An Introduction to Probability and Statistical Inference

Description

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

Front Cover 2
Inside Front Cover 3
An Introduction to Probability and Statistical Inference 7
Copyright 8
Dedication 9
Contents 11
Preface 15
- Features 15
- Overview 15
- Chapter Descriptions 15
- Brief Preface of the Revised Version 17
- Acknowledgments and Credits 17
Chapter 1: Some motivating examples and some fundamental concepts 19
- 1.1 Some motivating examples 19
- 1.2 Some fundamental concepts 26

Front Cover 2
Inside Front Cover 3
An Introduction to Probability and Statistical Inference 7
Copyright 8
Dedication 9
Contents 11
Preface 15
- Features 15
- Overview 15
- Chapter Descriptions 15
- Brief Preface of the Revised Version 17
- Acknowledgments and Credits 17
Chapter 1: Some motivating examples and some fundamental concepts 19
- 1.1 Some motivating examples 19
- 1.2 Some fundamental concepts 26
- 1.3 Random variables 38
Chapter 2: The concept of probability and basic results 41
- 2.1 Definition of probability and some basic results 41
  - 2.1.1 Some basic properties of a probability function 45
  - 2.1.2 Justification 46
- 2.2 Distribution of a random variable 52
- 2.3 Conditional probability and related results 60
- 2.4 Independent events and related results 73
- 2.5 Basic concepts and results in counting 82
Chapter 3: Numerical characteristics of a random variable, some special random variables 95
- 3.1 Expectation, variance, and moment generating function of a random variable 95
- 3.2 Some probability inequalities 108
- 3.3 Some special random variables 111
  - 3.3.1 The discrete case 111
    - Binomial distribution 111
    - Geometric distribution 113
    - Poisson Distribution 115
    - Hypergeometric distribution 117
  - 3.3.2 The continuous case 118
    - Gamma distribution 118
    - Negative Exponential distribution 120
    - Chi-square distribution 122
    - Normal distribution 122
    - Uniform (or Rectangular) distribution 127
  - 3.4 Median and mode of a random variable 143
- Chapter 4: Joint and conditional p.d.f.'s, conditional expectation and variance, moment generating function, covariance, a ... 153
  - 4.1 Joint d.f. and joint p.d.f. of two random variables 153
  - 4.2 Marginal and conditional p.d.f.'s, conditional expectation and variance 162
  - Expectation of a function of two r.v.'s, joint and marginal m.g.f.'s, covariance, and correlation coefficient 176
  - 4.3 Expectation of a function of two r.v.'s, joint and marginal m.g.f.'s 176
  - 4.4 Some generalizations to k random variables 189
  - 4.5 The Multinomial, the Bivariate Normal, and the multivariate Normal 191
    - 4.5.1 Multinomial distribution 191
    - 4.5.2 Bivariate Normal distribution 194
    - 4.5.3 Multivariate Normal distribution 199
  - Chapter 5: Independence of random variables and some applications 205
    - 5.1 Independence of random variables and criteria of independence 205
    - 5.2 The reproductive property of certain distributions 216
  - Chapter 6: Transformation of random variables 225
    - 6.1 Transforming a single random variable 225
    - 6.2 Transforming two or more random variables 230
    - 6.3 Linear transformations 244
    - 6.4 The probability integral transform 250
    - 6.5 Order statistics 251
  - Chapter 7: Some modes of convergence of random variables, applications 263
    - 7.1 Convergence in distribution or in probability and their relationship 263
    - 7.2 Some applications of convergence in distribution: WLLN and CLT 269
      - 7.2.1 Applications of the WLLN 271
      - 7.2.2 Applications of the CLT 275
      - 7.2.3 The continuity correction 277
    - 7.3 Further limit theorems 285
  - Chapter 8: An overview of statistical inference 291
    - 8.1 The basics of point estimation 292
    - 8.2 The basics of interval estimation 294
    - 8.3 The basics of testing hypotheses 295
    - 8.4 The basics of regression analysis 299
    - 8.5 The basics of analysis of variance 300
    - 8.6 The basics of nonparametric inference 302
  - Chapter 9: Point estimation 305
    - 9.1 Maximum likelihood estimation: Motivation and examples 305
    - 9.2 Some properties of MLE's 318
    - 9.3 Uniformly minimum variance unbiased estimates 326
    - 9.4 Decision-theoretic approach to estimation 335
    - 9.5 Other methods of estimation 342
  - Chapter 10: Confidence intervals and confidence regions 347
    - 10.1 Confidence intervals 347
    - 10.2 Confidence intervals in the presence of nuisance parameters 355
    - 10.3 A confidence region for ((μ, σ22) in the N((μ, σ2) distribution 358
    - 10.4 Confidence intervals with approximate confidence coefficient 361
  - Chapter 11: Testing hypotheses 367
    - 11.1 General concepts, formulation of some testing hypotheses 368
    - 11.2 Neyman–Pearson fundamental lemma, Exponential type families, UMP tests for some composite hypotheses 370
      - 11.2.1 Exponential type families of p.d.f.'s 377
      - 11.2.2 UMP Tests for some composite hypotheses 378
    - 11.3 Some applications of theorems 2 381
      - 11.3.1 Further uniformly most powerful tests for some composite hypotheses 390
      - 11.3.2 An application of Theorem 3 391
    - 11.4 Likelihood ratio tests 393
      - 11.4.1 Testing hypotheses for the parameters in a single Normal population 396
      - 11.4.2 Comparing the parameters of two Normal populations 402
    - Chapter 12: More about testing hypotheses 415
      - 12.1 Likelihood ratio tests in the Multinomial case and contingency tables 415
      - 12.2 A goodness-of-fit test 421
      - 12.3 Decision-theoretic approach to testing hypotheses 426
      - 12.4 Relationship between testing hypotheses and confidence regions 433
    - Chapter 13: A simple linear regression model 437
      - 13.1 Setting up the model—the principle of least squares 437
      - 13.2 The least squares estimates of 1 and 2 and some of their properties 440
      - 13.3 Normally distributed errors: mle’s of β1, β2, and σ2, some distributional results 448
      - 13.4 Confidence intervals and hypotheses testing problems 457
      - 13.5 Some prediction problems 463
      - 13.6 Proof of theorem 5 467
      - 13.7 Concluding remarks 469
    - Chapter 14: Two models of analysis of variance 471
      - 14.1 One-way layout with the same number of observations per cell 471
        
        14.1.1 The MLE's of the parameters of the model 472
        
        14.1.2 Testing the hypothesis of equality of means 473
        
        14.1.3 Proof of Lemmas in Section 14.1 479
      - 14.2 A multicomparison method 481
      - 14.3 Two-way layout with one observation per cell 487
        
        14.3.1 The MLE's of the parameters of the model 488
        
        14.3.2 Testing the hypothesis of no row or no column effects 489
        
        14.3.3 Proof of Lemmas in Section 14.3 495
      - Chapter 15: Some topics in nonparametric inference 503
        
        15.1 Some confidence intervals with given approximate confidence coefficient 504
        
        15.2 Confidence intervals for quantiles of a distribution function 506
        
        15.3 The two-sample sign test 508
        
        15.4 The rank sum and the Wilcoxon–Mann–Whitney two-sample tests 513
        
        15.4.1 Proofs of Lemmas 1 and 2 521
        
        15.5 Nonparametric curve estimation 523
        
        15.5.1 Nonparametric estimation of a probabilitydensity function 523
        
        15.5.2 Nonparametric regression estimation 528
        
        Tables 535
        
        Some notation and abbreviations 569
        
        Answers to even-numbered exercises 573
        
        Index 617
        
        Inside Back Cover 625

Author's affiliation

George G. Roussas: University of California, Davis, USA

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