2-criterion
..... 136
Probability Theory for Engineers, 1987, 173 pp., ISBN 0-911575-13-8, $30.00
A concise, authoritative introductory text specifically
designed to meet the needs of Engineers by three world-renowned experts
from the prestigious Steklov Institute of Mathematics, Moscow. A unique
feature is the systematic use of the notion of Random Variable rather than
the Distribution Law. Succinct and yet precise treatment with a profusion
of illustrative examples, and problems. Separate chapters on Least Squares
method, Hypothesis testing, and Statistical Estimation, including the Principle
of Maximum Likelihood. Foreword by G. Kallianpur.
TABLE OF CONTENTS
Foreword ..... xi
Preface ..... xiii
CHAPTER 1. MATHEMATICAL MODELS OF RANDOM PHENOMENA ..... 1
1.1. Mathematical Models ..... 1
1.2. Random Phenomena ..... 1
1.3. Space of Elementary Events ..... 3
1.4. Algebra of Events ..... 4
1.5. Probability: Basic Definitions ..... 6
1.6. Finite Probability Spaces ..... 7
1.7. Countable Probability Spaces ..... 12
1.8. Continuous Probability Spaces ..... 14
PROBLEMS ..... 18
CHAPTER 2. CONDITIONAL PROBABILITIES.
INDEPENDENCE OF EVENTS ..... 20
2.1. Conditional Probabilities ..... 20
2.2. The Total Probability Formula. The Bayes Formula. ..... 23
2.3. Independence of Events ..... 25
2.4. Applications of the Total Probability Formula ..... 26
PROBLEMS ..... 28
CHAPTER 3. SEQUENCES OF TRIALS ..... 31
3.1. Definition of a Sequence of Indpendent Trials ..... 31
3.2. General Definition of a Sequence of Trials ..... 35
PROBLEMS ..... 38
CHAPTER 4. LIMIT THEOREMS IN A BERNOULLI MODEL ..... 39
4.1. Poisson’s Theorem ..... 39
4.2. The De Moivre-Laplace Theorem ..... 41
PROBLEMS ..... 47
CHAPTER 5. RANDOM VARIABLES ..... 49
5.1. Random Variables: Finite Scheme ..... 49
5.2. Random Variables: Countable Scheme ..... 54
5.3. Random Variables: General Scheme. Distribution Function ..... 55
5.4. Functions of Random Variables ..... 63
PROBLEMS ..... 64
CHAPTER 6. JOINT DISTRIBUTION OF RANDOM VARIABLES ..... 66
6.1. Multidimensional Distribution Laws ..... 66
6.2. Independence of Random Variables ..... 69
6.3. Convolution of Distributions ..... 71
PROBLEMS ..... 75
CHAPTER 7. MATHEMATICAL EXPECTATION ..... 76
7.1. Mathematical Expectation: Finite Scheme ..... 76
7.2. Mathematical Expectation: Countable Scheme ..... 83
7.3. Mathematical Expectation: The General Case ..... 84
PROBLEMS ..... 89
CHAPTER 8. VARIANCE AND MOMENTS ..... 90
8.1. Definition of Variance ..... 90
8.2. Properties of Variance ..... 93
8.3. Moments of Higher Order ..... 96
PROBLEMS ..... 98
CHAPTER 9. COVARIANCE AND CORRELATION COEFFICIENTS ..... 99
PROBLEMS ..... 104
CHAPTER 10. THE LAW OF LARGE NUMBERS ..... 106
10.1. The Chebyshev Inequality ..... 106
10.2. The Weak Law of Large Numbers ..... 107
PROBLEMS ..... 109
CHAPTER 11. CENTRAL LIMIT THEOREM ..... 111
PROBLEMS ..... 116
CHAPTER 12. PROCESSING MEASUREMENT DATA ..... 117
12.1. Sample ..... 117
12.2. Estimation ..... 117
12.3. Interval Estimates ..... 123
12.4. Maximum Likelihood Method of Parameter Estimation.
Method of Moments ..... 126
CHAPTER 13. THE METHOD OF LEAST SQUARES ..... 129
CHAPTER 14. STATISTICAL TESTING OF HYPOTHESES ..... 136
14.1. The 2-criterion
..... 136
14.2. Choosing Between Two Hypotheses ..... 139
Tables ..... 142
Appendix. Proof of the Local De Moivre-Laplace Theorem ..... 147
Answers to the Problems ..... 149
References ..... 153
Index ..... 155
Transliteration Table (Russian-English) ..... 159