markov chain is introduced in Chapter 4 in connection

with dependent Bernoulli trials long before the comprehensive treatment of Chapter 22. Similarly, the random walk makes a cameo appearance in Chapter 9 preceding the much fuller exposition in Chapter 16. This gets the reader to ponder important ideas early on so that they will not seem so onerous when encountered again. The book contains abundant “asides” pointing out interesting probabilistic facts connected with the material under discussion. There are also innumerable cautions (set off by a triangle containing two exclamation points) issued throughout the book. The number of misconceptions a student can develop about this material is sobering, but happily, many (all?) of these were anticipated by the author (doubtless the result of his having taught this subject for many years in the classroom). For example, we are warned in Chapter 12 that, when factoring a joint PDF (to check for independence), we better make sure that the domain too is factorable, and in Chapter 16 we are advised that to declare a random process stationary on the basis of a single realization is pure folly. Users of this text should greatly appreciate the inclusion of “Real World Examples” at the ends of Chapters 2–22. Most are quite interesting and fresh. Many have an engineering flavor and are undoubtedly drawn from the author’s experience. Examples include data compression, assessing health risks, image coding, optical character recognition, retirement planning, signal detection, speech synthesis, and brain physiology research. The one on cluster recognition (Chapter 4) is interesting, but I found the solution to be a bit shaky. Each chapter contains at its end 30–60 problems that follow the order of the chapter sections (but are not broken down by section) and provide a nice mix with regard to content and difficulty. Some are marked with a happy face, which I finally figured out are there to indicate that they have solutions in the back (a happy occasion indeed for all concerned). The problems come in four varieties. Some are plug-ins requiring the use of a formula from the preceding chapter. Some are word problems requiring problem-solving skill. Some ask the student for a proof (why are these not marked with a sad face?), and some are computer problems. Some chapters include non-MATLAB appendices containing such material as a derivation of the Cauchy–

Schwarz inequality and a proof of the CLT. There are appendices at the end of the book covering “Linear and Matrix Algebra,” “Summary of Signals,” “Linear Transforms,” and “Linear Systems,” and (everyone’s favorite) “Answers to Selected Problems.” Kay’s book undoubtedly will see its greatest use in engineering schools, but I think it would work nicely in other settings as well. TheMATLAB angle probably sets Intuitive Probability apart in the panoply of introductory probability and random processes books. But no matter, even without the MATLAB, this is a well-thought-out and high-quality effort. It is written in a clear and informal style that students will appreciate, its coverage is excellent, and the author’s stated objective (to lessen the difficulty that students usually experience assimilating and applying probability and random processes) will, I predict, be met. Ralph P. RUSSO The University of Iowa

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