Probability and statistics

Description

Probability and Statistics Title Page Contents Preface 1 Introduction to Probability 1.1 The History of Probability 1.2 Interpretations of Probability 1.3 Experiments and Events 1.4 Set Theory 1.5 The Definition of Probability 1.6 Finite Sample Spaces 1.7 Counting Methods 1.8 Combinatorial Methods 1.9 Multinomial Coefficients 1.10 The Probability of a Union of Events 1.11 Statistical Swindles 1.12 Supplementary Exercises 2 Conditional Probability 2.1 The Definition of Conditional Probability 2.2 Independent Events 2.3 Bayes’ Theorem 2.4 The Gambler’s Ruin Problem 2.5 Supplementary Exercises 3 Random Variables and Distributions 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions 3.6 Conditional Distributions 3.7 Multivariate Distributions 3.8 Functions of a Random Variable 3.9 Functions of Two or More Random Variables 3.10 Markov Chains 3.11 Supplementary Exercises 4 Expectation 4.1 The Expectation of a Random Variable 4.2 Properties of Expectations 4.3 Variance 4.4 Moments 4.5 The Mean and the Median 4.6 Covariance and Correlation 4.7 Conditional Expectation 4.8 Utility 4.9 Supplementary Exercises 5 Special Distributions 5.1 Introduction 5.2 The Bernoulli and Binomial Distributions 5.3 The Hypergeometric Distributions 5.4 The Poisson Distributions 5.5 The Negative Binomial Distributions 5.6 The Normal Distributions 5.7 The Gamma Distributions 5.8 The Beta Distributions 5.9 The Multinomial Distributions 5.10 The Bivariate Normal Distributions 5.11 Supplementary Exercises 6 Large Random Samples 6.1 Introduction 6.2 The Law of Large Numbers 6.3 The Central Limit Theorem 6.4 The Correction for Continuity 6.5 Supplementary Exercises 7 Estimation 7.1 Statistical Inference 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions 7.4 Bayes Estimators 7.5 Maximum Likelihood Estimators 7.6 Properties of Maximum Likelihood Estimators 7.7 Sufficient Statistics 7.8 Jointly Sufficient Statistics 7.9 Improving an Estimator 7.10 Supplementary Exercises 8 Sampling Distributions of Estimators 8.1 The Sampling Distribution of a Statistic 8.2 The Chi-Square Distributions 8.3 Joint Distribution of the Sample Mean and Sample Variance 8.4 The t Distributions 8.5 Confidence Intervals 8.6 Bayesian Analysis of Samples from a Normal Distribution 8.7 Unbiased Estimators 8.8 Fisher Information 8.9 Supplementary Exercises 9 Testing Hypotheses 9.1 Problems of Testing Hypotheses 9.2 Testing Simple Hypotheses 9.3 Uniformly Most Powerful Tests 9.4 Two-Sided Alternatives 9.5 The t Test 9.6 Comparing the Means of Two Normal Distributions 9.7 The F Distributions 9.8 Bayes Test Procedures 9.9 Foundational Issues 9.10 Supplementary Exercises 10 Categorical Data and Nonparametric Methods 10.1 Tests of Goodness-of-Fit 10.2 Goodness-of-Fit for Composite Hypotheses 10.3 Contingency Tables 10.4 Tests of Homogeneity 10.5 Simpson’s Paradox 0.6 Kolmogorov-Smirnov Tests 10.7 Robust Estimation 10.8 Sign and Rank Tests 10.9 Supplementary Exercises 11 Linear Statistical Models 11.1 The Method of Least Squares 11.2 Regression 11.3 Statistical Inference in Simple Linear Regression 11.4 Bayesian Inference in Simple Linear Regression 11.5 The General Linear Model and Multiple Regression 11.6 Analysis of Variance 11.7 The Two-Way Layout 11.8 The Two-Way Layout with Replications 11.9 Supplementary Exercises 12 Simulation 12.1 What Is Simulation? 12.2 Why Is Simulation Useful? 12.3 Simulating Specific Distributions 12.4 Importance Sampling 12.5 Markov Chain Monte Carlo 12.6 The Bootstrap 12.7 Supplementary Exercises Tables Answers to Odd-Numbered Exercises References Index Corrections Instructor's Solutions Manual Contents Preface 1 Introduction to Probability 2 Conditional Probability 3 Random Variables and Distributions 4 Expectation 5 Special Distributions 6 Large Random Samples 7 Estimation 8 Sampling Distributions of Estimators 9 Testing Hypotheses 10 Categorical Data and Nonparametric Methods 11 Linear Statistical Models 12 Simulation