An Intermediate Course in Probability

by Allan Gut

Overview

The purpose of this book is to provide the reader with a solid background and understanding of the basic results and methods in probability theory before entering into more advanced courses (in probability and/or statistics). The presentation is fairly thorough and detailed with many solved examples. Several examples are solved with di erent methods in order to illustrate their di erent levels of sophistication, their pros, and their cons. The motivation for this style of exposition is that experience has proved that the hard part in courses of this kind usually is the application of the results and methods; to know how, when, and where to apply what; and then, technically, to solve a given problem once one knows how to proceed. Exercises are spread out along the way, and every chapter ends with a large selection of problems. Chapters 1 through 6 focus on some central areas of what might be called pure probability theory: multivariate random variables, conditioning, tra- forms, order variables, the multivariate normal distribution, and convergence.

Product Details

ISBN-13:	9781441901613
Publisher:	Springer New York
Publication date:	06/23/2009
Series:	Springer Texts in Statistics
Edition description:	2nd ed. 2009
Pages:	303
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

Preface to the First Edition v

Preface to the Second Edition vii

Notation and Symbols xiii

Introduction 1

1 Models 1

2 The Probability Space 2

3 Independence and Conditional Probabilities 4

4 Random Variables 5

5 Expectation, Variance, and Moments 7

6 Joint Distributions and Independence 8

7 Sums of Random Variables, Covariance, Correlation 9

8 Limit Theorems 10

9 Stochastic Processes 11

10 The Contents of the Book 11

1 Multivariate Random Variables 15

1 Introduction 15

2 Functions of Random Variables 19

2.1 The Transformation Theorem 20

2.2 Many-to-One 23

3 Problems 24

2 Conditioning 31

1 Conditional Distributions 31

2 Conditional Expectation and Conditional Variance 33

3 Distributions with Random Parameters 38

4 The Bayesian Approach 43

5 Regression and Prediction 46

6 Problems 50

3 Transforms 57

1 Introduction 57

2 The Probability Generating Function 59

3 The Moment Generating Function 63

4 The Characteristic Function 70

5 Distributions with Random Parameters 77

6 Sums of a Random Number of Random Variables 79

7 Branching Processes 85

8 Problems 91

4 Order Statistics 101

1 One-Dimensional Results 101

2 The Joint Distribution of the Extremes 105

3 The Joint Distribution of the Order Statistic 109

4 Problems 113

5 The Multivariate Normal Distribution 117

1 Preliminaries from Linear Algebra 117

2 The Covariance Matrix 119

3 A First Definition 120

4 The Characteristic Function: Another Definition 123

5 The Density: A Third Definition 125

6 Conditional Distributions 127

7 Independence 130

8 Linear Transformations 131

9 Quadratic Forms and Cochran's Theorem 136

10 Problems 140

6 Convergence 147

1 Definitions 147

2 Uniqueness 150

3 Relations Between the Convergence Concepts 152

4 Convergence via Transforms 158

5 The Law of Large Numbers and the Central Limit Theorem 161

6 Convergence of Sums of Sequences of Random Variables 165

7 The Galton-Watson Process Revisited 173

8 Problems 176

7 An Outlook on Further Topics 187

1 Extensions of the Main Limit Theorems 188

1.1 The Law of Large Numbers: The Non-i-i.d. Case 188

1.2 The Central Limit Theorem: The Non-i-i.d. Case 190

1.3 Sums of Dependent Random Variables 190

2 Stable Distributions 192

3 Domains of Attraction 193

4 Uniform Integrability 196

5 An Introduction to Extreme Value Theory 199

6 Records 201

7 The Borel-Cantelli Lemmas 204

7.1 Patterns 207

7.2 Records Revisited 210

7.3 Complete Convergence 211

8 Martingales 213

9 Problems 217

8 The Poisson Process 221

1 Introduction and Definitions 221

1.1 First Definition of a Poisson Process 221

1.2 Second Definition of a Poisson Process 222

1.3 The Lack of Memory Property 226

1.4 A Third Definition of the Poisson Process 231

2 Restarted Poisson Processes 233

2.1 Fixed Times and Occurrence Times 234

2.2 More General Random Times 236

2.3 Some Further Topics 240

3 Conditioning on the Number of Occurrences in an Interval 241

4 Conditioning on Occurrence Times 245

5 Several Independent Poisson Processes 246

5.1 The Superpositioned Poisson Process 247

5.2 Where Did the First Event Occur? 250

5.3 An Extension 252

5.4 An Example 254

6 Thinning of Poisson Processes 255

7 The Compound Poisson Process 260

8 Some Further Generalizations and Remarks 261

8.1 The Poisson Process at Random Time Points 261

8.2 Poisson Processes with Random Intensities 262

8.3 The Nonhomogeneous Poisson Process 264

8.4 The Birth Process 264

8.5 The Doubly Stochastic Poisson Process 265

8.6 The Renewal Process 265

8.7 The Life Length Process 267

9 Problems 269

A Suggestions for Further Reading 277

References 278

B Some Distributions and Their Characteristics 281

C Answers to Problems 287

Index 297

Editorial Reviews

From the reviews of the second edition:

"This is an excellent introductory book on random variables, with a wealth of examples and exercises. … The material is very well organized … . The text is remarkably well written, mathematically and aesthetically; layout and fonts make it a pleasant reading, and the examples are often enlightening. I think it will be a valuable support for students and instructors and it should definitely find a place in every good library." (Fabio Mainardi, The Mathematical Association of America, October, 2009)

“…A worthwhile addition to the textbook pool, one that will guide the student safely through to a point of competence and ability to embark on a more advanced study…” (International Statistical Review, 2010, 78, 1, 134-159)

“The book addresses a unique niche mathematically inclined students previously exposed to an introductory course in probability … . The writing style is lucid and easy to follow. … book is clearly directed toward mathematicians and the highly mathematically inclined scientist or engineer who might be induced to study the mathematics of probability or mathematical statistics. For those who find the classical mathematical pedagogy motivating or those requiring a comprehensive readable reference work on the mathematics of probability theory the book can be highly recommended.” (Thomas D. Sandry, Technometrics, Vol. 53 (1), February, 2011)

“This book … is intended as an introductory graduate level textbook in probability for statistics majors. … This book provides an elaborate description and a collection of results in probability theory. … The level of this book is suitable for a graduate course. Overall all concepts are well discussed with full mathematical rigor. … has a good collection of most of the results related to probability theory, the price is very reasonable, and I will recommend this book to university mathematics and statisticslibraries.” (Sounak Chakraborty, Journal of the American Statistical Association, Vol. 106 (495), September, 2011)

From the Publisher