Elements of Probability Theory (eBook)

Front Cover 1
Elements of Probability Theory 4
Copyright Page 5
Table of Contents 6
FOREWORD 8
TRANSLATOR'S PREFACE 9
INTRODUCTION 10
CHAPTER I. EVENTS AND PROBABILITIES 14
§ 1. EVENTS. RELATIVE FREQUENCY AND PROBABILITY 14
§ 2. THE CLASSICAL DEFINITION OF PROBABILITY 15
§ 3. FUNDAMENTAL PROPERTIES OF PROBABILITIES. RULE FOR THE ADDITION OF PROBABILITIES 17
§ 4. THE INTERSECTION OF EVENTS. INDEPENDENT EVENTS 21
§ 5. CONDITIONAL PROBABILITIES. THE GENERAL RULE FOR THE MULTIPLICATION OF PROBABILITIES. THE FORMULA OF TOTAL PROBABILITY 26
EXERCISES 30
CHAPTER II. RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 32
§ 6. DISCRETE RANDOM VARIABLES 32
§ 7. THE BINOMIAL DISTRIBUTION 38
§ 8. CONTINUOUS RANDOM VARIABLES 44
§9. FUNCTIONS OF RANDOM VARIABLES 52
EXERCISES 60
CHAPTER .II. NUMERICAL CHARACTERISTICS OF PROBABILITY DISTRIBUTIONS 63
§ 10. MEAN VALUE. THE MATHEMATICAL EXPECTATION OF A RANDOM VARIABLE 63
§ 11. THE CENTRE OF THE PROBABILITY DISTRIBUTION OF A RANDOM VARIABLE 69
§ 12. THE MEASURE OF THE DISPERSION OF A RANDOM VARIABLE. THE MOMENTS OF A DISTRIBUTION 73
EXERCISES 80
CHAPTER IV. THE LAW OF LARGE NUMBERS 82
§ 13. ON EVENTS WITH VERY SMALL PROBABILITY 82
§ 14. BERNOULLI'S THEOREM AND THE STABILITY OF RELATIVE FREQUENCIES 84
§ 15. CHEBYSHEV'S THEOREM 86
§ 16. THE STABILITY OF THE SAMPLE MEAN AND THE METHOD OF MOMENTS 89
EXERCISES 95
CHAPTER V. LIMIT THEOREMS AND ESTIMATES OF THE MEAN 97
§ 17. THE CHARACTERISTIC FUNCTION 97
§ 18. THE LIMIT THEOREM OF DE MOIVRE-LAPLACE. ESTIMATION OF THE RELATIVE FREQUENCY 101
§ 19. RELIABILITY INTERVALS FOR MEANS. THE CENTRAL LIMIT THEOREM OF LYAPUNOV 106
EXERCISES 112
CHAPTER VI. APPLICATIONS OF PROBABILITY TO THE THEORY OF OBSERVATIONS 114
§ 20. RANDOM ERRORS OF MEASUREMENT AND THEIR DISTRIBUTION 114
§21. THE SOLUTION OF TWO FUNDAMENTAL PROBLEMS IN THE THEORY OF ERRORS. ESTIMATION OF THE TRUE VALUE OF THE QUANTITY BEING MEASURED, AND ESTIMATION OF THE ACCURACY OF THE APPARATUS 116
EXERCISES 127
CHAPTER VII. LINEAR CORRELATION 130
§ 22. ON DIFFERENT TYPES OF DEPENDENCE 130
§23. CONDITIONAL EXPECTATIONS AND THEIR PROPERTIES 131
§ 24. LINEAR CORRELATION 135
§ 25. THE COEFFICIENT OF CORRELATION 139
§26. THE BEST LINEAR APPROXIMATION TO THE REGRESSION FUNCTION 141
§ 27. THE ANALYSIS OF LINEAR CORRELATION IN A GIVEN RANDOM SAMPLE. THE SIGNIFICANCE OF THE VALUE OF THE COEFFICIENT OF CORRELATION 143
EXERCISES 150
ANSWERS TO EXERCISES 152
APPENDIX: TABLES 166
VALUES OF THE NORMAL PROBABILITY INTEGRAL 166
VALUES OF THE NORMAL PROBABILITY DENSITY 168
THE POISSON DISTRIBUTION 169
INDEX 172