Theory of Probability

This book is the sixth edition of a classic text that was first published in 1950 in the former Soviet Union. The clear presentation of the subject and extensive applications supported with real data helped establish the book as a standard for the field. To date, it has been published into more that ten languages and has gone through five editions.The sixth edition is a major revision over the fifth. It contains new material and results on the Local Limit Theorem, the Integral Law of Large Numbers, and Characteristic Functions. The new edition retains the feature of developing the subject from intuitive concepts and demonstrating techniques and theory through large numbers of examples. The author has, for the first time, included a brief history of probability and its development. Exercise problems and examples have been revised and new ones added.

Translator's Preface

ix

(2)

Author's Preface to the English Translation of the Russian Sixth Edition

xi

(2)

Author's preface to the Russian Sixth Edition

xiii

(2)

From the Author's Preface to the Russian First Edition

xv

(2)

Foreword

xvii

(2)

Biography of Boris V. Gnedenko

xix

1 INTRODUCTION

1

(4)

2 RANDOM EVENTS AND THEIR PROBABILITIES

5

(58)

1 Intuitive Understanding of Random Events

5

(3)

2 Sample Space. Classical Definition of Probability

8

(8)

3 Examples

16

(10)

4 Geometrical Probability

26

(8)

5 On Statistical Estimation of Unknown Probability

34

(3)

6 Axiomatic Construction of the Theory of Probability

37

(6)

7 Conditional Probability and the Simplest Basic Formulae

43

(9)

8 Examples

52

(7)

9 Exercises

59

(4)

3 SEQUENCES OF INDEPENDENT TRIALS

63

(42)

10 Introduction

63

(5)

11 The Local Limit Theorem

68

(7)

12 The Integral Limit Theorem

75

(9)

13 Applications of the Integral Theorem of DeMoivre-Laplace

84

(6)

14 Poisson's Theorem

90

(5)

15 Illustration of the Scheme of Independent Trials

95

(4)

16 Exercises

99

(6)

4 MARKOV CHAINS

105

(8)

17 Definition of a Markov Chain

105

(1)

18 Transition Matrix

106

(2)

19 Theorem on Limiting Probabilities

108

(4)

20 Exercises

112

(1)

5 RANDOM VARIABLES AND DISTRIBUTION FUNCTIONS

113

(46)

21 Fundamental Properties of Distribution Functions

113

(7)

22 Continuous and Discrete Distributions

120

(5)

23 Multidimensional Distribution Functions

125

(9)

24 Functions of Random Variables

134

(13)

25 The Stieltjes Integral

147

(6)

26 Exercises

153

(6)

6 NUMERICAL CHARACTERISTICS OF RANDOM VARIABLES

159

(32)

27 Mathematical Expectation

159

(7)

28 Variance

166

(6)

29 Theorems on Mathematical Expectation and Variance

172

(7)

30 Moments

179

(7)

31 Exercises

186

(5)

7 THE LAW OF LARGE NUMBERS

191

(28)

32 Mass Phenomenon and the Law of Large Numbers

191

(3)

33 Chebyshev's Form of the Law of Large Numbers

194

(5)

34 A Necessary and Sufficient Condition for the Law of Large Numbers

199

(4)

35 The Strong Law of Large Numbers

203

(7)

36 Glivenko's Theorem

210

(7)

37 Exercises

217

(2)

8 CHARACTERISTIC FUNCTIONS

219

(44)

38 The Definition and Simplest Properties of Characteristic Functions

219

(6)

39 The Inversion Formula and the Uniqueness Theorem

225

(6)

40 Helly's Theorem

231

(6)

41 Limit Theorems for Characteristic Functions

237

(4)

42 Positive-Semidefinite Functions

241

(7)

43 Characteristic Functions of Multi-Dimensional Random Variables

248

(5)

44 Laplace-Stieltjes Transform

253

(6)

45 Exercises

259

(4)

9 THE CLASSICAL LIMIT THEOREM

263

(18)

46 Statement of the Problem

263

(3)

47 Lindeberg's Theorem

266

(6)

48 The Local Limit Theorem

272

(7)

49 Exercises

279

(2)

10 THE THEORY OF INFINITELY DIVISIBLE DISTRIBUTIONS

281

(30)

50 Infinitely Divisible Distributions and Their Fundamental Properties

282

(3)

51 Canonical Representation of Infinitely Divisible Distributions

285

(6)

52 A Limit Theorem for Infinitely Divisible Distributions

291

(3)

53 Limit Theorems for Sums: Formulation of the Problem

294

(2)

54 Limit Theorems for Sums

296

(4)

55 Conditions for Convergence to the Normal and Poisson Distributions

300

(3)

56 Sum of a Random Number of Random Variables

303

(5)

57 Exercises

308

(3)

11 THE THEORY OF STOCHASTIC PROCESSES

311

(68)

58 Introduction

311

(4)

59 The Poisson Process

315

(7)

60 Death and Birth Processes

322

(11)

61 Conditional Distribution Functions and Bayes' Formula

333

(4)

62 The Generalized Markov Equation

337

(2)

63 Continuous Stochastic Processes. Kolmogorov's Equations

339

(9)

64 Purely Discontinuous Stochastic Process. The Kolmogorov-Feller Equations

348

(9)

65 Homogeneous Stochastic Processes with Independent Increments

357

(6)

66 The Concept of a Stationary Stochastic Process. Khinchine's Theorem on the Correlation Coefficient

363

(6)

67 The Notion of a Stochastic Integral. Spectral Decomposition of Stationary Processes

369

(4)

68 The Birkhoff-Khinchine Ergodic Theorem

373

(6)

12 ELEMENTS OF STATISTICS

379

(34)

69 Some Problems of Mathematical Statistics

379

(4)

70 The Classical Procedure for Estimating the Distribution Parameters

383

(11)

71 Exhaustive Statistics

394

(2)

72 Confidence Limits and Confidence Probabilities

396

(7)

73 Test of Statistical Hypothesis

403

(10)

APPENDIX: THE HISTORY OF PROBABILITY THEORY

413

(72)

Part 1 Main Concepts of Probability and Random Event

413

(22)

Part 2 Formation of the Foundation of Probability Theory

435

(19)

Part 3 Formation of the Concept of Random Variable

454

(24)

Part 4 History of Stochastic Processes

478

(7)

Tables of Function Values

485

(4)

References

489

(2)

Index

491

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Theory of Probability

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