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*

For magnet / torrent links go *here.*

Password if needed: *mirtitles*

4-shared link 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**

Super book, thanks

Very nice post, thanks

I was quite lucky this Monday because i got this book for 150 in a bookStore. Was happy to see a 1977 book.

access denied. please upload new link