随机模型概论(英文版.第4版)[图书]

1.3.2 Binomial Distribution
1.3.3 Geometric and Negative Binominal Distributions
1.3.4 The Poisson Distribution
1.3.5 The Multinomial Distribution
1.4 Important Continuous Distributions
1.4.1 The Normal Distribution
1.4.2 The Exponential Distribution
1.4.3 The Uniform Distribution
1.4.4 The Gamma Distribution
1.4.5 The Beta Distribution
1.4.6 The Joint Normal Distribution
1.5 Some Elementary Exercises
1.5.1 Tail Probabilities
1.5.2 The Exponential Distribution
1.6 Useful Functions, Integrals, and Sums
2 Conditional Probability and Conditional Expectation
2.1 The Discrete Case
2.2 The Dice Game Craps
2.3 Random Sums
2.3.1 Conditional Distributions: The Mixed Case
2.3.2 The Moments of a Random Sum
2.313 The Distribution of a Random Sum
2.4 Conditioning on a Continuous Random Variable
2.5 Martingales
2.5.1 The Definition
2.5.2 The Markov Inequality
2.5.3 The Maximal Inequality for Nonnegative Martingales
3 Markov Chains: Introduction
3.1 Definitions
3.2 Transition Probability Matrices of a Markov Chain
3.3 Some Markov Chain Models
3.3.1 An Inventory Model
3.3.2 The Ehrenfest Urn Model
3.3.3 Markov Chains in Genetics
3.3.4 A Discrete Queueing Markov Chain
3.4 First Step Analysis
3.4.1 Simple First Step Analyses
3.4.2 The General Absorbing Markov Chain
3.5 Some Spe Markov Chains
3.5.1 The Two-State Markov Chain
3.5.2 Markov Chains Defined by Independent
Random Variables
3.5.3 One-Dimensional Random Walks
3.5.4 Success Runs
3.6 Functionals of Random Walks and Success Runs
3.6.1 The General Random Walk
3.6.2 Cash Management
3.6.3 The Success Runs Markov Chain
3.7 Another Look at First Step Analysis
3.8 Branching Processes
3.8.1 Examples of Branching Processes
3.8.2 The Mean and Variance of a Branching Process
3.8.3 Extinction Probabilities
3.9 Branching Processes and Generating Functions
3.9.1 Generating Functions and Extinction Probabilities
3.9.2 Probability Generating Functions and Sums of
Independent Random Variables
3.9.3 Multiple Branching Processes
4 The Long Run Behavior of Markov Chains
4.1 Regular Transition Probability Matrices
4.1.1 Doubly Stochastic Matrices
4.1.2 Interpretation of the Limiting Distribution
4.2 Examples
4.2.1 Including History in the State Description
4.2.2 Reliability and Redundancy
4.2.3 A Continuous Sampling Plan
4.2.4 Age Replacement Policies
4.2.5 Optimal Replacement Rules
4.3 The Classification of States
4.3.1 Irreducible Markov Chains
4.3.2 Periodicity of a Markov Chain
4.3.3 Recurrent and Transient States
4.4 The Basic Limit Theorem of Markov Chains
4.5 Reducible Markov Chains
5 Poisson Processes
5.1 The Poisson Distribution and the Poisson Process
5.1.1 The Poisson Distribution
5.1.2 The Poisson Process
5.1.3 Nonhomogeneous Processes
5.1.4 Cox Processes
5.2 The Law of Rare Events
5.2.1 The Law of Rare Events and the Poisson Process
5.2.2 Proof of Theorem 5.3
5.3 Distributions Associated with the Poisson Process
5.4 The Uniform Distribution and Poisson Processes
5.4.1 Shot Noise
5.4.2 Sum Quota Sampling
5.5 Spatial Poisson Processes
5.6 Compound and Marked Poisson Processes
5.6.1 Compound Poisson Processes
5.6.2 Marked Poisson Processes
6 Continuous Time Markov Chains
6.1 Pure Birth Processes
6.1.1 Postulates for the Poisson Process
6.1.2 Pure Birth Process
6.1.3 The Yule Process
6.2 Pure Death Processes
6.2.1 The Linear Death Process
6.2.2 Cable Failure Under Static Fatigue
6.3 Birth and Death Processes
6.3.1 Postulates
6.3.2 Sojourn Times
6.3.3 Differential Equations of Birth and Death Processes
6.4 The Limiting Behavior of Birth and Death Processes
6.5 Birth and Death Processes with Absorbing States
6.5.1 Probability of Absorption into State 0
6.5.2 Mean Time Until Absorption
6.6 Finite-State Continuous Time Markov Chains
6.7 A Poisson Process with a Markov Intensity
7 Renewal Phenomena
7.1 Definition of a Renewal Process and Related Concepts
7.2 Some Examples of Renewal Processes
7.2.1 Brief Sketches of Renewal Situations
7.2.2 Block Replacement
7.3 The Poisson Process Viewed as a Renewal Process
7.4 The Asymptotic Behavior of Renewal Processes
7.4.1 The Elementary Renewal Theorem
7.4.2 The Renewal Theorem for Continuous Lifetimes
7.4.3 The Asymptotic Distribution of N(t)
7.4.4 The Limiting Distribution of Age and Excess Life
7.5 Generalizations and Variations on Renewal Processes
7.5.1 Delayed Renewal Processes
7.5.2 Stationary Renewal Processes
7.5.3 Cumulative and Related Processes
7.6 Discrete Renewal Theory
7.6.1 The Discrete Renewal Theorem
7.6.2 Deterministic Population Growth with Age Distribution
8 Brownian Motion and Related Processes
8.1 Brownian Motion and Gaussian Processes
8.1.1 A Little History
8.1.2 The Brownian Motion Stochastic Process
8.1.3 The Central Limit Theorem and the Invariance Principle
8.1.4 Gaussian Processes
8.2 The Maximum Variable and the Reflection Principle
8.2.1 The Reflection Principle
8.2.2 The Time to First Reach a Level
8.2.3 The Zeros of Brownian Motion
8.3 Variations and Extensions
8.3.1 Reflected Brownian Motion
8.3.2 Absorbed Brownian Motion
8.3.3 The Brownian Bridge
8.3.4 Brownian Meander
8.4 Brownian Motion with Drift
8.4.1 The Gambler's Ruin Problem
8.4.2 Geometric Brownian Motion
8.5 The Ornstein-Uhlenbeck Process
8.5.1 A Second Approach to Physical Brownian Motion
8.5.2 The positi0n Process
8.5.3 The Long Run Behavior
8.5.4 Brownian Measure and Integration
9 Queueing Systems
9.1 Queueing Processes
9.1.1 The Queueing Formula L = ~.W
9.1.2 A Sampling of Queueing Models
9.2 Poisson Arrivals, Exponential Service Times
9.2.1 The M/M/1 System
9.2.2 The M/M/oo System
9.2.3 The M/M/s System
9.3 General Service Time Distributions
9.3.1 TheM~G~1 System
9.3.2 The M/G/oo System
9.4 Variations and Extensions
9.4.1 Systems with Balking
9.4.2 Variable Service Rates
9.4.3 A System with Feedback
9.4.4 A Two-Server Overflow Queue
9.4.5 Preemptive Priority Queues
9.5 Open Acyclic Queueing Networks
9.5.1 The Basic Theorem
9.5.2 Two Queues in Tandem
9.5.3 Open Acyclic Networks
9.5.4 Appendix: Time Reversibility
9.5.5 Proof of Theorem 9.1
9.6 General Open Networks
9.6.1 The General Open Network
10 Random Evolutions
10.1 Two-State Velocity Model
10.1.1 Two-State Random Evolution
10.1.2 The Telegraph Equation
10.1.3 Distribution Functions and Densities in the
Two-State Model
10.1.4 Passage Time Distributions
10.2 N-State Random Evolution
10.2.1 Finite Markov Chains and Random Velocity Models
10.2.2 Constructive Approach of Random Velocity Models
10.2.3 Random Evolution Processes
10.2.4 Existence-Uniqueness of the First-Order
System (10.26)
10.2.5 Single Hyperbolic Equation
10.2.6 Spectral Properties of the Transition Matrix
10.2.7 Recurrence Properties of Random Evolution
10.3 Weak Law and Central Limit Theorem
10.4 Isotropic Transport in Higher Dimensions
10.4.1 The Rayleigh Problem of Random Flights
10.4.2 Three-Dimensional Rayleigh Model
11 Characteristic Functions and Their Applications
11.1 Definition of the Characteristic Function
11.1.1 Two Basic Properties of the Characteristic Function
11.2 Inversion Formulas for Characteristic Functions
11.2.1 Fourier Reciprocity/Local Non-Uniqueness
11.2.2 Fourier Inversion and Parseval's Identity
11.3 Inversion Formula for General Random Variables
11.4 The Continuity Theorem
11.4.1 Proof of the Continuity Theorem
11.5 Proof of the Central Limit Theorem
11.6 Stirling's Formula and Applications
11.6.1 Poisson Representation of n!
11.6.2 Proof of Stirling's Formula
11.7 Local deMoivre-Laplace Theorem
Further Reading
Answers to Exercises
Index