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Introductio to probability Model
Introduction to Probability Models, Tenth Edition, provides an introduction to elementary probability theory and stochastic processes. There are two approaches to the study of probability theory. One is heuristic and nonrigorous, and attempts to develop in students an intuitive feel for the subje...
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| Format: | Printed Book |
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Academic Press
2010
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| Edition: | 10th Edition |
| Series: | Probability and mathematical statistics, v. 10.
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Table of Contents:
- Preface 1 Introduction to Probability Theory 1.1 Introduction 1.2 Sample Space and Events 1.3 Probabilities Defined on Events 1.4 Conditional Probabilities 1.5 Independent Events 1.6 Bayes' Formula Exercises References 2 Random Variables 2.1 Random Variables 2.2 Discrete Random Variables 2.2.1 The Bernoulli Random Variable 2.2.2 The Binomial Random Variable 2.2.3 The Geometric Random Variable 2.2.4 The Poisson Random Variable 2.3 Continuous Random Variables 2.3.1 The Uniform Random Variable 2.3.2 Exponential Random Variables 2.3.3 Gamma Random Variables 2.3.4 Normal Random Variables 2.4 Expectation of a Random Variable 2.4.1 The Discrete Case 2.4.2 The Continuous Case 2.4.3 Expectation of a Function of a Random Variable 2.5 Jointly Distributed Random Variables 2.5.1 Joint Distribution Functions 2.5.2 Independent Random Variables 2.5.3 Covariance and Variance of Sums of Random Variables 2.5.4 Joint Probability Distribution of Functions of Random Variables 2.6 Moment Generating Functions 2.6.1 The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population 2.7 The Distribution of the Number of Events that Occur 2.8 Limit Theorems 2.9 Stochastic Processes Exercises References 3 Conditional Probability and Conditional Expectation 3.1 Introduction 3.2 The Discrete Case 3.3 The Continuous Case 3.4 Computing Expectations by Conditioning 3.4.1 Computing Variances by Conditioning 3.5 Computing Probabilities by Conditioning 3.6 Some Applications 3.6.1 A List Model 3.6.2 A Random Graph 3.6.3 Uniform Priors, Polyas Urn Model, and Bose-Einstein Statistics 3.6.4 Mean Time for Patterns 3.6.5 The k-Record Values of Discrete Random Variables 3.6.6 Left Skip Free Random Walks 3.7 An Identity for Compound Random Variables 3.7.1 Poisson Compounding Distribution 3.7.2 Binomial Compounding Distribution 3.7.3 A Compounding Distribution Related to the Negative Binomial Exercises 4 Markov Chains 4.1 Introduction 4.2 Chapman-Kolmogorov Equations 4.3 Classification of States 4.4 Limiting Probabilities 4.5 Some Applications 4.5.1 The Gamblers Ruin Problem 4.5.2 A Model for Algorithmic Efficiency 4.5.3 Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem 4.6 Mean Time Spent in Transient States 4.7 Branching Processes 4.8 Time Reversible Markov Chains 4.9 Markov Chain Monte Carlo Methods 4.10 Markov Decision Processes 4.11 Hidden Markov Chains 4.11.1 Predicting the States Exercises References 5 The Exponential Distribution and the Poisson Process 5.1 Introduction 5.2 The Exponential Distribution 5.2.1 Definition 5.2.2 Properties of the Exponential Distribution 5.2.3 Further Properties of the Exponential Distribution 5.2.4 Convolutions of Exponential Random Variables 5.3 The Poisson Process 5.3.1 Counting Processes 5.3.2 Definition of the Poisson Process 5.3.3 Interarrival and Waiting Time Distributions 5.3.4 Further Properties of Poisson Processes 5.3.5 Conditional Distribution of the Arrival Times 5.3.6 Estimating Software Reliability 5.4 Generalizations of the Poisson Process 5.4.1 Nonhomogeneous Poisson Process 5.4.2 Compound Poisson Process 5.4.3 Conditional or Mixed Poisson Processes Exercises References 6 Continuous-Time Markov Chains 6.1 Introduction 6.2 Continuous-Time Markov Chains 6.3 Birth and Death Processes 6.4 The Transition Probability Function Pij(t) 6.5 Limiting Probabilities 6.6 Time Reversibility 6.7 Uniformization 6.8 Computing the Transition Probabilities Exercises References 7 Renewal Theory and Its Applications 7.1 Introduction 7.2 Distribution of N(t) 7.3 Limit Theorems and Their Applications 7.4 Renewal Reward Processes 7.5 Regenerative Processes 7.5.1 Alternating Renewal Processes 7.6 Semi-Markov Processes 7.7 The Inspection Paradox 7.8 Computing the Renewal Function 7.9 Applications to Patterns 7.9.1 Patterns of Discrete Random Variables 7.9.2 The Expected Time to a Maximal Run of Distinct Values 7.9.3 Increasing Runs of Continuous Random Variables 7.10 The Insurance Ruin Problem Exercises References 8 Queueing Theory