Fundamentals of Applied Probability and Random Processes

Preface AcknowledgmentsChapter 1 Basic Probability Concepts 1.1 Introduction 1.2 Sample Space and Events 1.3 Definitions of Probability 1.3.1 Axiomatic Definition 1.3.2 Relative-Frequency Definition 1.3.3 Classical Definition 1.4 Applications of Probability 1.4.1 Reliability Engineering 1.4.2 Quality Control 1.4.3 Channel Noise 1.4.4 System Simulation 1.5 Elementary Set Theory 1.5.1 Set Operations 1.5.2 Number of Subsets of a Set 1.5.3 Venn Diagram 1.5.4 Set Identities 1.5.5 Duality Principle 1.6 Properties of Probability 1.7 Conditional Probability 1.7.1 Total Probability and the Bayes’ Theorem 1.7.2 Tree Diagram 1.8 Independent Events1.9 Combined Experiments 1.10 Basic Combinatorial Analysis 1.10.1 Permutations 1.10.2 Circular Arrangement 1.10.3 Applications of Permutations in Probability 1.10.4 Combinations 1.10.5 The Binomial Theorem 1.10.6 Stirling’s Formula 1.10.7 Applications of Combinations in Probability 1.11 Reliability Applications 1.12 Summary 1.13 Problems 1.14 ReferencesChapter 2 Random Variables 2.1 Introduction 2.2 Definition of a Random Variable 2.3 Events Defined by Random Variables2.4 Distribution Functions 2.5 Discrete Random Variables 2.5.1 Obtaining the PMF from the CDF 2.6 Continuous Random Variables 2.7 Chapter Summary 2.8 ProblemsChapter 3 Moments of Random Variables 3.1 Introduction 3.2 Expectation 3.3 Expectation of Nonnegative Random Variables 3.4 Moments of Random Variables and the Variance 3.5 Conditional Expectations 3.6 The Chebyshev Inequality3.7 The Markov Inequality 3.8 Chapter Summary 3.9 ProblemsChapter 4 Special Probability Distributions 4.1 Introduction4.2 The Bernoulli Trial and Bernoulli Distribution 4.3 Binomial Distribution4.4 Geometric Distribution 4.4.1 Modified Geometric Distribution 4.4.2 “Forgetfulness” Property of the Geometric Distribution 4.5 Pascal (or Negative Binomial) Distribution 4.6 Hypergeometric Distribution 4.7 Poisson Distribution 4.7.1 Poisson Approximation to the Binomial Distribution 4.8 Exponential Distribution 4.8.1 “Forgetfulness” Property of the Exponential Distribution 4.8.2 Relationship between the Exponential and Poisson Distributions 4.9 Erlang Distribution4.10 Uniform Distribution 4.10.1 The Discrete Uniform Distribution 4.11 Normal Distribution 4.11.1 Normal Approximation to the Binomial Distribution 4.11.2 The Error Function 4.11.3 The Q-Function 4.12 The Hazard Function 4.13 Chapter Summary 4.14 ProblemsChapter 5 Multiple Random Variables 5.1 Introduction 5.2 Joint CDFs of Bivariate Random Variables 5.2.1 Properties of the Joint CDF 5.3 Discrete Random Variables5.4 Continuous Random Variables 5.5 Determining Probabilities from a Joint CDF 5.6 Conditional Distributions 5.6.1 Conditional PMF for Discrete Random Variables 5.6.2 Conditional PDF for Continuous Random Variables 5.6.3 Conditional Means and Variances 5.6.4 Simple Rule for Independence 5.7 Covariance and Correlation Coefficient 5.8 Many Random Variables 5.9 Multinomial Distributions 5.10 Chapter Summary 5.11 Problems Chapter 6 Functions of Random Variables 6.1 Introduction 6.2 Functions of One Random Variable 6.2.1 Linear Functions 6.2.2 Power Functions 6.3 Expectation of a Function of One Random Variable 6.3.1 Moments of a Linear Function 6.4 Sums of Independent Random Variables 6.4.1 Moments of the Sum of Random Variables 6.4.2 Sum of Discrete Random Variables 6.4.3 Sum of Independent Binomial Random Variables 6.4.4 Sum of Independent Poisson Random Variables 6.4.5 The Spare Parts Problem 6.5 Minimum of Two Independent Random Variables6.6 Maximum of Two Independent Random Variables6.7 Comparison of the Interconnection Models 6.8 Two Functions of Two Random Variables 6.8.1 Application of the Transformation Method6.9 Laws of Large Numbers 6.10 The Central Limit Theorem 6.11 Order Statistics 6.12 Chapter Summary 6.13 Problems Chapter 7 Transform Methods 7.1 Introduction 7.2 The Characteristic Function 7.2.1 Moment-Generating Property of the Characteristic Function 7.3 The s-Transform 7.3.1 Moment-Generating Property of the s-Transform 7.3.2 The s-Transforms of Some Well-Known PDFs 7.3.2.1 The s-Transform of the Exponential Distribution 7.3.2.2 The s-Transform of the Uniform Distribution 7.3.3 The s-Transform of the PDF of the Sum of Independent Random Variables 7.3.3.1 The s-Transform of the Erlang Distribution7.4 The z-Transform 7.4.1 Moment-Generating Property of the z-Transform 7.4.2 The z-Transform of the Bernoulli Distribution 7.4.3 The z-Transform of the Binomial Distribution 7.4.4 The z-Transform of the Geometric Distribution 7.4.5 The z-Transform of the Poisson Distribution 7.4.6 The z-Transform of the PMF of Sum of Independent Random Variables 7.4.7 The z-Transform of the Pascal Distribution 7.5 Random Sum of Random Variables 7.6 Chapter Summary 7.7 ProblemsChapter 8 Introduction to Random Processes 8.1 Introduction8.2 Classification of Random Processes 8.3 Characterizing a Random Process 8.3.1 Mean and Autocorrelation Function of a Random Process 8.3.2 The Autocovariance Function of a Random Process 8.4 Crosscorrelation and Crosscovariance Functions 8.4.1 Review of Some Trigonometric Identities8.5 Stationary Random Processes 8.5.1 Strict-Sense Stationary Processes 8.5.2 Wide-Sense Stationary Processes 8.5.2.1 Properties of Autocorrelation Functions for WSS Processes 8.5.2.2 Autocorrelation Matrices for WSS Processes 8.5.2.3 Properties of Crosscorrelation Functions for WSS Processes8.6 Ergodic Random Processes8.7 Power Spectral Density 8.7.1 White Noise 8.8 Discrete-Time Random Processes 8.8.1 Mean, Autocorrelation Function and Autocovariance Function 8.8.2 Power Spectral Density 8.8.3 Sampling of Continuous-Time Processes 8.9 Chapter Summary8.10 ProblemsChapter 9 Linear Systems with Random Inputs9.1 Introduction9.2 Overview of Linear Systems with Deterministic Inputs 9.2.Linear Systems with Continuous-Time Random Inputs 9.3 Linear Systems with Discrete-Time Random Inputs 9.4 Auto regressive Moving Average Process 9.4.1 Moving Average Process 9.4.2 Auto regressive Process 9.4.3 ARMA Process 9.5 Chapter Summary 9.6 ProblemsChapter 10 Some Models of Random Processes 10.1 Introduction10.2 The Bernoulli Process 10.3 Random Walk 10.3.1 Gambler’s Ruin 10.4 The Gaussian Process 10.4.1 White Gaussian Noise Process 10.5 Poisson Process 10.5.1 Counting Processes 10.5.2 Independent Increment Processes 10.5.3 Stationary Increments 10.5.4 Definitions of a Poisson Process 10.5.5 Interarrival Times for the Poisson Process 10.5.6 Conditional and Joint PMFs for Poisson Processes 10.5.7 Compound Poisson Process 10.5.8 Combinations of Independent Poisson Processes 10.5.9 Competing Independent Poisson Processes 10.5.10 Subdivision of a Poisson Process and the Filtered Poisson Process 4 10.5.11 Random Incidence 10.5.12 Nonhomogeneous Poisson Process 10.6 Markov Processes 10.7 Discrete-time Markov Chains 10.7.1 State Transition Probability Matrix 10.7.2 The n-step State Transition Probability 10.7.3 State Transition Diagrams 10.7.4 Classification of States 10.7.5 Limiting-state Probabilities 10.7.6 Doubly Stochastic Matrix10.8 Continuous-time Markov Chains 10.8.1 Birth and Death Processes 10.9 Gambler’s Ruin as a Markov Chain 10.10 Chapter Summary 10.11 Problems Chapter 11 Introduction to Statistics 11.1 Introduction 11.2 Sampling Theory 11.2.1 The Sample Mean 11.2.2 The Sample Variance 11.2.3 Sampling Distributions 11.3 Estimation Theory 11.3.1 Point Estimate, Interval Estimate and Confidence Interval 11.3.2 Maximum Likelihood Estimation 11.3.3 Minimum Mean Squared Error Estimation 11.4 Hypothesis Testing 11.4.1 Hypothesis Test Procedure 11.4.2 Type I and Type II Errors 11.4.3 One-Tailed and Two-Tailed Tests 11.5 Curve Fitting and Linear Regression 11.6 Problems Appendix 1: Table for the CDF of the Standard Normal Random Variable