Intermediate Course in Probability (eBook)
303 Seiten
Springer New York (Verlag)
978-1-4419-0162-0 (ISBN)
This is the only book that gives a rigorous and comprehensive treatment with lots of examples, exercises, remarks on this particular level between the standard first undergraduate course and the first graduate course based on measure theory.
There is no competitor to this book.
The book can be used in classrooms as well as for self-study.
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.
Preface to the First Edition 5
Preface to the Second Edition 7
Contents 8
Notation and Symbols 12
Introduction 15
1 Models 15
2 The Probability Space 16
3 Independence and Conditional Probabilities 18
4 Random Variables 19
5 Expectation, Variance, and Moments 21
6 Joint Distributions and Independence 22
7 Sums of Random Variables, Covariance, Correlation 23
8 Limit Theorems 24
9 Stochastic Processes 25
10 The Contents of the Book 25
1 Multivariate Random Variables 29
1 Introduction 29
2 Functions of Random Variables 33
3 Problems 38
2 Conditioning 44
1 Conditional Distributions 44
2 Conditional Expectation and Conditional Variance 46
3 Distributions with Random Parameters 51
4 The Bayesian Approach 56
5 Regression and Prediction 59
6 Problems 63
3 Transforms 69
1 Introduction 69
2 The Probability Generating Function 71
3 The Moment Generating Function 75
4 The Characteristic Function 82
5 Distributions with Random Parameters 89
6 Sums of a Random Number of Random Variables 91
7 Branching Processes 97
8 Problems 103
4 Order Statistics 112
1 One-Dimensional Results 112
2 The Joint Distribution of the Extremes 116
3 The Joint Distribution of the Order Statistic 120
4 Problems 124
5 The Multivariate Normal Distribution 128
1 Preliminaries from Linear Algebra 128
2 The Covariance Matrix 130
3 A First Definition 131
4 The Characteristic Function: Another Definition 134
5 The Density: A Third Definition 136
6 Conditional Distributions 138
7 Independence 141
8 Linear Transformations 142
9 Quadratic Forms and Cochran’s Theorem 147
10 Problems 151
6 Convergence 157
1 Definitions 157
2 Uniqueness 160
3 Relations Between the Convergence Concepts 162
4 Convergence via Transforms 168
5 The Law of Large Numbers and the Central Limit Theorem 171
6 Convergence of Sums of Sequences of Random Variables 175
7 The Galton–Watson Process Revisited 183
8 Problems 186
7 An Outlook on Further Topics 196
1 Extensions of the Main Limit Theorems 197
2 Stable Distributions 201
3 Domains of Attraction 202
4 Uniform Integrability 205
5 An Introduction to Extreme Value Theory 208
6 Records 210
7 The Borel–Cantelli Lemmas 213
8 Martingales 222
9 Problems 226
8 The Poisson Process 230
1 Introduction and Definitions 230
2 Restarted Poisson Processes 242
3 Conditioning on the Number of Occurrences in an Interval 250
4 Conditioning on Occurrence Times 254
5 Several Independent Poisson Processes 255
6 Thinning of Poisson Processes 264
7 The Compound Poisson Process 269
8 Some Further Generalizations and Remarks 270
9 Problems 278
A Suggestions for Further Reading 285
References 286
B Some Distributions and Their Characteristics 289
C Answers to Problems 294
Index 304
Erscheint lt. Verlag | 6.6.2009 |
---|---|
Sprache | englisch |
Themenwelt | Mathematik / Informatik ► Informatik ► Theorie / Studium |
Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
Mathematik / Informatik ► Mathematik ► Statistik | |
Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik | |
Naturwissenschaften ► Biologie ► Ökologie / Naturschutz | |
Technik | |
ISBN-10 | 1-4419-0162-0 / 1441901620 |
ISBN-13 | 978-1-4419-0162-0 / 9781441901620 |
Haben Sie eine Frage zum Produkt? |
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