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
1.7 Probability Is a Continuous Event Function
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 50 Properties of Covariance
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 Limit Theorems
2.8 Proof of the Strong Law of Large Numbers
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 Long-Run Proportions and Limiting Probabilities
4.4.1 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
5.2.5 The Dirichlet Distribution
5.3 The Poisson Process
5.3.1 Counting Processes
5.3.2 Definition of the Poisson Process
5.3.3 Further Properties of Poisson Processes
5.3.4 Conditional Distribution of the Arrival Times
5.3.5 Estimating Software Reliability
5.4 Generalizations of the Poisson Process
5.4.1 Nonhomogeneous Poisson Process
5.4.2 Compound Poisson Process Examplesof Compound Poisson Processes
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 F
谢尔登·M.罗斯(Sheldon M. Ross),靠前知名概率与统计学家,南加州大学工业与系统工程系的教授。1968年博士毕业于斯坦福大学统计系,曾在加州大学伯克利分校任教多年。他是靠前数理统计协会会士、运筹学与管理学研究协会(INFORMS)会士、美国洪堡资深科学家奖获得者。罗斯教授著述颇丰,他的多本畅销数学和统计教材均产生了世界性的影响,如《概率论基础教程》《随机过程》《统计模拟》等。