1 Experiments, Sample Spaces, and Events 1
1.1 Introduction 1
1.2 Sample Space 2
1.3 Algebra of Events 9
1.4 Infinite Operations on Events 15
2 Probability 25
2.1 Introduction 25
2.2 Probability as a Frequency 25
2.3 Axioms of Probability 26
2.4 Consequences of the Axioms 31
2.5 Classical Probability 35
2.6 Necessity of the Axioms* 36
2.7 Subjective Probability* 41
3 Counting 45
3.1 Introduction 45
3.2 Product Sets, Orderings, and Permutations 45
3.3 Binomial Coefficients 51
3.4 Multinomial Coefficients 64
4 Conditional Probability, Independence, and Markov Chains 67
4.1 Introduction 67
4.2 Conditional Probability 68
4.3 Partitions; Total Probability Formula 74
4.4 Bayes Formula 79
4.5 Independence 84
4.6 Exchangeability; Conditional Independence 90
4.7 Markov Chains* 93
5 Random Variables: Univariate Case 107
5.1 Introduction 107
5.2 Distributions of Random Variables 108
5.3 Discrete and Continuous Random Variables 117
5.4 Functions of Random Variables 129
5.5 Survival and Hazard Functions 136
6 Random Variables: Multivariate Case 141
6.1 Bivariate Distributions 141
6.2 Marginal Distributions; Independence 148
6.3 Conditional Distributions 160
6.4 Bivariate Transformations 167
6.5 Multidimensional Distributions 176
7 Expectation 183
7.1 Introduction 183
7.2 Expected Value 184
7.3 Expectation as an Integral* 192
7.4 Properties of Expectation 199
7.5 Moments 207
7.6 Variance 215
7.7 Conditional Expectation 227
7.8 Inequalities 231
8 Selected Families of Distributions 237
8.1 Bernoulli Trials and Related Distributions 237
8.2 Hypergeometric Distribution 251
8.3 Poisson Distribution and Poisson Process 256
8.4 Exponential, Gamma and Related Distributions 269
8.5 Normal Distribution 276
8.6 Beta Distribution 286
9 Random Samples 293
9.1 Statistics and Sampling Distributions 293
9.2 Distributions Related to Normal 295
9.3 Order Statistics 300
9.4 Generating Random Samples 307
9.5 Convergence 312
9.6 Central Limit Theorem 322
10 Introduction to Statistical Inference 331
10.1 Overview 331
10.2 Basic Models 334
10.3 Sampling 336
10.4 Measurement Scales 342
11 Estimation 347
11.1 Introduction 347
11.2 Consistency 352
11.3 Loss, Risk, and Admissibility 355
11.4 Efficiency 361
11.5 Methods of Obtaining Estimators 368
11.6 Sufficiency 387
11.7 Interval Estimation 403
12 Testing Statistical Hypotheses 419
12.1 Introduction 419
12.2 Intuitive Background 423
12.3 Most Powerful Tests 432
12.4 Uniformly Most Powerful Tests 445
12.5 Unbiased Tests 452
12.6 Generalized Likelihood Ratio Tests 456
12.7 Conditional Tests 463
12.8 Tests and Confidence Intervals 466
12.9 Review of Tests for Normal Distributions 467
12.10 Monte Carlo, Bootstrap, and Permutation Tests 477
14 Linear Models 483
14.1 Introduction 483
14.2 Regression of the First and Second Kind 485
14.3 Distributional Assumptions 491
14.4 Linear Regression in the Normal Case 494
14.5 Testing Linearity 500
14.6 Prediction 503
14.7 Inverse Regression 505
14.8 BLUE 508
14.9 Regression Toward the Mean 510
14.10 Analysis of Variance 512
14.11 One-Way Layout 512
14.12 Two-Way Layout 516
14.13 ANOVA Models with Interaction 518
14.14 Further Extensions 522
15 Rank Methods 525
15.1 Introduction 525
15.2 Glivenko-Cantelli Theorem 526
15.3 Kolmogorov-Smirnov Tests 530
15.4 One-Sample Rank Tests 537
15.5 Two-Sample Rank Tests 544
15.6 Kruskal-Wallis Test 548
16 Analysis of Categorical Data 551
16.1 Introduction 551
16.2 Chi-Square Tests 553
16.3 Homogeneity and Independence 559
16.4 Consistency and Power 565
16.5 2×2 Contingency Tables 570
16.6 r × c Contingency Tables 578
17 Basics of Bayesian Statistics 583
17.1 Introduction 583
17.2 Prior and Posterior Distributions 584
17.3 Bayesian Inference 592
17.4 Final Comments 608
Appendix 1 609
Appendix 2 616
Bibliography 619
Answers to Odd-Numbered Problems 624
Index 632
Index 632
