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## Discrete Time Models: Basic Theory

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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

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1st Edition - September 28, 1983

Author: Jeffrey J. Hunter

Editor: J. William Schmidt

Language: EnglisheBook ISBN:

9 7 8 - 1 - 4 8 3 2 - 6 3 8 8 - 5

Mathematical Techniques of Applied Probability, Volume 1: Discrete Time Models: Basic Theory provides information pertinent to the basic theory of discrete time models. This book… Read more

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Mathematical Techniques of Applied Probability, Volume 1: Discrete Time Models: Basic Theory provides information pertinent to the basic theory of discrete time models. This book introduces the tools of generating functions and matrix theory to facilitate a detailed study of such models. Organized into five chapters, this volume begins with an overview of the elementary theory of probability for discrete random variables. This text then reviews the concepts of convergence, absolute convergence, uniform convergence, continuity, differentiation, and integration. Other chapters consider the occurrence of general patterns of successes and failures in Bernoulli trials. This book discusses as well the matrix theory, which is used in the study of stochastic processes, particularly in the analysis of the behavior of Markov chains. The final chapter deals with the properties of a special class of discrete time chains. This book is a valuable resource for students and teachers.

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

- No. of pages: 254
- Language: English
- Edition: 1
- Published: September 28, 1983
- Imprint: Academic Press
- eBook ISBN: 9781483263885

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