An Introduction to Probability and Statistics

Content: 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