Probability and Mathematical Statistics

PrefaceIntroductionPart I. Probability Theory Chapter 1. The Probability Space 1.1 The Outcome Space 1.2 Probabilities 1.3 The Axioms 1.4 Problems References Chapter 2. Elementary Properties of Probability Spaces 2.1 Simple Consequences of the Axioms 2.2 Conditional Probability and Independence 2.3 Finite Probability Spaces 2.4 Problems Chapter 3. Random Variables and Their Probability Distributions 3.1 Random Variables 3.2 Distribution Functions 3.3 Examples of Discrete Distributions 3.4 Examples of Absolutely Continuous Distributions 3.5 Multivariate Distributions 3.6 Problems References Chapter 4. Typical Values 4.1 The Mathematical Expectation of a Random Variable 4.2 Expectations of Functions of Random Variables 4.3 Properties of Expectations 4.4 Moments 4.5 Regression 4.6 Problems Reference Chapter 5. Limit Theorems 5.1 Laws of Large Numbers 5.2 The Central Limit Theorem 5.3 The Poisson Approximation to the Binomial 5.4 Problems References Chapter 6. Some Important Distributions 6.1 The Distribution of the Sum of Independent, Absolutely Continuous Random Variables 6.2 Addition of Independent Normal Random Variables 6.3 The Chi-Square Distribution 6.4 Student's Distribution 6.5 ProblemsPart II. Mathematical Statistics Chapter 7. Sampling 7.1 Statistical Data 7.2 Sample Characteristics 7.3 Moments and Distributions of Sample Characteristics 7.4 Problems References Chapter 8. Estimation 8.1 Properties of Estimates 8.2 Point Estimation 8.3 Interval Estimation 8.4 Problems References Chapter 9. Testing Hypotheses 9.1 Statistical Hypotheses 9.2 The Power of a Test 9.3 The t-Test 9.4 Nonparametric Methods 9.5 Problems ReferencesAppendix A. Some Combinatorial FormulasAppendix B. The Gamma FunctionAppendix C. Proof of the Central Limit TheoremAppendix D. TablesAnswers to Selected ProblemsIndex