An intermediate course in probability:
Summary: 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. The first six chapters focus on some central areas of what might be called pure probability theory:...
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht
Springer
[2009]
|
| Ausgabe: | second edition |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | Summary: 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. The first six chapters focus on some central areas of what might be called pure probability theory: multivariate random variables, conditioning, transforms, order variables, the multivariate normal distribution, convergence. A final chapter is devoted to the Poisson process as a means both to introduce stochastic processes and to apply many of... |
| Beschreibung: | XV, 303 Seiten |
| ISBN: | 9781441901620 9781441901613 |
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Datensatz im Suchindex
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| adam_text | Titel: An intermediate course in probability
Autor: Gut, Allan
Jahr: 2009
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
Contents
6 Problems.............................................. 50
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
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
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
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
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
Contents xi
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
xii Contents
C Answers to Problems ....................................287
Index..........................................................297
|
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| dewey-ones | 519 - Probabilities and applied mathematics |
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| discipline | Informatik Mathematik |
| edition | second edition |
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| spelling | Gut, Allan 1944- (DE-588)110855329 aut An intermediate course in probability Allan Gut second edition Dordrecht Springer [2009] © 2009 XV, 303 Seiten txt rdacontent n rdamedia nc rdacarrier Summary: 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. The first six chapters focus on some central areas of what might be called pure probability theory: multivariate random variables, conditioning, transforms, order variables, the multivariate normal distribution, convergence. A final chapter is devoted to the Poisson process as a means both to introduce stochastic processes and to apply many of... Matematiskstatistik Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Erscheint auch als Online-Ausgabe 10.1007/978-1-441-90162-0 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017601588&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gut, Allan 1944- An intermediate course in probability Matematiskstatistik Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| subject_GND | (DE-588)4079013-7 (DE-588)4064324-4 (DE-588)4123623-3 |
| title | An intermediate course in probability |
| title_auth | An intermediate course in probability |
| title_exact_search | An intermediate course in probability |
| title_full | An intermediate course in probability Allan Gut |
| title_fullStr | An intermediate course in probability Allan Gut |
| title_full_unstemmed | An intermediate course in probability Allan Gut |
| title_short | An intermediate course in probability |
| title_sort | an intermediate course in probability |
| topic | Matematiskstatistik Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| topic_facet | Matematiskstatistik Wahrscheinlichkeitstheorie Wahrscheinlichkeitsrechnung Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017601588&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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