An introduction to probability and statistics

An Introduction to Probability and Statistics......Page 5
Contents......Page 9
Preface to the Second Edition......Page 13
Preface to the First Edition......Page 15
1.1 Introduction......Page 21
1.2 Sample Space......Page 22
1.3 Probability Axioms......Page 27
1.4 Combinatorics: Probability on Finite Sample Spaces......Page 41
1.5 Conditional Probability and Bayes Theorem......Page 48
1.6 Independence of Events......Page 53
2.2 Random Variables......Page 60
2.3 Probability Distribution of a Random Variable......Page 63
2.4 Discrete and Continuous Random Variables......Page 68
2.5 Functions of a Random Variable......Page 77
3.2 Moments of a Distribution Function......Page 89
3.3 Generating Functions......Page 105
3.4 Some Moment Inequalities......Page 115
4.2 Multiple Random Variables......Page 122
4.3 Independent Random Variables......Page 139
4.4 Functions of Several Random Variables......Page 147
4.5 Covariance, Correlation, and Moments......Page 169
4.6 Conditional Expectation......Page 184
4.7 Order Statistics and Their Distributions......Page 191
5.2 Some Discrete Distributions......Page 200
5.3 Some Continuous Distributions......Page 224
5.4 Bivariate and Multivariate Normal Distributions......Page 258
5.5 Exponential Family of Distributions......Page 271
6.2 Modes of Convergence......Page 276
6.3 Weak Law of Large Numbers......Page 294
6.4 Strong Law of Large Numbers......Page 301
6.5 Limiting Moment Generating Functions......Page 309
6.6 Central Limit Theorem......Page 313
7.1 Introduction......Page 326
7.2 Random Sampling......Page 327
7.3 Sample Characteristics and Their Distributions......Page 330
7.4 Chi-Square, t-, and F-Distributions: Exact Sampling Distributions......Page 344
7.5 Large-Sample Theory......Page 354
7.6 Distribution of (X, S2) in Sampling from a Normal Population......Page 359
7.7 Sampling from a Bivariate Normal Distribution,......Page 364
8.1 Introduction......Page 373
8.2 Problem of Point Estimation......Page 374
8.3 Sufficiency, Completeness, and Ancillarity......Page 378
8.4 Unbiased Estimation......Page 397
8.5 Unbiased Estimation (Continued): Lower Bound for the Variance of an Estimator......Page 411
8.6 Substitution Principle (Method of Moments)......Page 426
8.7 Maximum Likelihood Estimators......Page 429
8.8 Bayes and Minimax Estimation......Page 444
8.9 Principle of Equivariance......Page 462
9.2 Some Fundamental Notions of Hypotheses Testing......Page 474
9.3 Neyman–Pearson Lemma......Page 484
9.4 Families with Monotone Likelihood Ratio......Page 492
9.5 Unbiased and Invariant Tests......Page 499
9.6 Locally Most Powerful Tests......Page 506
10.2 Generalized Likelihood Ratio Tests......Page 510
10.3 Chi-Square Tests......Page 520
10.4 t-Tests......Page 532
10.5 F-Tests......Page 538
10.6 Bayes and Minimax Procedures......Page 540
11.2 Some Fundamental Notions of Confidence Estimation......Page 547
11.3 Methods of Finding Confidence Intervals......Page 552
11.4 Shortest-Length Confidence Intervals......Page 566
11.5 Unbiased and Equivariant Confidence Intervals......Page 573
12.2 General Linear Hypothesis......Page 581
12.3 Regression Model......Page 589
12.4 One-Way Analysis of Variance......Page 597
12.5 Two-Way Analysis of Variance with One Observation per Cell......Page 603
12.6 Two-Way Analysis of Variance with Interaction......Page 610
13.2 U-Statistics......Page 618
13.3 Some Single-Sample Problems......Page 628
13.4 Some Two-Sample Problems......Page 644
13.5 Tests of Independence......Page 653
13.6 Some Applications of Order Statistics......Page 664
13.7 Robustness......Page 670
References......Page 683
Frequently Used Symbols and Abbreviations......Page 689
Statistical Tables......Page 693
Answers to Selected Problems......Page 713
Author Index......Page 725
Subject Index......Page 727