An Introduction to Probability and Statistics

A well-balanced introduction to probability theory and mathematical statistics

Featuring updated material, An Introduction to Probability and Statistics, Third Edition remains a solid overview to probability theory and mathematical statistics. Divided intothree parts, the Third Edition begins by presenting the fundamentals and foundationsof probability. The second part addresses statistical inference, and the remainingchapters focus on special topics.

An Introduction to Probability and Statistics, Third Edition includes:

• A new section on regression analysis to include multiple regression, logistic regression, and Poisson regression
• A reorganized chapter on large sample theory to emphasize the growing role of asymptotic statistics
• Additional topical coverage on bootstrapping, estimation procedures, and resampling
• Discussions on invariance, ancillary statistics, conjugate prior distributions, and invariant confidence intervals
• Over 550 problems and answers to most problems, as well as 350 worked out examples and 200 remarks
• Numerous figures to further illustrate examples and proofs throughout

An Introduction to Probability and Statistics, Third Edition is an ideal reference and resource for scientists and engineers in the fields of statistics, mathematics, physics, industrial management, and engineering. The book is also an excellent text for upper-undergraduate and graduate-level students majoring in probability and statistics.

Vijay K. Rohatgi, PhD, is Professor Emeritus in the Department of Mathematics and Statistics at Bowling Green State University.  An Investment Research Consultant for PRI Investments, he is also the author of several books and over 100 research papers. 

A. K. Md. Ehsanes Saleh, PhD, is Distinguished Research Professor in the School of Mathematics and Statistics at Carleton University. Dr. Saleh is the author of more than 200 journal articles, and his research interests include nonparametric statistics, order statistics, and robust estimation. 

PREFACE TO THE THIRD EDITION xiii

PREFACE TO THE SECOND EDITION xv

PREFACE TO THE FIRST EDITION xvii

ACKNOWLEDGMENTS xix

ENUMERATION OF THEOREMS AND REFERENCES xxi

1 Probability 1

1.1 Introduction 1

1.2 Sample Space 2

1.3 Probability Axioms 7

1.4 Combinatorics: Probability on Finite Sample Spaces 20

1.5 Conditional Probability and Bayes Theorem 26

1.6 Independence of Events 31

2 Random Variables and Their Probability Distributions 39

2.1 Introduction 39

2.2 Random Variables 39

2.3 Probability Distribution of a Random Variable 42

2.4 Discrete and Continuous Random Variables 47

2.5 Functions of a Random Variable 55

3 Moments and Generating Functions 67

3.1 Introduction 67

3.2 Moments of a Distribution Function 67

3.3 Generating Functions 83

3.4 Some Moment Inequalities 93

4 Multiple Random Variables 99

4.1 Introduction 99

4.2 Multiple Random Variables 99

4.3 Independent Random Variables 114

4.4 Functions of Several Random Variables 123

4.5 Covariance, Correlation and Moments 143

4.6 Conditional Expectation 157

4.7 Order Statistics and Their Distributions 164

5 Some Special Distributions 173

5.1 Introduction 173

5.2 Some Discrete Distributions 173

5.2.1 Degenerate Distribution 173

5.2.2 Two-Point Distribution 174

5.2.3 Uniform Distribution on n Points 175

5.2.4 Binomial Distribution 176

5.2.5 Negative Binomial Distribution (Pascal or Waiting Time Distribution) 178

5.2.6 Hypergeometric Distribution 183

5.2.7 Negative Hypergeometric Distribution 185

5.2.8 Poisson Distribution 186

5.2.9 Multinomial Distribution 189

5.2.10 Multivariate Hypergeometric Distribution 192

5.2.11 Multivariate Negative Binomial Distribution 192

5.3 Some Continuous Distributions 196

5.3.1 Uniform Distribution (Rectangular Distribution) 199

5.3.2 Gamma Distribution 202

5.3.3 Beta Distribution 210

5.3.4 Cauchy Distribution 213

5.3.5 Normal Distribution (the Gaussian Law) 216

5.3.6 Some Other Continuous Distributions 222

5.4 Bivariate and Multivariate Normal Distributions 228

5.5 Exponential Family of Distributions 240

6 Sample Statistics and Their Distributions 245

6.1 Introduction 245

6.2 Random Sampling 246

6.3 Sample Characteristics and Their Distributions 249

6.4 Chi-Square, t-, and F-Distributions: Exact Sampling Distributions 262

6.5 Distribution of (X,S2) in Sampling from a Normal Population 271

6.6 Sampling from a Bivariate Normal Distribution 276

7 Basic Asymptotics: Large Sample Theory 285

7.1 Introduction 285

7.2 Modes of Convergence 285

7.3 Weak Law of Large Numbers 302

7.4 Strong Law of Large Numbers 308

7.5 Limiting Moment Generating Functions 316

7.6 Central Limit Theorem 321

7.7 Large Sample Theory 331

8 Parametric Point Estimation 337

8.1 Introduction 337

8.2 Problem of Point Estimation 338

8.3 Sufficiency, Completeness and Ancillarity 342

8.4 Unbiased Estimation 359

8.5 Unbiased Estimation (Continued): A Lower Bound for the Variance of An Estimator 372

8.6 Substitution Principle (Method of Moments) 386

8.7 Maximum Likelihood Estimators 388

8.8 Bayes and Minimax Estimation 401

8.9 Principle of Equivariance 418

9 Neyman–Pearson Theory of Testing of Hypotheses 429

9.1 Introduction 429

9.2 Some Fundamental Notions of Hypotheses Testing 429

9.3 Neyman–Pearson Lemma 438

9.4 Families with Monotone Likelihood Ratio 446

9.5 Unbiased and Invariant Tests 453

9.6 Locally Most Powerful Tests 459

10 Some Further Results on Hypotheses Testing 463

10.1 Introduction 463

10.2 Generalized Likelihood Ratio Tests 463

10.3 Chi-Square Tests 472

10.4 t-Tests 484

10.5 F-Tests 489

10.6 Bayes and Minimax Procedures 491

11 Confidence Estimation 499

11.1 Introduction 499

11.2 Some Fundamental Notions of Confidence Estimation 499

11.3 Methods of Finding Confidence Intervals 504

11.4 Shortest-Length Confidence Intervals 517

11.5 Unbiased and Equivariant Confidence Intervals 523

11.6 Resampling: Bootstrap Method 530

12 General Linear Hypothesis 535

12.1 Introduction 535

12.2 General Linear Hypothesis 535

12.3 Regression Analysis 543

12.3.1 Multiple Linear Regression 543

12.3.2 Logistic and Poisson Regression 551

12.4 One-Way Analysis of Variance 554

12.5 Two-Way Analysis of Variance with One Observation Per Cell 560

12.6 Two-Way Analysis of Variance with Interaction 566

13 Nonparametric Statistical Inference 575

13.1 Introduction 575

13.2 U-Statistics 576

13.3 Some Single-Sample Problems 584

13.3.1 Goodness-of-Fit Problem 584

13.3.2 Problem of Location 590

13.4 Some Two-Sample Problems 599

13.4.1 Median Test 601

13.4.2 Kolmogorov–Smirnov Test 602

13.4.3 The Mann–Whitney–Wilcoxon Test 604

13.5 Tests of Independence 608

13.5.1 Chi-square Test of Independence—Contingency Tables 608

13.5.2 Kendall’s Tau 611

13.5.3 Spearman’s Rank Correlation Coefficient 614

13.6 Some Applications of Order Statistics 619

13.7 Robustness 625

13.7.1 Effect of Deviations from Model Assumptions on Some Parametric Procedures 625

13.7.2 Some Robust Procedures 631

FREQUENTLY USED SYMBOLS AND ABBREVIATIONS 637

REFERENCES 641

STATISTICAL TABLES 647

ANSWERS TO SELECTED PROBLEMS 667

AUTHOR INDEX 677

SUBJECT INDEX 679

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