Elements of Probability Theory

Elements of Probability Theory presents the methods of the theory of probability. This book is divided into seven chapters that discuss the general rule for the multiplication of probabilities, the fundamental properties of the subject matter, and the classical definition of probability. The introductory chapters deal with the functions of random variables; continuous random variables; numerical characteristics of probability distributions; center of the probability distribution of a random variable; definition of the law of large numbers; stability of the sample mean and the method of moments; and Chebyshev’s theorem. The next chapters consider the limit theorem of de Moivre-Laplace and the solution of two fundamental problems in the theory of errors. The discussion then shifts to the best linear approximation to the regression function. The concluding chapters look into the central limit theorem of Lyapunov and the significance of the value of the coefficient of correlation. The book can provide useful information to the statisticians, students, and researchers.

Foreword

Translator's Preface

Introduction

Chapter I. Events and Probabilities

§ 1. Events. Relative Frequency and Probability

§ 2. The Classical Definition of Probability

§ 3. Fundamental Properties of Probabilities. Rule for the Addition of Probabilities

§ 4. The Intersection of Events. Independent Events

§ 5. Conditional Probabilities. The General Rule for the Multiplication of Probabilities. The Formula of Total Probability

Exercises

Chapter II. Random Variables and Probability Distributions

§ 6. Discrete Random Variables

§ 7. The Binomial Distribution

§ 8. Continuous Random Variables

§ 9. Functions of Random Variables

Exercises

Chapter III. Numerical Characteristics of Probability Distributions

§ 10. Mean Value. The Mathematical Expectation of a Random Variable

§ 11. The Center of the Prooadiiity Distribution of a Random Variable

§ 12. The Measure of the Dispersion of a Random Variable. The Moments of a Distribution

Exercises

Chapter IV. The Law of Large Numbers

§ 13. On Events with very Small Probability

§ 14. Bernoulli's Theorem and the Stability of Relative Frequencies

§ 15. Chebyshev's Theorem

§ 16. The Stability of the Sample Méafa and the Method of Moments

Exercises

Chapter V. Limit Theorems and Estimates of the Mean

§ 17. The Characteristic Function

§ 18. The Limit Theorem of De Moivre-Laplace. Estimation of the Relative Frequency

§ 19. Reliability Intervals For Means. The Central Limit Theorem of Lyapunov

Exercises

Chapter VI. Applications of Probability to the Theory of Observations

§ 20. Random Errors of Measurement and Their Distribution

§ 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

Exercises

Chapter VII Linearcorrelation

§ 22. On Different Types of Dependence

§ 23. Conditional Expectations and Their Properties

§ 24. Linear Correlation

§ 25. The Coefficient of Correlation

§ 26. The Best Linear Approximation to the Regression Function

§ 27. The Analysis of Linear Correlation in a Given Random Sample. The Significance of the Value of the Coefficient of Correlation

Exercises

Answers to Exercises

Appendix. Tables

I. Values of the Normal Probability Integral

II. Values of the Normal Probability Integral

III. Values of the Normal Probability Density

IV. The Poisson Distribution

Index