Introduction to Probability and Statistical Inference (eBook)

George G. Roussas (Autor)

eBook Download: PDF | EPUB

2014 | 2. Auflage
624 Seiten
Elsevier Science (Verlag)
978-0-12-800437-1 (ISBN)

Infos

An Introduction to Probability and Statistical Inference, Second Edition, guides you through probability models and statistical methods and helps you to think critically about various concepts. Written by award-winning author George Roussas, this book introduces readers with no prior knowledge in probability or statistics to a thinking process to help them obtain the best solution to a posed question or situation. It provides a plethora of examples for each topic discussed, giving the reader more experience in applying statistical methods to different situations. This text contains an enhanced number of exercises and graphical illustrations where appropriate to motivate the reader and demonstrate the applicability of probability and statistical inference in a great variety of human activities. Reorganized material is included in the statistical portion of the book to ensure continuity and enhance understanding. Each section includes relevant proofs where appropriate, followed by exercises with useful clues to their solutions. Furthermore, there are brief answers to even-numbered exercises at the back of the book and detailed solutions to all exercises are available to instructors in an Answers Manual. This text will appeal to advanced undergraduate and graduate students, as well as researchers and practitioners in engineering, business, social sciences or agriculture. - Content, examples, an enhanced number of exercises, and graphical illustrations where appropriate to motivate the reader and demonstrate the applicability of probability and statistical inference in a great variety of human activities - Reorganized material in the statistical portion of the book to ensure continuity and enhance understanding - A relatively rigorous, yet accessible and always within the prescribed prerequisites, mathematical discussion of probability theory and statistical inference important to students in a broad variety of disciplines - Relevant proofs where appropriate in each section, followed by exercises with useful clues to their solutions - Brief answers to even-numbered exercises at the back of the book and detailed solutions to all exercises available to instructors in an Answers Manual

George G. Roussas earned a B.S. in Mathematics with honors from the University of Athens, Greece, and a Ph.D. in Statistics from the University of California, Berkeley. As of July 2014, he is a Distinguished Professor Emeritus of Statistics at the University of California, Davis. Roussas is the author of five books, the author or co-author of five special volumes, and the author or co-author of dozens of research articles published in leading journals and special volumes. He is a Fellow of the following professional societies: The American Statistical Association (ASA), the Institute of Mathematical Statistics (IMS), The Royal Statistical Society (RSS), the American Association for the Advancement of Science (AAAS), and an Elected Member of the International Statistical Institute (ISI); also, he is a Corresponding Member of the Academy of Athens. Roussas was an associate editor of four journals since their inception, and is now a member of the Editorial Board of the journal Statistical Inference for Stochastic Processes. Throughout his career, Roussas served as Dean, Vice President for Academic Affairs, and Chancellor at two universities; also, he served as an Associate Dean at UC-Davis, helping to transform that institution's statistical unit into one of national and international renown. Roussas has been honored with a Festschrift, and he has given featured interviews for the Statistical Science and the Statistical Periscope. He has contributed an obituary to the IMS Bulletin for Professor-Academician David Blackwell of UC-Berkeley, and has been the coordinating editor of an extensive article of contributions for Professor Blackwell, which was published in the Notices of the American Mathematical Society and the Celebratio Mathematica.

Inhalt

Front Cover 1
Inside Front Cover 2
An Introduction to Probability and Statistical Inference 6
Copyright 7
Dedication 8
Contents 10
Preface 14
Overview 14
Chapter Descriptions 14
Features 14
Brief Preface of the Revised Version 16
Acknowledgments and Credits 16
Chapter 1: Some motivating examples and some fundamental concepts 18
1.1 Some motivating examples 18
1.2 Some fundamental concepts 25
1.3 Random variables 37
Chapter 2: The concept of probability and basic results 40
2.1 Definition of probability and some basic results 40
2.1.1 Some basic properties of a probability function 44
2.1.2 Justification 45
2.2 Distribution of a random variable 51
2.3 Conditional probability and related results 59
2.4 Independent events and related results 72
2.5 Basic concepts and results in counting 81
Chapter 3: Numerical characteristics of a random variable, some special random variables 94
3.1 Expectation, variance, and moment generating function of a random variable 94
3.2 Some probability inequalities 107
3.3 Some special random variables 110
3.3.1 The discrete case 110
Binomial distribution 110
Geometric distribution 112
Poisson Distribution 114
Hypergeometric distribution 116
3.3.2 The continuous case 117
Gamma distribution 117
Negative Exponential distribution 119
Chi-square distribution 121
Normal distribution 121
Uniform (or Rectangular) distribution 126
3.4 Median and mode of a random variable 142
Chapter 4: Joint and conditional p.d.f.'s, conditional expectation and variance, moment generating function, covariance, a ... 152
4.1 Joint d.f. and joint p.d.f. of two random variables 152
4.2 Marginal and conditional p.d.f.'s, conditional expectation and variance 161
4.3 Expectation of a function of two r.v.'s, joint and marginal m.g.f.'s 175
Expectation of a function of two r.v.'s, joint and marginal m.g.f.'s, covariance, and correlation coefficient 175
4.4 Some generalizations to k random variables 188
4.5 The Multinomial, the Bivariate Normal, and the multivariate Normal 190
4.5.1 Multinomial distribution 190
4.5.2 Bivariate Normal distribution 193
4.5.3 Multivariate Normal distribution 198
Chapter 5: Independence of random variables and some applications 204
5.1 Independence of random variables and criteria of independence 204
5.2 The reproductive property of certain distributions 215
Chapter 6: Transformation of random variables 224
6.1 Transforming a single random variable 224
6.2 Transforming two or more random variables 229
6.3 Linear transformations 243
6.4 The probability integral transform 249
6.5 Order statistics 250
Chapter 7: Some modes of convergence of random variables, applications 262
7.1 Convergence in distribution or in probability and their relationship 262
7.2 Some applications of convergence in distribution: WLLN and CLT 268
7.2.1 Applications of the WLLN 270
7.2.2 Applications of the CLT 274
7.2.3 The continuity correction 276
7.3 Further limit theorems 284
Chapter 8: An overview of statistical inference 290
8.1 The basics of point estimation 291
8.2 The basics of interval estimation 293
8.3 The basics of testing hypotheses 294
8.4 The basics of regression analysis 298
8.5 The basics of analysis of variance 299
8.6 The basics of nonparametric inference 301
Chapter 9: Point estimation 304
9.1 Maximum likelihood estimation: Motivation and examples 304
9.2 Some properties of MLE's 317
9.3 Uniformly minimum variance unbiased estimates 325
9.4 Decision-theoretic approach to estimation 334
9.5 Other methods of estimation 341
Chapter 10: Confidence intervals and confidence regions 346
10.1 Confidence intervals 346
10.2 Confidence intervals in the presence of nuisance parameters 354
10.3 A confidence region for ((µ, s22) in the N((µ, s2) distribution 357
10.4 Confidence intervals with approximate confidence coefficient 360
Chapter 11: Testing hypotheses 366
11.1 General concepts, formulation of some testing hypotheses 367
11.2 Neyman–Pearson fundamental lemma, Exponential type families, UMP tests for some composite hypotheses 369
11.2.1 Exponential type families of p.d.f.'s 376
11.2.2 UMP Tests for some composite hypotheses 377
11.3 Some applications of theorems 2 380
11.3.1 Further uniformly most powerful tests for some composite hypotheses 389
11.3.2 An application of Theorem 3 390
11.4 Likelihood ratio tests 392
11.4.1 Testing hypotheses for the parameters in a single Normal population 395
11.4.2 Comparing the parameters of two Normal populations 401
Chapter 12: More about testing hypotheses 414
12.1 Likelihood ratio tests in the Multinomial case and contingency tables 414
12.2 A goodness-of-fit test 420
12.3 Decision-theoretic approach to testing hypotheses 425
12.4 Relationship between testing hypotheses and confidence regions 432
Chapter 13: A simple linear regression model 436
13.1 Setting up the model—the principle of least squares 436
13.2 The least squares estimates of 1 and 2 and some of their properties 439
13.3 Normally distributed errors: mle’s of ß1, ß2, and s2, some distributional results 447
13.4 Confidence intervals and hypotheses testing problems 456
13.5 Some prediction problems 462
13.6 Proof of theorem 5 466
13.7 Concluding remarks 468
Chapter 14: Two models of analysis of variance 470
14.1 One-way layout with the same number of observations per cell 470
14.1.1 The MLE's of the parameters of the model 471
14.1.2 Testing the hypothesis of equality of means 472
14.1.3 Proof of Lemmas in Section 14.1 478
14.2 A multicomparison method 480
14.3 Two-way layout with one observation per cell 486
14.3.1 The MLE's of the parameters of the model 487
14.3.2 Testing the hypothesis of no row or no column effects 488
14.3.3 Proof of Lemmas in Section 14.3 494
Chapter 15: Some topics in nonparametric inference 502
15.1 Some confidence intervals with given approximate confidence coefficient 503
15.2 Confidence intervals for quantiles of a distribution function 505
15.3 The two-sample sign test 507
15.4 The rank sum and the Wilcoxon–Mann–Whitney two-sample tests 512
15.4.1 Proofs of Lemmas 1 and 2 520
15.5 Nonparametric curve estimation 522
15.5.1 Nonparametric estimation of a probabilitydensity function 522
15.5.2 Nonparametric regression estimation 527
Tables 534
Some notation and abbreviations 568
Answers to even-numbered exercises 572
Index 616
Inside Back Cover 624

Produkt-Details

• Erscheint lt. Verlag: 21.10.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• ISBN-10: 0-12-800437-1 / 0128004371
• ISBN-13: 978-0-12-800437-1 / 9780128004371