Author(s): Chin Long Chiang
Year: 1980
Preface
Preface to The 1968 Book
Table of Contents
Chapter 1. Random Variables
1. Introduction
2. Random Variables
3. Multivariate Probability Distributions
4. Mathematical Expectation
4.1. A useful inequality
4.2. Conditional expectation
5. Moments, Variances and Covariances
5.1. Variance of a linear function of random variables
5.2. Covariance between two linear functions of random variables
5.3. Variance of a product of random variables
5.4. Approximate variance of a function of random variables
5.5. Conditional variance and covariance
5.6. Correlation coefficient
6. Chebyshev’s Inequality and Laws of Large Numbers
6.1. Chebyshev’s inequality
6.2. Bernoulli’s theorem
6.3. Laws of large numbers
6.4. The central Emit theorem
7. Problems for Solution
Chapter 2. Probability Generating Functions
1. Introduction
2. General Properties
3. Convolutions
4. Examples
4.1. Binomial distribution
4.2. Poisson distribution
4.3. Geometric and negative binomial distributions
5. The Continuity Theorem
6. Partial Fraction Expansions
7. Multivariate Probability Generating Functions
7.1. Probability generating functions of marginal distributions
7.2. Probability generating functions of conditional distributions
8. Sum of a Random Number of Random Variables
9. Problems for Solution
Chapter 3. Exponential-Type Distributions and Maximum Likelihood Estimation
1. Introduction
2. Gamma Function
3. Convolutions
4. Moment Generating Functions
4.1. The Central limit theorem
5. Sum of Non-Identically Distributed Random Variables
6. Sum of Consecutive Random Variables
7. Maximum Likelihood Estimation
7.1. Optimum properties of the m.1. estimator
8. Problems for Solution
Chapter 4. Branching Process, Random Walk and Ruin Problem
1. A Simple Branching Process
1.1. Probability of extinction
2. Random Walk and Diffusion Process
2.1. Random walk
2.2. Diffusion process
3. Gambler’s Ruin
3.1. Expected number of games
3.2. Ruin probability in a finite number of games
4. Problems for Solution
Chapter 5. Markov Chains
1. Introduction
2. Definition of Markov Chains and Transition Probabilities
3. Higher Order Transition Probabilities, pij(n)
4. Classification of States
5. Asymptotic Behavior of pij(n)
6. Closed Sets and Irreducible Chains
7. Stationary Distribution
8. An Application to Genetics
8.1. Two loci each with two alleles
9. Problems for Solution
Chapter 6. Algebraic Treatment of Finite Markov Chains
1. Introduction
2. Eigenvalues of Stochastic Matrix P and a Useful Lemma
2.1. A useful lemma
3. Formulas for Higher Order Transition Probabilities
4. Limiting Probability Distribution
5. Examples
6. Problems for Solution
Chapter 7. Renewal Processes
1. Introduction
2. Discrete Time Renewal Processes
2.1. Renewal probability and classification of events
2.2. Probability of occurrence of E
2.3. Probability generating functions of {f(n)} and {p(n)}
2.4. An application to random walk
2.5. Delayed renewal processes
3. Continuous Time Renewal Processes
3.1. Some specific distributions
3.2. Number of renewals in a time interval (0, t]
3.3. Renewal function and renewal density
3.4. Illustrative examples
4. Problems for Solution
Chapter 8. Some Stochastic Models of Population Growth
1. Introduction
2. The Poisson Process
2.1. The method of probability generating functions
2.2. Some generalizations of the Poisson process
3. Pure Birth Process
3.1. The Yule process
3.2. Time dependent Yule process
3.3. Joint distribution in the time dependent Yule process
4. The Polya Process
5. Pure Death Process
6. Migration Process
6.1. A simple immigration-emigration process
6.1.1. An alternative derivation
7. An Appendix – First Order Differential Equations
7.1. Ordinary differential equations
7.2. Partial differential equations
7.2.1. Preliminaries
7.2.2. Auxiliary equations
7.2.3. The general solution
8. Problems for Solution
Chapter 9. A General Birth Process, an Equality and an Epidemic Model
1. Introduction
2. A General Birth Process
3. An Equality in Stochastic Processes
4. A Simple Stochastic Epidemic – McKendrick’s Model
4.1. Solution for the probability p1,n(0,t)
4.2. Infection time and duration of an epidemic
5. Problems for Solution
Chapter 10. Birth-Death Processes and Queueing Processes
1. Introduction
2. Linear Growth
3. A Time-Dependent General Birth-Death Process
4. Queueing Processes
4.1. Differential equations for an M/M/s, queue
4.2. The M/M/1 queue
4.2.1. The length of queue
4.2.2. Service time and waiting time
4.2.3. Interarrival time and traffic intensity
4.3. The M/M/s queue
4.3.1. The length of queue
4.3.2. Service time and waiting time
5. Problems for Solution
Chapter 11. A Simple Illness-Death Process – Fix-Neyman Process
1. Introduction
2. Health Transition Probability Pαβ(0,t)) and Death Transition Probability Qαδ(0,t)
3. Chapman-Kolmogorov Equations
4. Expected Duration of Stay
5. Population Sizes in Health States and Death States
5.1. The limiting distribution
6. Generating Function of Population Sizes
7. Survival and Stages of Disease
7.1. Distribution of survival time
7.2. Maximum-likelihood estimates
8. Problems for Solution
Chapter 12. Multiple Transition Probabilities in the Simple Illness Death Process
1. Introduction
2. Identities and Multiple Transition Probabilities
2.1. Formula for the multiple passage probabilities Pαβ(m)(0,t)
2.2. Three equalities
2.3. Inductive proof of formula for Pαβ(m)(0,t)
2.4. Formula for the multiple renewal probabilities Pαα(m)(0,t)
3. Differential Equations and Multiple Transition Probabilities
4. Probability Generating Functions
4.1. Multiple transition probabilities
4.2. Equivalence of formulas
5. Verification of the Stochastic Identities
6. Chapman-Kolmogorov Equations
7. Conditional Probability Distribution of the Number of Transitions
8. Multiple Transitions Leading to Death
9. Multiple Entrance Transition Probabilities pαβ (n)(0,t)
10. Problems for Solution
Chapter 13. Multiple Transition Time in the Simple Illness Death Process – an Alternating Renewal Process
1. Introduction
2. Density Functions of Multiple Transition Times
3. Convolution of Multiple Transition Time
4. Distribution Function of Multiple Transition Time
4.1. Distribution function of the m-th renewal time
4.2. Distribution function of the m-th passage time
5. Survival Time
6. A Two-state Stochastic Process
6.1. Multiple transition probabilities
6.2. Multiple transition time
6.3. Number of renewals and renewal time
7. Problems for Solution
Chapter 14. The Kolmogorov Differential Equations and Finite Markov Processes
1. Markov Processes and the Chapman-Kolmogorov Equation
1.1 The Chapman-Kolmogorov equation
2. Kolmogorov Differential Equations
2.1. Derivation of the Kolmogorov differential equations
2.2. Examples
3. Matrices, Eigenvalues, and Diagonalization
3.1. Eigenvalues and eigenvectors
3.2. Diagonalization of a matrix
3.3. A useful lemma
3.4. Matrix of eigenvectors
4. Explicit Solutions for Kolmogorov Differential Equations
4.1. Intensity matrix V and its eigenvalues
4.2. First solution for individual transition probabilities Pij(0,t)
4.3. Second solution for individual transition probabilities Pij(0,t)
4.4. Identity of the two solutions
4.5. Chapman-Kolmogorov equations
4.6. Limiting probabilities
5. Problems for Solution
Chapter 15. Kolmogorov Differential Equations and Finite Markov Processes – Continuation
1. Introduction
2. First Solution for Individual Transition Probabilities Pij(0,t)
3. Second Solution for Individual Transition Probabilities Pij(0,t)
4. Problems for Solution
Chapter 16. A General Illness-Death Process
1. Introduction
2. Transition Probabilities
2.1. Health transition probabilities Pαβ (0,t)
2.2. Transition probabilities leading to death Qαδ (0,t)
2.3. An equality concerning transition probabilities
2.4. Limiting transition probabilities
2.5. Expected duration of stay in health and death states
2.6. Population sizes in health states and death states
3. Multiple Transition Probabilities
3.1. Multiple exit transition probabilities, Pαβ(m) (0,t)
3.2. Multiple transition probabilities leading to death Qαδ(m)(0,t)
3.3. Multiple entrance transition probabilities, pαβ(n) (0,t)
4. An Annual Health Index
5. Problems for Solution
Chapter 17. Migration Processes and Birth-Illness-Death Processes
1. Introduction
2. Emigration-Immigration Processes – Poisson-Markov Processes
2.1. First approach
2.2. Second approach
2.2.1. Differential equation of the probability generating function
2.2.2. Solution of the probability generating function
3. A Birth-Illness-Death Process
4. Problems for Solution
Bibliography
Author Index
Subject Index
Errata
