Intermediate Course in Probability (eBook)

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