An Introduction to Probability and Statistical Inference
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Table of contents
- Cover
- Contentsvii
- Prefacexi
- Chapter 1. SOME MOTIVATING EXAMPLES AND SOME FUNDAMENTAL CONCEPTS1
- 1.1 Some Motivating Examples1
- 1.2 Some Fundamental Concepts8
- 1.3 Random Variables19
- Chapter 2. THE CONCEPT OF PROBABILITY AND BASIC RESULTS23
- 2.1 Definition of Probability and Some Basic Results24
- 2.2 Distribution of a Random Variable33
- 2.3 Conditional Probability and Related Results41
- 2.4 Independent Events and Related Results51
- 2.5 Basic Concepts and Results in Counting59
- Chapter 3. NUMERICAL CHARACTERISTICS OF A RANDOM VARIABLE, SOME SPECIAL RANDOM VARIABLES68
- 3.1 Expectation, Variance, and Moment Generating Function of a Random Variable68
- 3.2 Some Probability Inequalities77
- 3.3 Some Special Random Variables79
- 3.4 Median and Mode of a Random Variable102
- Chapter 4. JOINT AND CONDITIONAL P.D.F.’S, CONDITIONAL EXPECTATION AND VARIANCE, MOMENT GENERATING109
- 4.1 Joint d.f. and Joint p.d.f. of Two Random Variables110
- 4.2 Marginal and Conditional p.d.f.’s, Conditional Expectation and Variance117
- 4.3 Expectation of a Function of Two r.v.’s, Joint and Marginal m.g.f.’s, Covariance, and Correl126
- 4.4 Some Generalizations to k Random Variables137
- 4.5 The Multinomial, the Bivariate Normal, and the Multivariate Normal Distributions139
- Chapter 5. INDEPENDENCE OF RANDOM VARIABLES AND SOME APPLICATIONS150
- 5.1 Independence of Random Variables and Criteria of Independence150
- 5.2 The Reproductive Property of Certain Distributions159
- Chapter 6. TRANSFORMATION OF RANDOM VARIABLES168
- 6.1 Transforming a Single Random Variable168
- 6.2 Transforming Two or More Random Variables173
- 6.3 Linear Transformations185
- 6.4 The Probability Integral Transform192
- 6.5 Order Statistics193
- Chapter 7. SOME MODES OF CONVERGENCE OF RANDOM VARIABLES, APPLICATIONS202
- 7.1 Convergence in Distribution or in Probability and Their Relationship202
- 7.2 Some Applications of Convergence in Distribution: The Weak Law of Large Numbers and the Central208
- 7.3 Further Limit Theorems222
- Chapter 8. AN OVERVIEW OF STATISTICAL INFERENCE227
- 8.1 The Basics of Point Estimation228
- 8.2 The Basics of Interval Estimation230
- 8.3 The Basics of Testing Hypotheses231
- 8.4 The Basics of Regression Analysis235
- 8.5 The Basics of Analysis of Variance236
- 8.6 The Basics of Nonparametric Inference238
- Chapter 9. POINT ESTIMATION240
- 9.1 Maximum Likelihood Estimation: Motivation and Examples240
- 9.2 Some Properties of Maximum Likelihood Estimates253
- 9.3 Uniformly Minimum Variance Unbiased Estimates261
- 9.4 Decision-Theoretic Approach to Estimation270
- 9.5 Other Methods of Estimation277
- Chapter 10. CONFIDENCE INTERVALS AND CONFIDENCE REGIONS281
- 10.1 Confidence Intervals282
- 10.2 Confidence Intervals in the Presence of Nuisance Parameters289
- 10.3 A Confidence Region for (µ, s2) in the N(µ, s2) Distribution292
- 10.4 Confidence Intervals with Approximate Confidence Coefficient294
- Chapter 11. TESTING HYPOTHESES299
- 11.1 General Concepts, Formulation of Some Testing Hypotheses300
- 11.2 Neyman–Pearson Fundamental Lemma, Exponential Type Families, Uniformly Most Powerful Tests fo302
- 11.3 Some Applications of Theorems 2 and 3315
- 11.4 Likelihood Ratio Tests324
- Chapter 12. MORE ABOUT TESTING HYPOTHESES343
- 12.1 Likelihood Ratio Tests in the Multinomial Case and Contingency Tables343
- 12.2 A Goodness-of-Fit Test349
- 12.3 Decision-Theoretic Approach to Testing Hypotheses353
- 12.4 Relationship Between Testing Hypotheses and Confidence Regions360
- Chapter 13. A SIMPLE LINEAR REGRESSION MODEL363
- 13.1 Setting-up the Model„The Principle of Least Squares364
- 13.2 The Least Squares Estimates of β1 and β2, and Some of Their Properties366
- 13.3 Normally Distributed Errors: MLE’s of β1, β2, and σ2, Some Distributional Results374
- 13.4 Confidence Intervals and Hypotheses Testing Problems383
- 13.5 Some Prediction Problems389
- 13.6 Proof of Theorem 5393
- 13.7 Concluding Remarks395
- Chapter 14. TWO MODELS OF ANALYSIS OF VARIANCE397
- 14.1 One-Way Layout with the Same Number of Observations per Cell398
- 14.2 A Multicomparison Method407
- 14.3 Two-Way Layout with One Observation per Cell412
- Chapter 15. SOME TOPICS IN NONPARAMETRIC INFERENCE428
- 15.1 Some Confidence Intervals with Given Approximate Confidence Coefficient429
- 15.2 Confidence Intervals for Quantiles of a Distribution Function431
- 15.3 The Two-Sample Sign Test433
- 15.4 The Rank Sum and the Wilcoxon–Mann–Whitney Two-Sample Tests435
- 15.5 Nonparametric Curve Estimation442
- APPENDIX450
- SOME NOTATION AND ABBREVIATIONS480
- ANSWERS TO EVEN-NUMBERED EXERCISES483
- INDEX515
- Distributions and Some of their Characteristics525
Roussas introduces readers with no prior knowledge in probability or statistics, to a thinking process to guide them toward the best solution to a posed question or situation. An Introduction to Probability and Statistical Inference provides a plethora of examples for each topic discussed, giving the reader more experience in applying statistical methods to different situations.
"The text is wonderfully written and has the most
comprehensive range of exercise problems that I have ever seen." — Tapas K. Das, University of South Florida
"The exposition is great; a mixture between conversational tones and formal mathematics; the appropriate combination for a math text at [this] level. In my examination I could find no instance where I could improve the book." — H. Pat Goeters, Auburn, University, Alabama
* Contains more than 200 illustrative examples discussed in detail, plus scores of numerical examples and applications
* Chapters 1-8 can be used independently for an introductory course in probability
* Provides a substantial number of proofs
"The text is wonderfully written and has the most
comprehensive range of exercise problems that I have ever seen." — Tapas K. Das, University of South Florida
"The exposition is great; a mixture between conversational tones and formal mathematics; the appropriate combination for a math text at [this] level. In my examination I could find no instance where I could improve the book." — H. Pat Goeters, Auburn, University, Alabama
* Contains more than 200 illustrative examples discussed in detail, plus scores of numerical examples and applications
* Chapters 1-8 can be used independently for an introductory course in probability
* Provides a substantial number of proofs