The Theory of Probability

Translator's Preface......Page 4
FROM THE PREFACE TO THE FIRST EDITION......Page 6
PREFACE TO THE SECOND EDITION......Page 5
TABLE OF CONTENTS......Page 8
Table of Contents 2......Page 9
Table of Contents 3
......Page 10
Table of Contents 4......Page 11
INTRODUCTION......Page 12
1. Certain, Impossible, and Random Events......Page 18
2. Different Approaches to the Definition of Probability......Page 21
3. The Field of Events......Page 24
4. The Classical Definition of Probability......Page 29
5. Examples......Page 33
6. Geometrical Probability......Page 40
7. The Statistical Definition of Probability......Page 47
8. Axiomatic Construction of the Theory of Probability......Page 54
9. Conditional Probability and the Simplest Basic Formulas......Page 61
10. Examples......Page 70
Exercises......Page 79
CHAPTER II - SEQUENCES OF INDEPENDENT TRIALS......Page 82
11. The Probability Pn( m1, m2, ..., mk)......Page 83
12. The Local Limit Theorem......Page 87
13. The Integral Limit Theorem......Page 96
14. Applications of the Integral Theorem of DeMoivre-LaPlace......Page 110
15. Poisson’s Theorem......Page 115
16. Illustration of the Scheme of Independent Trials......Page 121
Exercises......Page 125
17. Definition of a Markov Chain. Transition Matrix......Page 128
18. Classification of Possible States......Page 133
19. A Theorem on Limiting Probabilities......Page 135
20. Generalization of the DeMoivre-LaPlace Theorem to a Sequence of Chain-Dependent Trials......Page 139
Exercises......Page 147
21. Fundamental Properties of Distribution Functions......Page 148
22. Continuous and Discrete Distributions......Page 155
23. Multi-Dimensional Distribution Functions......Page 160
24. Functions of Random Variables......Page 169
25. The Stieltjes Integral......Page 183
Exercises......Page 188
26. Mathematical Expectation......Page 192
27. Variance......Page 198
28. Theorems on Expectation and Variance......Page 205
29. The Definition of Mathematical Expectation in Kolmogorov’s Axiomatic Treatment......Page 213
30. Moments......Page 216
Exercises......Page 222
31. Mass Phenomena and the Law of Large Numbers......Page 225
32. Tchebychev’s Form of the Law of Large Numbers......Page 228
33. A Necessary and Sufficient Condition for the Law of Large Numbers......Page 238
34. The Strong Law of Large Numbers......Page 242
Exercises......Page 254
35. The Definition and Simplest Properties of Characteristic Functions......Page 255
36. The Inversion Formula and The Uniqueness Theorem......Page 261
37. Helly’s Theorems......Page 268
38. Limit Theorems for Characteristic Functions......Page 273
39. Positive-Semidefinite Functions......Page 278
40. Characteristic Functions of Multi-Dimensional Random Variables......Page 282
Exercises......Page 288
41. Statement of the Problem......Page 291
42. Liapounov’s Theorem......Page 295
43. The Local Limit Theorem......Page 300
Exercises......Page 306
CHAPTER IX - THE THEORY OF INFINITELY DIVISIBLE DISTRIBUTION LAWS......Page 308
44. Infinitely Divisible Laws and Their Fundamental Properties......Page 309
45. Canonical Representation of Infinitely Divisible Laws......Page 312
46. A Limit Theorem for Infinitely Divisible Laws......Page 317
47. Limit Theorems for Sums: Formulation of the Problem......Page 320
48. Limit Theorems for Sums......Page 321
49. Conditions for Convergence to the Normal and Poisson Laws......Page 325
Exercises......Page 328
50. Introductory Remarks......Page 330
51. The Poisson Process......Page 335
52. Conditional Distribution Functions and Bayes’ Formula......Page 343
53. The Generalized Markov Equation......Page 347
54. Continuous Stochastic Processes. Kolmogorov’s Equations......Page 349
55. Purely Discontinuous Stochastic Processes. The Kolmogorov-Feller Equations......Page 358
56. Homogeneous Stochastic Processes with Independent Increments......Page 365
57. The Concept of a Stationary Stochastic Process. Khintchine’s Theorem on the Correlation Coefficient......Page 371
58. The Notion of a Stochastic Integral. Spectral Decomposition of Stationary Processes......Page 379
59. The Birkhoff-Khintchine Ergodic Theorem......Page 383
60. Some Problems of Mathematical Statistics......Page 389
61. Variational Series and Empirical Distribution Functions......Page 392
62. Glivenko’s Theorem and Kolmogorov’s Compatibility Criterion......Page 394
63. Comparison of Two Distribution Functions......Page 400
64. The Concept of Critical Region. Type I and Type II Errors. Comparison of Two Statistical Hypotheses......Page 406
65. The Classical Procedure for Estimating the Distribution Parameters......Page 414
66. Confidence Limits......Page 424
Tables......Page 434
BIBLIOGRAPHY......Page 450
INDEX......Page 458