Probability, random variables, and stochastic processes. 2nd ed.

*(English)*Zbl 0558.60001
McGraw-Hill Series in Electrical Engineering. Communications and Information Theory. New York etc.: McGraw-Hill Book Company. XV, 576 p. DM 162.95 (1984).

This book is an extensively revised edition, reflecting the pertinent developments of the last two decades, of the first edition of 1965. The first ten chapters are essentially the same, the new material of the later chapters includes the following: (i) discrete-time processes with applications in system theory; (ii) innovations, factorization, spectral representation; (iii) queuing theory, level crossings, spectra of FM signals, sampling theory; (iv) mean-square estimation, orthonormal expansions, Levinson’s algorithm, Wold’s decomposition, Wiener, lattice and Kalman filters; (v) spectral estimation, windows, extrapolation, Burg’s method, detection of line spectra; (vi) entropy.

The book is aimed, according to the author, at ”the majority of engineers and physicists who have sufficient maturity to appreciate and follow a logical presentation, but, because of their limited mathematical background, would find a book such as Doob’s too difficult for a beginning text” (p. XiV). Thus the main thrust is random processes, with strong emphasis from the beginning on applications to illustrate basic (and also more sophisticated) concepts. For the purposes of this book, according to the author, the emphasis is placed ”on explanation, facility, and economy” (p. XV), and in this aim he succeeds. There are, however, certain small matters which might be mentioned, and attention to which would serve to improve further editions of this work.

In example 1.2 the author finds the probability p of obtaining a 7 when two dice are rolled. Considering as possible outcomes the sums 2,3,...,12 and noting that only one outcome is favourable, we might conclude that \(p=1/11\). Then the author says ”this result is of course wrong.” But why? The example precedes the ”equally likely” definition of probability, and it is by no means clear why the suggested answer should be viewed as incorrect - particularly since, in writing three pages further on Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics, the author states ”the three models proposed are actually only hypotheses and the physicist accepts the one whose consequences agree with experience” (p. 11).

In chapter 2 the discussion of set operations merits a little tidying up. Firstly, it is more usual to denote the empty set by \(\{\) \(\}\) or \(\emptyset\), rather than \(\{\) \(\emptyset \}\)- indeed, it might well be argued that the latter denotes a set whose only element is the empty set. Secondly, the statements: \(Transitivity:=\) If \(C\subset B\) and \(B\subset A\) then \(C\subset A\); \(Equality:=\) \(A=B\) iff \(A\subset B\) and \(B\subset A\) although presented as being on the same footing, are respectively an axiom and a definition.

In the first edition of his book the author states the necessity of having each \(P(A_ i)>0\) in Bayes’s theorem. This caveat is omitted from the second edition, as is the proof that independence of A and B implies that \(P(A| B)=P(A)\). A statistician might also be perturbed by the definition of erf x as \[ \frac{1}{\sqrt{2\pi}}\int^{x}_{0}\exp (- y^ 2/2)dy \] though far less use is made of this function here than in the first edition. Perhaps too the sense of \(\simeq\) in \[ \left( \begin{matrix} n\\ k\end{matrix} \right)p^ kq^{n-k}\simeq \frac{1}{\sqrt{2\pi npq}}\exp [-(k-np)^ 2/2npq] \] could be more carefully considered.

It would have been useful if the practice of the first edition had been adopted in printing the subscripts in things like \(F_ x(x)\) in bold print, thus keeping clear the distinction between an RV and its value. The author is also careless in defining \(f(x)=dF(x)/dx:\) no mention of continuity is made (this too was satisfactorily handled in the earlier edition). Some justification should also be given for the interchange of orders of integration and differentiation in various places.

A curiosity is the notation N(\(\mu\),\(\sigma)\) to denote a normally distributed random variable: this could cause confusion to a student used to the far more customary \(N(\mu,\sigma^ 2)\). The sketch of the normal distribution in table 4.1 is gravely misleading (it appears to have a finite domain). In section 5.4 mention is made of ”distribution functions”, though prior to that only distribution functions of random variables have been defined. There is also a curious phrase, viz. ”the area of differential” on page 135.

Other comments in similar vein might be mentioned, but enough has probably been said to indicate where the presentation might be improved. The book is relatively free of misprints: I noticed ”methaphysical”, ”subjects” (for ”subsets”), ”inprecise”, and the occasional use of \(<\) for \(\leq\). There are also some incorrect references to earlier sections or equations, and the frequent omission of definite or indefinite articles, which make for slightly stilted reading. The bibliography would be improved by placing the references given as footnotes in the bibliography itself.

All-in-all the book is a useful one and a welcome improvement on the first edition. It can be recommended to its intended audience.

The book is aimed, according to the author, at ”the majority of engineers and physicists who have sufficient maturity to appreciate and follow a logical presentation, but, because of their limited mathematical background, would find a book such as Doob’s too difficult for a beginning text” (p. XiV). Thus the main thrust is random processes, with strong emphasis from the beginning on applications to illustrate basic (and also more sophisticated) concepts. For the purposes of this book, according to the author, the emphasis is placed ”on explanation, facility, and economy” (p. XV), and in this aim he succeeds. There are, however, certain small matters which might be mentioned, and attention to which would serve to improve further editions of this work.

In example 1.2 the author finds the probability p of obtaining a 7 when two dice are rolled. Considering as possible outcomes the sums 2,3,...,12 and noting that only one outcome is favourable, we might conclude that \(p=1/11\). Then the author says ”this result is of course wrong.” But why? The example precedes the ”equally likely” definition of probability, and it is by no means clear why the suggested answer should be viewed as incorrect - particularly since, in writing three pages further on Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics, the author states ”the three models proposed are actually only hypotheses and the physicist accepts the one whose consequences agree with experience” (p. 11).

In chapter 2 the discussion of set operations merits a little tidying up. Firstly, it is more usual to denote the empty set by \(\{\) \(\}\) or \(\emptyset\), rather than \(\{\) \(\emptyset \}\)- indeed, it might well be argued that the latter denotes a set whose only element is the empty set. Secondly, the statements: \(Transitivity:=\) If \(C\subset B\) and \(B\subset A\) then \(C\subset A\); \(Equality:=\) \(A=B\) iff \(A\subset B\) and \(B\subset A\) although presented as being on the same footing, are respectively an axiom and a definition.

In the first edition of his book the author states the necessity of having each \(P(A_ i)>0\) in Bayes’s theorem. This caveat is omitted from the second edition, as is the proof that independence of A and B implies that \(P(A| B)=P(A)\). A statistician might also be perturbed by the definition of erf x as \[ \frac{1}{\sqrt{2\pi}}\int^{x}_{0}\exp (- y^ 2/2)dy \] though far less use is made of this function here than in the first edition. Perhaps too the sense of \(\simeq\) in \[ \left( \begin{matrix} n\\ k\end{matrix} \right)p^ kq^{n-k}\simeq \frac{1}{\sqrt{2\pi npq}}\exp [-(k-np)^ 2/2npq] \] could be more carefully considered.

It would have been useful if the practice of the first edition had been adopted in printing the subscripts in things like \(F_ x(x)\) in bold print, thus keeping clear the distinction between an RV and its value. The author is also careless in defining \(f(x)=dF(x)/dx:\) no mention of continuity is made (this too was satisfactorily handled in the earlier edition). Some justification should also be given for the interchange of orders of integration and differentiation in various places.

A curiosity is the notation N(\(\mu\),\(\sigma)\) to denote a normally distributed random variable: this could cause confusion to a student used to the far more customary \(N(\mu,\sigma^ 2)\). The sketch of the normal distribution in table 4.1 is gravely misleading (it appears to have a finite domain). In section 5.4 mention is made of ”distribution functions”, though prior to that only distribution functions of random variables have been defined. There is also a curious phrase, viz. ”the area of differential” on page 135.

Other comments in similar vein might be mentioned, but enough has probably been said to indicate where the presentation might be improved. The book is relatively free of misprints: I noticed ”methaphysical”, ”subjects” (for ”subsets”), ”inprecise”, and the occasional use of \(<\) for \(\leq\). There are also some incorrect references to earlier sections or equations, and the frequent omission of definite or indefinite articles, which make for slightly stilted reading. The bibliography would be improved by placing the references given as footnotes in the bibliography itself.

All-in-all the book is a useful one and a welcome improvement on the first edition. It can be recommended to its intended audience.

Reviewer: A.Dale