Theory of Probability

by Gnedenko; Boris V.

• ISBN13: 9789056995850
• ISBN10: 9056995855
• Edition: 6th
• Format: Hardcover
• Copyright: 1998-05-13
• Publisher: CRC Press

Summary

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.

Table of Contents

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