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An Elementary Introduction to the Theory of Probability

B. V. Gnedenko and A. Ya. Khinchin
Publisher: 
Dover Publications
Publication Date: 
1962
Number of Pages: 
130
Format: 
Paperback
Edition: 
5
Price: 
9.95
ISBN: 
9780486601557
Category: 
Textbook
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Michael Berg
, on
08/15/2011
]

As a number theorist, I have of course heard of Khinchin (consider, e.g., his famous “three pearls” and his book on continued fractions). But it is in fact the case that he was also a major player in the former Soviet Union in the area of probability. His co-author for the book under review, Gnedenko, was a student of Kolmogorov and was therefore only one degree of separation away from one of the very founders of the modern theory of probability. Thus, it comes as no surprise that An Elementary Introduction to the Theory of Probability is exceedingly useful at the same time that it is both idiosyncratic and even anachronistic (or so one would hope).

The anachronism is largely contained in the nature of the examples chosen to introduce and illustrate certain themes in the book. For example, already on p. 3, the first example concerns the success rate of a marksman, while the second example, half a page down, concerns inspections at a factory. Then, on p. 9, two marksmen enter the arena, and on p. 11, at the start of the second chapter, we encounter a competition of … marksmen. All a bit different from the slew of examples I have encountered in introductory texts on probability that I have used over the last twenty years in courses aimed at a certain part of the mathematically hostile undergraduate population at my university, namely the psychology majors.

But this is not to say that I was seriously considering Gnedenko and Khinchin’s book as a candidate for primary text for my next outing in “Prob/Stat’s for Psych Majors,” of course, even if the book’s Foreword to the First Edition (1945) advertises it as meant for “leaders (and occasionally also rank and file workers) in the military, industry, agricultural economy, economy, etc., whose mathematical training is very limited.” Indeed, even though Gnedenko and Khinchin go on to say that their little book “is completely accessible to all those who have completed the 10-year secondary school (ages 7–17),” I would not foist it on an American undergraduate populace. It is not that the material presented is inaccessible; quite the contrary, it is eminently accessible and very well crafted. My reasons for classing An Elementary Introduction to the Theory of Probability as unsuitable for us today is that it is entirely devoid of exercises, and, more importantly, its style is truly aimed at the Soviet proletariat, not the (post?)modern Western youths we try to teach some probability to as part of our service charter — youths headed for careers in, say clinical psychology or medicine, instead of service to the Soviet state. In other words, it just wouldn’t fly.

On the other hand, every one who does duty in the trenches of “Prob/Stat’s for Psych Majors” is likely to find this book hugely valuable as a pedagogical resource. For the authors, who are indeed past masters of the subject, certainly do a wonderful job introducing elementary probability themes via very well chosen (if idiosyncratic) examples, worked out in great detail. I, who never had any formal training in prob/stat, will be served greatly by these discussions not only in order to bolster my own limited understanding but to provide me with material for motivation and examples during my lectures — making certain changes, perhaps, although I must say that I find the whole business of destroying enemy planes with rifle shots fascinating (cf. p.24). (Was it G. H. Hardy or David Hilbert or Edmund Landau who said about World War I that his experience was essentially that of his counterpart on the other side “except for trivial changes of sign”? Is a Mosin-Nagant to an ME 109 like a Lee-Enfield to a Stuka? Oh, well… God-willing these are now rhetorical questions.)

And there is an additional reason to get hold of this little (and cheap!) book: it’s got proofs! Bernoulli’s Theorem is dealt with starting on p.50, and the proof of the law of large numbers starts on p.97 (preceded by a critical discussion).

So, I’m taking this book to my office and will place it right next to the text I use for “Prob/Stat’s for Psych Majors,” which I’ll doubtless teach again before too long.


Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

PART I. PROBABILITIES
CHAPTER I. THE PROBABILITY OF AN EVENT
1. The concept of probability
2. Impossible and certain events
3. Problem
CHAPTER 2. RULE FOR THE ADDITION OF PROBABILITIES
4. Derivation of the rule for the addition of probabilities
5. Complete system of events
6. Examples
CHAPTER 3. CONDITIONAL PROBABILITIES AND THE MULTIPLICATION RULE
7. The concept of conditional probability
8. Derivation of the rule for the multiplication of probabilities
9. Independent events
CHAPTER 4. CONSEQUENCES OF THE ADDITION AND MULTIPLICATION RULES
10. Derivation of certain inequalities
11. Formula for total probability
12. Bayes's formula
CHAPTER 5. BERNOULLI'S SCHEME
13. Examples
14. The Bernoulli formulas
15. The most probable number of occurrences of an event
CHAPTER 6 BERNOULLI'S THEOREM
16. Content of Bernoulli's theorem
17. Proof of Bernoulli's theorem
PART II. RANDOM VARIABLES
CHAPTER 7. RANDOM VARIABLES AND DISTRIBUTION LAWS
18. The concept of random variable
19. The concept of law of distribution
CHAPTER 8. MEAN VALUES
20. Determination of the mean value of a random variable
CHAPTER 9. MEAN VALUE OF A SUM AND OF A PRODUCT
21. Theorem on the mean value of a sum
22. Theorem on the mean value of a product
CHAPTER 10. DISPERSION AND MEAN MEAN DEVIATIONS
23. Insufficiency of the mean value for the characterization of a random variable
24. Various methods of measuring the dispersion of a random variable
25. Theorems on the standard deviation
CHAPTER 11. LAW OF LARGE NUMBERS
26. Chebyshev's inequality
27. Law of large numbers
28. Proof of the law of large numbers
CHAPTER 12. NORMAL LAWS
29. Formulation of the problem
30. Concept of a distribution curve
31. Properties of normal distribution curves
32. Solution of problems
CONCLUSION
APPENDIX. Table of values of the function F (a)
BIBLIOGRAPHY
INDEX

Dummy View - NOT TO BE DELETED