An Introduction to Probability and Statistics

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