We will now see Theory of Probability by B. V. Gnedenko.

This book aims to give an exposition of the fundamentals of the
theory of probability, a mathematical science that treats of the
regularities of random phenomena.

This book was translated from the Russian by George Yankovsky. The
book was published by first Mir Publishers in 1969, with reprints in
1973, 1976 and 1978. The book below is from the 1978 reprint.

All credits to the original uploader.

DJVU | OCR | 15.1 MB | Pages: 390 |
You can get the book here
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Table of Contents

Introduction 7
Chapter 1.
THE CONCEPT OF PROBABILITY 13

Sec.1. Certain, Impossible, and Random Events 13
Sec.2. Different Approaches to the Definition of Probability 16
Sec.3. The Sample Space 19
Sec.4. The Classical Definition of Probability 23
Sec.5. The Classical Definition of Probability. Examples. 26
Sec.6. Geometrical Probability 33
Sec.7. Frequency and Probability 39
Sec.8. An Axiomatic Construction of the Theory of Probability 45
Sec.9. Conditional ProbabiHty and the Most Elementary Basic Formulas 51
Sec.10. Examples 59
Exercises 67

Chapter 2.
SEQUENCES OF INDEPENDENT TRIALS 70
Sec.11. Independent Trials. Bernoulli’s Formulas 70
Sec.12. The Local Limit Theorem 76
Sec.13. The Integral Limit Theorem 85
Sec.14. Applications of the Integral Theorem of DeMoivre-Laplace 92
Sec.15. Poisson’s Theorem 97
Sec.16. An Illustration of the Scheme of Independent Trials 102
Exercises 104

Chapter 3.
MARKOV CHAINS 107

Sec.17. Markov Chains Defined. Transition Matrix 107
Sec.18. Classification of Possible States 111
Sec.19. Theorem on Limiting Probabilities 113
Sec.20. Generalizing the DeMoivre-Laplace Theorem to a Sequence of
Chain-Dependent Trials 116

Exercises 123

Chapter 4.
RANDOM VARIABLES AND DISTRIBUTION FUNCTIONS 124

Sec.21. Basic Properties of Distribution Functions 124
Sec.22. Continuous and Discrete Distributions 130
Sec.23. Multidimensional Distribution Functions 134
Sec.24. Functions of Random Variables 142
Sec.25. The Stieltjes Integral 155
Exercises 160

Chapter 5.
NUMERICAL CHARACTERISTICS OF RANDOM VARIABLES 164

Sec.26. Mathematical Expectation 164
Sec.27. Variance 169
Sec.28. Theorems on Expectation and Variance 176
Sec.29. Mathematical Expectation Defined in the Axiomatics of Kolmogorov 182
Sec.30. Moments 185
Exercises 191

Chapter6.
THE LAW OF LARGE NUMBERS 195

Sec.31. Mass-Scale. Phenomena and the Law of Large Numbers 195
Sec.32. Chebyshev’s Form of the Law of Large Numbers 198
Sec.33. A Necessary and Sufficient Condition for the Law of Large Numbers 206
Sec.34. The Strong Law of Large Numbers 209
Exercises 218

Chapter 7.
CHARACTERISTIC FUNCTIONS 219

Sec.35. Definition and Elementary Properties of Characteristic Functions 219
Sec.36. The Inversion Formula and the Uniqueness Theorem 224
Sec.37. Helly’s Theorems 230
Sec.38. Limit Theorems for Characteristic Functions 235
Sec.39. Positive Definite Functions 239
Sec.40. Characteristic Functions of Multidimensional Random Variables  243
Exercises 248

Chapter 8.
THE CLASSICAL LIMIT THEOREM 251

Sec.41. Statement of the Problem 251
Sec.42. Lyapunov’s Theorem 254
Sec.43. The Local Limit Theorem 259
Exercises 266

Chapter 9.
THE THEORY OF INFINITELY DIVISIBLE DISTRIBUTION LAWS 267

Sec.44. Infinitely Divisible Laws and Their Basic Properties 268
Sec.45. The Canonical Representation of Infinitely Divisible Laws 270
Sec.46: A Limit Theorem for Infinitely Divisible Laws 275
Sec.47. Statement of the Problem of Limit Theorems for Sums 278
Sec.48. Limit Theorems for Sums 279
Sec.49. Conditions for Convergence to the Normal and Poisson Laws 282
Exercises 285

Chapter10.
THE THEORY OF STOCHASTIC PROCESSES. 287
Sec.50. Introductory Remarks. 287
Sec.51. The Poisson Process 291
Sec.52. Conditional Distribution Functions and Bayes’ Formula. 298
Sec.53. Generalized Markov Equation 302
Sec.54. Continuous Stochastic Processes. Kolmogorov’s Equations 303
Sec.55. Purely Discontinuous Stochastic Processes. The Kolmogorov-Feller Equations 311
Sec.56. Homogeneous Stochastic Processes with Independent Increments 318
Sec.57. The Concept of a Stationary Stochastic Process. Khinchin’s
Theorem on the Correlation Coefficient 323
Sec.58. The Concept of a Stochastic Integral. The Spectral
Decomposition of Stationary Processes 331
Sec.59. The Birkhoff-Khinchin Ergodic Theorem 334

Chapter11.
ELEMENTS OF QUEUEING THEORY 339

Sec.60. A General Description of the Problems of the Theory 339
Sec.61. Birth and Death Processes 346
Sec.62. Single-Server Queueing System 355
Sec.63. A Limit Theorem for Flows 361
Sec.64. Elements of the Theory of Stand by Systems 367

APPENDIX376
BIBLIOGRAPHY.382
SUBJECT INDEX388