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Probability and Statistical Inference
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Preface to Third Edition xi

Preface to Second Edition xiii

1 Experiments, Sample Spaces, and Events 1

1.1 Introduction 1

1.2 Sample Space 2

1.3 Algebra of Events 8

1.4 Infinite Operations on Events 13

2 Probability 21

2.1 Introduction 21

2.2 Probability as a Frequency 21

2.3 Axioms of Probability 22

2.4 Consequences of the Axioms 26

2.5 Classical Probability 30

2.6 Necessity of the Axioms 31

2.7 Subjective Probability 35

3 Counting 39

3.1 Introduction 39

3.2 Product Sets, Orderings, and Permutations 39

3.3 Binomial Coefficients 44

3.4 Multinomial Coefficients 56

4 Conditional Probability, Independence, and Markov Chains 59

4.1 Introduction 59

4.2 Conditional Probability 60

4.3 Partitions; Total Probability Formula 65

4.4 Bayes' Formula 69

4.5 Independence 74

4.6 Exchangeability; Conditional Independence 80

4.7 Markov Chains* 82

5 Random Variables: Univariate Case 93

5.1 Introduction 93

5.2 Distributions of Random Variables 94

5.3 Discrete and Continuous Random Variables 102

5.4 Functions of Random Variables 112

5.5 Survival and Hazard Functions 118

6 Random Variables: Multivariate Case 123

6.1 Bivariate Distributions 123

6.2 Marginal Distributions; Independence 129

6.3 Conditional Distributions 140

6.4 Bivariate Transformations 147

6.5 Multidimensional Distributions 155

7 Expectation 163

7.1 Introduction 163

7.2 Expected Value 164

7.3 Expectation as an Integral 171

7.4 Properties of Expectation 177

7.5 Moments 184

7.6 Variance 191

7.7 Conditional Expectation 202

7.8 Inequalities 206

8 Selected Families of Distributions 211

8.1 Bernoulli Trials and Related Distributions 211

8.2 Hypergeometric Distribution 223

8.3 Poisson Distribution and Poisson Process 228

8.4 Exponential, Gamma, and Related Distributions 240

8.5 Normal Distribution 246

8.6 Beta Distribution 255

9 Random Samples 259

9.1 Statistics and Sampling Distributions 259

9.2 Distributions Related to Normal 261

9.3 Order Statistics 266

9.4 Generating Random Samples 272

9.5 Convergence 276

9.6 Central Limit Theorem 287

10 Introduction to Statistical Inference 295

10.1 Overview 295

10.2 Basic Models 298

10.3 Sampling 299

10.4 Measurement Scales 305

11 Estimation 309

11.1 Introduction 309

11.2 Consistency 313

11.3 Loss, Risk, and Admissibility 316

11.4 Efficiency 321

11.5 Methods of Obtaining Estimators 328

11.6 Sufficiency 345

11.7 Interval Estimation 359

12 Testing Statistical Hypotheses 373

12.1 Introduction 373

12.2 Intuitive Background 377

12.3 Most Powerful Tests 384

12.4 Uniformly Most Powerful Tests 396

12.5 Unbiased Tests 402

12.6 Generalized Likelihood Ratio Tests 405

12.7 Conditional Tests 412

12.8 Tests and Confidence Intervals 415

12.9 Review of Tests for Normal Distributions 416

12.10 Monte Carlo, Bootstrap, and Permutation Tests 424

13 Linear Models 429

13.1 Introduction 429

13.2 Regression of the First and Second Kind 431

13.3 Distributional Assumptions 436

13.4 Linear Regression in the Normal Case 438

13.5 Testing Linearity 444

13.6 Prediction 447

13.7 Inverse Regression 449

13.8 BLUE 451

13.9 Regression Toward the Mean 453

13.10 Analysis of Variance 455

13.11 One-Way Layout 455

13.12 Two-Way Layout 458

13.13 ANOVA Models with Interaction 461

13.14 Further Extensions 465

14 Rank Methods 467

14.1 Introduction 467

14.2 Glivenko-Cantelli Theorem 468

14.3 Kolmogorov-Smirnov Tests 471

14.4 One-Sample Rank Tests 478

14.5 Two-Sample Rank Tests 484

14.6 Kruskal-Wallis Test 488

15 Analysis of Categorical Data 491

15.1 Introduction 491

15.2 Chi-Square Tests 492

15.3 Homogeneity and Independence 499

15.4 Consistency and Power 504

15.5 2 x 2 Contingency Tables 509

15.6 r x c Contingency Tables 516

16 Basics of Bayesian Statistics 521

16.1 Introduction 521

16.2 Prior and Posterior Distributions 522

16.3 Bayesian Inference 529

Appendix A Supporting R Code 545

Appendix B Statistical Tables 551

Bibliography 555

Index 571  