应用随机过程 概率模型导论

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, Polya's 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 Long-Run Proportions and Limiting Probabilities
　　4.4.1 Limiting Probabilities
　4.5 Some Applications
　　4.5.1 The Gambler's 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
　5.5 Random Intensity Functions and Hawkes 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 The Reversed Chai