Mathematical Techniques of Applied Probability

Preface

Acknowledgments

Chapter 1 Basic Probability


1.1 Introduction


1.2 Probability Measures


1.3 Conditional Probability and Independence


1.4 Random Variables and Their Distributions


1.5 Multivariate Random Variables and Their Distributions


1.6 Stochastic Processes

Chapter 2 Generating Functions


2.1 Introduction


2.2 Real Analysis—Sequences of Functions and Double Sequences


2.3 Properties of Generating Functions


2.4 Tail Probabilities


2.5 Convolutions


2.6 Compound Distributions


2.7 Derivation of Probability Distributions from Probabihty-Generating Functions


2.8 The Convergence of Sequences of Probability-Generating Functions


2.9 Solution of Difference Equations


2.10 Bivariate Generating Functions

Chapter 3 Recurrent Event Theory


3.1 Introduction and Definitions


3.2 Classification of Events


3.3 Basic Theorems for Recurrent Events


3.4 The Number of Occurrences of a Recurrent Event


3.5 General Recurrent Event Processes


3.6 An Application of Recurrent Event Theory: Success Runs in Bernoulli Trials


3.7 An Application of Recurrent Event Theory: Patterns in Bernoulli Trials

Chapter 4 Matrix Techniques


4.1 Introduction and Basic Operations


4.2 Determinants, Inverses, Ranks, and Generalized Inverses


4.3 Solving Systems of Linear Equations


4.4 Vector Spaces and Diagonalization of Matrices


4.5 Matrix Analysis


4.6 Nonnegative Matrices


4.7 Infinite Matrices

Chapter 5 Markov Chains in Discrete Time—Basic Theory


5.1 Introduction and Basic Definitions


5.2 Classification of States


5.3 Decomposition of the State Space


5.4 Canonical Forms of the Transition Matrix of a Markov Chain


5.5 Limiting Distributions

References

Index