ABOUT THE BOOK The objective of this book is to help students interested in probability and statistics, and their applications to understand the basic concepts of stochastic process and to equip them with skills necessary to conduct simple stochastic analysis of data in the field of business, management, social science, life science, physics, and many other disciplines.The book contains such standard topics as probability, random variables and probability distributions, generating functions, stochastic processes, deterministic and stochastic approaches to mathematically model real life phenomena, finite and infinite Markov chains and stationary distributions, continuous time Markov processes with discrete state space, branching processes, and applications of stochastic processes in queueing systems, business and manpower planning.
Efforts have been made to draw examples from a large array of disciplines to demonstrate the wide applicability of stochastic processes. Although a good amount of knowledge in elementary probability and calculus would be of great help, nonmathematically oriented readers can also use the book to apply the ideas of stochastic process since the examples are straightforward and have been organized in a logical step-by-step fashion.
CONTENTS PREFACE Chapter 1 : PROBABILITY * Introduction * Experiment and Sample Space * Events * Complements of an Event Mutually Exclusive Events Definitions of Probability * Classical Definition of Probability * Frequency Definition of Probability * Definition of Subjective Probability * Geometric Definition of Probability * Compound Events * Conditional Probability * Probability of Joint Occurrence of Events: Intersection of Events * Dependent and Independent Events * Union of Events * Venn Diagram * Bayes' Theorem
Chapter 2: RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS * Random Variables * Probability Distribution * Examples of Probability Distribution * Graphical Presentation of Probability Distribution * Mathematical Expectation * Expected Values of Functions of Random Variables * Joint Probability Distribution * Covariance * Independence * Variance of the Sum and Difference of Two Random Variables
Chapter 3 : Special Probability Distributions * Binomial Distribution * Poisson Distribution * Continuous Random Variables and Their Distributions * Exponential Distribution * Memoryless Property of the Exponential Random Variable * Normal Distribution
Chapter 4 : GENERATING FUNCTIONS * Generating Functions * Examples * Probability Generating Function of Linear Functions of a Random Variable * Probability Generating Functions of the Sum of Random Variables * Probability Generating Functions When Bernoulli Trials Have Variable Probabilities * Moment Generating Function * Moment Generating Function of the Sum of Independent * Random Variables * Characteristic Function
Chapter 5 : STOCHASTIC PROCESSES * Introduction * Examples of Applications of Stochastic Processes * Deterministic and Stochastic Models * Definition and Description * Examples of Different Types of Stochastic Processes * Specification of Stochastic Processes
Chapter 6 : MARKOV CHAINS * Definition * Initial Distribution * Examples of Application of P(n) = P( ) Pn * n-Step Transition Probabilities * Chapman-Kolmogorov Equations * Examples of Markov Chains * Two-state Markov Chain * Graphs of Markov Chains * Tree Diagram * State Occupation Times
Chapter 7 : CLASSIFICATION OF STATES * Definition * Periodicity * How to Determine the Period of a State i Recurrence and Transience * Classification Criteria Based on Generating Functions
Chapter 8 : STATIONARY DISTRIBUTION * Introduction * Definition of a Stationary Distribution Spectral Representation of a Transition Probability Matrix * General Case * Theorems * Examples
Chapter 9 : INFINITE MARKOV CHAINS * Introduction * Taboo Probabilities * Theorems on Criteria For Transience and Recurrence * Examples
Chapter 10 : CONTINUOUS TIME MARKOV PROCESSES WITH DISCRETE STATE SPACE * Introduction * The Poisson Process * Postulates * Some Properties of the Poisson Process The Pure Birth Process * The Linear Birth Process (Yule Process) * The Pure Death Process * The Birth and Death Process * Postulates * Special Cases of Birth and Death Processes * Immigration-Emigration Process * * Immigration-Death Process * Infinitesimal Generator of the Process * Solutions of Differential Equations of Birth and Death Processes Discussion
Chapter 11 : QUEUEING PROCESSES * Introduction * Basic Components of a Queueing System Input or arrival process * Output or service process * Queue discipline * Notation * System Performance * Number of Customers in the System * Waiting Time of Customers * Birth-death Processes and Queueing Systems * The M/M/ Queueing System * Waiting Time Distribution * Performance Measures of M/M/ System * Examples of M/M/ System * The M/M/S Queueing System * Performance Measures of M/M/S System * Waiting Time Distribution for the M/M/S System * Probability That a Given Server is Busy at Any Time Point in M/M/S System * Examples of M/M/S System * The M/M/ Queueing System * M/M/ /K Queueing System * Performance Measures of M/M/ /K System * Blocking Probability * M/M/S/K Queueing System
Chapter 12: BRANCHING PROCESSES * Introduction * Mean and Variance of Branching Processes * Branching Processes and Probability Generating Functions * Probability of Extinction * Examples of Branching Processes
Chapter 13 : APPLICATIONS OF STOCHASTIC PROCESSES * Introduction * Manpower Planning * Manpower Planning in a Bank * General Case * Manpower Planning Models * Accounts Receivable Analysis * A Market Analysis of Food Items * Analysis of Market Shares * Computation of the Steady-state Probabilities