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