An Intermediate Course in Probability

by Gut, Allan

• ISBN13: 9781441901613
• ISBN10: 1441901612
• Edition: 2nd
• Format: Hardcover
• Copyright: 2009-08-30
• Publisher: Springer Nature

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 the techniques introduced earlier in the text.Students are assumed to have taken a first course in probability, though no knowledge of measure theory is assumed. Throughout, the presentation is thorough and includes many examples that are discussed in detail. Thus, students considering more advanced research in probability theory will benefit from this wide-ranging survey of the subject that provides them with a foretaste of the subject's many treasures.The present second edition offers updated content, one hundred additional problems for solution, and a new chapter that provides an outlook on further areas and topics, such as stable distributions and domains of attraction, extreme value theory and records, and martingales. The main idea is that this chapter may serve as an appetizer to the more advanced theory.

Author Biography

Allan Gut is Professor of Mathematical Statistics at Uppsala University, Uppsala, Sweden. He is a member of the International Statistical Institute, the Bernoulli Society, the Institute of Mathematical Statistics, and the Swedish Statistical Society. He is an Associate Editor of the Journal of Statistical Planning and Inference and Sequential Analysis, a former Associate Editor of the Scandinavian Journal of Statistics, and the author of five other books including Probability: A Graduate Course (Springer, 2005) and Stopped Random Walks: Limit Theorems and Applications, Second Edition (Springer, 2009).

Table of Contents

Preface to the First Edition	p. v
Preface to the Second Edition	p. vii
Notation and Symbols	p. xiii
Introduction	p. 1
Models	p. 1
The Probability Space	p. 2
Independence and Conditional Probabilities	p. 4
Random Variables	p. 5
Expectation, Variance, and Moments	p. 7
Joint Distributions and Independence	p. 8
Sums of Random Variables, Covariance, Correlation	p. 9
Limit Theorems	p. 10
Stochastic Processes	p. 11
The Contents of the Book	p. 11
Multivariate Random Variables	p. 15
Introduction	p. 15
Functions of Random Variables	p. 19
The Transformation Theorem	p. 20
Many-to-One	p. 23
Problems	p. 24
Conditioning	p. 31
Conditional Distributions	p. 31
Conditional Expectation and Conditional Variance	p. 33
Distributions with Random Parameters	p. 38
The Bayesian Approach	p. 43
Regression and Prediction	p. 46
Problems	p. 50
Transforms	p. 57
Introduction	p. 57
The Probability Generating Function	p. 59
The Moment Generating Function	p. 63
The Characteristic Function	p. 70
Distributions with Random Parameters	p. 77
Sums of a Random Number of Random Variables	p. 79
Branching Processes	p. 85
Problems	p. 91
Order Statistics	p. 101
One-Dimensional Results	p. 101
The Joint Distribution of the Extremes	p. 105
The Joint Distribution of the Order Statistic	p. 109
Problems	p. 113
The Multivariate Normal Distribution	p. 117
Preliminaries from Linear Algebra	p. 117
The Covariance Matrix	p. 119
A First Definition	p. 120
The Characteristic Function: Another Definition	p. 123
The Density: A Third Definition	p. 125
Conditional Distributions	p. 127
Independence	p. 130
Linear Transformations	p. 131
Quadratic Forms and Cochran's Theorem	p. 136
Problems	p. 140
Convergence	p. 147
Definitions	p. 147
Uniqueness	p. 150
Relations Between the Convergence Concepts	p. 152
Convergence via Transforms	p. 158
The Law of Large Numbers and the Central Limit Theorem	p. 161
Convergence of Sums of Sequences of Random Variables	p. 165
The Galton-Watson Process Revisited	p. 173
Problems	p. 176
An Outlook on Further Topics	p. 187
Extensions of the Main Limit Theorems	p. 188
The Law of Large Numbers: The Non-i-i.d. Case	p. 188
The Central Limit Theorem: The Non-i-i.d. Case	p. 190
Sums of Dependent Random Variables	p. 190
Stable Distributions	p. 192
Domains of Attraction	p. 193
Uniform Integrability	p. 196
An Introduction to Extreme Value Theory	p. 199
Records	p. 201
The Borel-Cantelli Lemmas	p. 204
Patterns	p. 207
Records Revisited	p. 210
Complete Convergence	p. 211
Martingales	p. 213
Problems	p. 217
The Poisson Process	p. 221
Introduction and Definitions	p. 221
First Definition of a Poisson Process	p. 221
Second Definition of a Poisson Process	p. 222
The Lack of Memory Property	p. 226
A Third Definition of the Poisson Process	p. 231
Restarted Poisson Processes	p. 233
Fixed Times and Occurrence Times	p. 234
More General Random Times	p. 236
Some Further Topics	p. 240
Conditioning on the Number of Occurrences in an Interval	p. 241
Conditioning on Occurrence Times	p. 245
Several Independent Poisson Processes	p. 246
The Superpositioned Poisson Process	p. 247
Where Did the First Event Occur?	p. 250
An Extension	p. 252
An Example	p. 254
Thinning of Poisson Processes	p. 255
The Compound Poisson Process	p. 260
Some Further Generalizations and Remarks	p. 261
The Poisson Process at Random Time Points	p. 261
Poisson Processes with Random Intensities	p. 262
The Nonhomogeneous Poisson Process	p. 264
The Birth Process	p. 264
The Doubly Stochastic Poisson Process	p. 265
The Renewal Process	p. 265
The Life Length Process	p. 267
Problems	p. 269
Suggestions for Further Reading	p. 277
References	p. 278
Some Distributions and Their Characteristics	p. 281
Answers to Problems	p. 287
Index	p. 297
Table of Contents provided by Ingram. All Rights Reserved.