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1 Basic Probability Theory1

1.1 Introduction1

1.2 Sample Spaces and Events3

1.3 The Axioms of Probability7

1.4 Finite Sample Spaces and Combinatorics15

1.4.1 Combinatorics17

1.5 Conditional Probability and Independence27

1.5.1 Independent Events33

1.6 The Law of Total Probability and Bayes’ Formula41

1.6.1 Bayes’ Formula47

1.6.2 Genetics and Probability54

1.6.3 Recursive Methods55

Problems63

2 Random Variables76

2.1 Introduction76

2.2 Discrete Random Variables77

2.3 Continuous Random Variables82

2.3.1 The Uniform Distribution90

2.3.2 Functions of Random Variables92

2.4 Expected Value and Variance95

2.4.1 The Expected Value of a Function of a Random Variable100

2.4.2 Variance of a Random Variable104

2.5 Special Discrete Distributions111

2.5.1 Indicators111

2.5.2 The Binomial Distribution112

2.5.3 The Geometric Distribution116

2.5.4 The Poisson Distribution117

2.5.5 The Hypergeometric Distribution121

2.5.6 Describing Data Sets121

2.6 The Exponential Distribution123

2.7 The Normal Distribution127

2.8 Other Distributions131

2.8.1 The Lognormal Distribution131

2.8.2 The Gamma Distribution133

2.8.3 The Cauchy Distribution134

2.8.4 Mixed Distributions135

2.9 Location Parameters137

2.10 The Failure Rate Function139

2.10.1 Uniqueness of the Failure Rate Function141

Problems144

3 Joint Distributions156

3.1 Introduction156

3.2 The Joint Distribution Function156

3.3 Discrete Random Vectors158

3.4 Jointly Continuous Random Vectors160

3.5 Conditional Distributions and Independence164

3.5.1 Independent Random Variables168

3.6 Functions of Random Vectors172

3.6.1 Real-Valued Functions of Random Vectors172

3.6.2 The Expected Value and Variance of a Sum176

3.6.3 Vector-Valued Functions of Random Vectors182

3.7 Conditional Expectation185

3.7.1 Conditional Expectation as a Random Variable189

3.7.2 Conditional Expectation and Prediction191

3.7.3 Conditional Variance192

3.7.4 Recursive Methods193

3.8 Covariance and Correlation196

3.8.1 The Correlation Coefficient201

3.9 The Bivariate Normal Distribution209

3.10 Multidimensional Random Vectors216

3.10.1 Order Statistics218

3.10.2 Reliability Theory223

3.10.3 The Multinomial Distribution225

3.10.4 The Multivariate Normal Distribution226

3.10.5 Convolution227

3.11 Generating Functions231

3.11.1 The Probability Generating Function231

3.11.2 The Moment Generating Function237

3.12 The Poisson Process240

3.12.1 Thinning and Superposition244

Problems247

4 Limit Theorems263

4.1 Introduction263

4.2 The Law of Large Numbers264

4.3 The Central Limit Theorem268

4.3.1 The Delta Method273

4.4 Convergence in Distribution275

4.4.1 Discrete Limits275

4.4.2 Continuous Limits277

Problems278

5 Simulation281

5.1 Introduction281

5.2 Random Number Generation282

5.3 Simulation of Discrete Distributions283

5.4 Simulation of Continuous Distributions285

5.5 Miscellaneous290

Problems292

6 Statistical Inference294

6.1 Introduction294

6.2 Point Estimators294

6.2.1 Estimating the Variance302

6.3 Confidence Intervals304

6.3.1 Confidence Interval for the Mean in the Normal Distribution with Known Variance307

6.3.2 Confidence Interval for an Unknown Probability308

6.3.3 One-Sided Confidence Intervals312

6.4 Estimation Methods312

6.4.1 The Method of Moments312

6.4.2 Maximum Likelihood315

6.4.3 Evaluation of Estimators with Simulation322

6.4.4 Bootstrap Simulation324

6.5 Hypothesis Testing327

6.5.1 Large Sample Tests332

6.5.2 Test for an Unknown Probability333

6.6 Further Topics in Hypothesis Testing334

6.6.1 P-Values334

6.6.2 Data Snooping335

6.6.3 The Power of a Test336

6.6.4 Multiple Hypothesis Testing338

6.7 Goodness of Fit339

6.7.1 Goodness-of-Fit Test for Independence346

6.7.2 Fisher’s Exact Test349

6.8 Bayesian Statistics351

6.8.1 Noninformative priors359

6.8.2 Credibility Intervals362

6.9 Nonparametric Methods363

6.9.1 Nonparametric Hypothesis Testing363

6.9.2 Comparing Two Samples370

6.9.3 Nonparametric Confidence Intervals375

Problems378

7 Linear Models391

7.1 Introduction391

7.2 Sampling Distributions392

7.3 Single Sample Inference395

7.3.1 Inference for the Variance396

7.3.2 Inference for the Mean399

7.4 Comparing Two Samples402

7.4.1 Inference about Means402

7.4.2 Inference about Variances407

7.5 Analysis of Variance409

7.5.1 One-Way Analysis of Variance409

7.5.2 Multiple Comparisons：Tukey’s Method412

7.5.3 Kruskal-Wallis Test413

7.6 Linear Regression415

7.6.1 Prediction422

7.6.2 Goodness of Fit424

7.6.3 The Sample Correlation Coefficient425

7.6.4 Spearman’s Correlation Coefficient429

7.7 The General Linear Model431

Problems436

8 Stochastic Processes444

8.1 Introduction444

8.2 Discrete-Time Markov Chains445

8.2.1 Time Dynamics of a Markov Chain447

8.2.2 Classification of States450

8.2.3 Stationary Distributions454

8.2.4 Convergence to the Stationary Distribution460

8.3 Random Walks and Branching Processes464

8.3.1 The Simple Random Walk464

8.3.2 Multidimensional Random Walks468

8.3.3 Branching Processes469

8.4 Continuous-Time Markov Chains475

8.4.1 Stationary Distributions and Limit Distributions480

8.4.2 Birth-Death Processes484

8.4.3 Queueing Theory488

8.4.4 Further Properties of Queueing Systems491

8.5 Martingales494

8.5.1 Martingale Convergence495

8.5.2 Stopping Times497

8.6 Renewal Processes502

8.6.1 Asymptotic Properties504

8.7 Brownian Motion509

8.7.1 Hitting Times512

8.7.2 Variations of the Brownian Motion515

Problems517

Appendix A Tables527

Appendix B Answers to Selected Problems535

Further Reading551

Index553