In this post, we will see the book An Elementary Introduction To The Theory Of Probability by B. V. Gnedenko and A. Ya. Khinchin

About the book

This compact volume equips the reader with all the facts and principles essential to a fundamental understanding of the theory of probability. It is an introduction, no more: throughout the book the authors discuss the theory of probability for situations having only a finite number of possibilities, and the mathematics employed is held to the elementary level. But within its purpose is  restricted range it is extremely thorough, well organized, and absolutely authoritative. It is the only English translation of the latest revised Russian edition; and it is the only current translation on the market that has been checked and approved by Gnedenko himself.

After explaining in simple terms the meaning of the concept of probability and the means by which an event is declared to be in practice, impossible, the authors take up the processes involved in the calculation of probabilities. They survey the rules for addition and multiplication of probabilities, the concept of conditional probability, the formula for total probability, Bayes’s formula, Bernoulli’s scheme and theorem, the concepts of random variables, insuffciency of the mean value for the characterization of a random variable, methods of measuring the variance of a random variable, theorems on the standard deviation, the Chebyshev inequality, normal laws of distribution, distribution curves, properties of normal distribution curves, and related topics.

The book is unique in that, while there are several high school and college textbooks available on this subject, there is no other popular treatment for the layman that contains quite the same material presented with the same degree of clarity and authenticity. The reader who shies away from oversimplified popularizations may be sure that in this book he is getting a perfectly reliable scientific treatment. Anyone who desires a fundamental grasp of this increasingly important subject cannot do better than to start with this book.

The book was translated from Russian by Leo F. Boron an edited by Sidney F. Mack and was published in 1962.

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You can get the book here.

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Contents

PART I. PROBABILITIES

CHAPTER 1. THE PROBABILITY OF AN EVENT 3

1. The concept of probability 3
2. Impossible and certain events 8
3. Problem 9

CHAPTER 2. RULE FOR THE ADDITION OF PROBABILITIES 11

4. Derivation of the rule for the addition of probabilities 11
5. Complete system of events 13
6. Examples 16

CHAPTER 3. CONDITIONAL PROBABILITIES AND THE MULTIPLICATION RULE 18

7. The concept of conditional probability 18
8. Derivation of the rule for the multiplication of probabilities 20
9. Independent events 21

CHAPTER 4. CONSEQUENCES OF THE ADDITION AND MULTIPLICATION RULES 27

10. Derivation of certain inequalities 27
11. Formula for total probability 29
12. Bayes’s formula 32

CHAPTER 5. BERNOULLI’s SCHEME 38

13. Examples 38
14. The Bernoulli formulas 40
15. The most probable number of occurrences of an event 43

CHAPTER 6. BERNOULLI’S THEOREM 49

16. Content of Bernoulli’s theorem 49
17. Proof of Bernoulli’s theorem 50

PART II. RANDOM VARIABLES

CHAPTER 7. RANDOM VARIABLES AND DISTRIBUTION LAWS 59

18. The concept of random variable 59
19. The concept of law of distribution 61

CHAPTER 8. MEAN VALUES 65

20. Determination of the mean value of a random variable 65

CHAPTER 9. MEAN VALUE OF A SUM AND OF A PRODUCT 74

21. Theorem on the mean value of a sum 74
22. Theorem on the mean value of a product 77

CHAPTER 10. DISPERSION AND MEAN DEVIATIONS 80

23. Insufficiency of the mean value for the characterization of a random variable 80
24. Various methods of measuring the dissension of a random variable 81
25. Theorems on the standard deviation 87

CHAPTER 11. LAW OF LARGE NUMBERS 93

26. Chebyshev’s inequality 93
27. Law of large numbers 94
28. Proof of the law of large numbers 97

CHAPTER 12. NORMAL LAWS 100

29. Formulation of the problem 100
30. Concept of a distribution curve 102
31. Properties of normal distribution curves 105
32. Solution of problems 111

CONCLUSION 118

APPENDIX. Table of values of the function 𝛷(a) 123

BIBLIOGRAPHY 125

INDEX 129