Fundamentals of Applied Probability and Random Processes

Fundamentals of Applied Probability and Random Processes,nd 2 na Edition Oliver C. Ibe University of Massachusetts, LoweLL, Massachusetts ip^ W >!^ AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO P. j ^ I""*, V X_t/XV Academic Press is an imprint of Elsevier

Contents ACKNOWLEDGMENT PREFACE TO THE SECOND EDITION PREFACE TO FIRST EDITION xiv xvi xix CHAPTER 1 Basic Probability Concepts 1 1.1 Introduction 1 1.2 Sample Space and Events 2 1.3 Definitions of Probability 4 1.3.1 Axiomatic Definition 4 1.3.2 Relative-Frequency Definition 4 1.3.3 Classical Definition 4 1.4 Applications of Probability 6 1.4.1 Information Theory 6 1.4.2 Reliability Engineering 7 1.4.3 Quality Control 7 1.4.4 Channel Noise 8 1.4.5 System Simulation 8 1.5 Elementary Set Theory 9 1.5.1 Set Operations 9 1.5.2 Number of Subsets of a Set 10 1.5.3 Venn Diagram 10 1.5.4 Set Identities 11 1.5.5 Duality Principle 13 1.6 Properties of Probability 13 1.7 Conditional Probability 14 1.7.1 Total Probability and the Bayes' Theorem 16 1.7.2 Tree Diagram 22 1.8 Independent Events 26 1.9 Combined Experiments 29

1.10 Basic Combinatorial Analysis 30 1.10.1 Permutations 30 1.10.2 Circular Arrangement 32 1.10.3 Applications of Permutations in Probability 33 1.10.4 Combinations 34 1.10.5 The Binomial Theorem 37 1.10.6 Stirling's Formula 37 1.10.7 The Fundamental Counting Rule 38 1.10.8 Applications of Combinations in Probability 40 1.11 Reliability Applications 41 1.12 Chapter Summary 46 1.13 Problems 46 Section 1.2 Sample Space and Events 46 Section 1.3 Definitions of Probability 47 Section 1.5 Elementary Set Theory 48 Section 1.6 Properties of Probability 50 Section 1.7 Conditional Probability 50 Section 1.8 Independent Events 52 Section 1.10 Combinatorial Analysis 52 Section 1.11 Reliability Applications 53 CHAPTER 2 Random Variables 57 2.1 Introduction 57 2.2 Definition of a Random Variable 57 2.3 Events Defined by Random Variables 58 2.4 Distribution Functions 59 2.5 Discrete Random Variables 61 2.5.1 Obtaining the PMF from the CDF 65 2.6 Continuous Random Variables 67 2.7 Chapter Summary 72 2.8 Problems 73 Section 2.4 Distribution Functions 73 Section 2.5 Discrete Random Variables 75 Section 2.6 Continuous Random Variables 77 CHAPTER 3 Moments of Random Variables 81 3.1 Introduction 81 3.2 Expectation 82 3.3 Expectation of Nonnegative Random Variables 84 3.4 Moments of Random Variables and the Variance 86 3.5 Conditional Expectations 95 3.6 The Markov Inequality 96 3.7 The Chebyshev Inequality 97

Contents 3.8 Chapter Summary 98 3.9 Problems 98 Section 3.2 Expected Values 98 Section 3.4 Moments of Random Variables and the Variance 100 Section 3.5 Conditional Expectations 101 Sections 3.6 and 3.7 Markov and Chebyshev Inequalities 102 CHAPTER 4 Special Probability Distributions 103 4.1 Introduction 103 4.2 The Bernoulli Trial and Bernoulli Distribution 103 4.3 Binomial Distribution 105 4.4 Geometric Distribution 108 4.4.1 CDF of the Geometric Distribution 111 4.4.2 Modified Geometric Distribution 111 4.4.3 "Forgetfulness" Property of the Geometric Distribution 112 4.5 Pascal Distribution 113 4.5.1 Distinction Between Binomial and Pascal Distributions 117 4.6 Hypergeometric Distribution 118 4.7 Poisson Distribution 122 4.7.1 Poisson Approximation of the Binomial Distribution 123 4.8 Exponential Distribution 124 4.8.1 "Forgetfulness" Property of the Exponential Distribution 126 4.8.2 Relationship between the Exponential and Poisson Distributions 127 4.9 Erlang Distribution 128 4.10 Uniform Distribution 133 4.10.1 The Discrete Uniform Distribution 134 4.11 Normal Distribution 135 4.11.1 Normal Approximation of the Binomial Distribution 138 4.11.2 The Error Function 139 4.11.3 The Q-Function 140 4.12 The Hazard Function 141 4.13 Truncated Probability Distributions 143 4.13.1 Truncated Binomial Distribution 145 4.13.2 Truncated Geometric Distribution 145 0

SSSiiSMefSSSftS' J -fe v 4.13.3 Truncated Poisson Distribution 145 4.13.4 Truncated Normal Distribution 146 4.14 Chapter Summary 146 4.15 Problems 147 Section 4.3 Binomial Distribution 147 Section 4.4 Geometrie Distribution 151 Section 4.5 Pascal Distribution 152 Section 4.6 Hypergeometric Distribution 153 Section 4.7 Poisson Distribution 154 Section 4.8 Exponential Distribution 154 Section 4.9 Erlang Distribution 156 Section 4.10 Uniform Distribution 157 Section 4.11 Normal Distribution 158 CHAPTER 5 Multiple Random Variables 159 5.1 Introduction 159 5.2 Joint CDFs of Bivariate Random Variables 159 5.2.1 Properties of the Joint CDF 159 5.3 Discrete Bivariate Random Variables 160 5.4 Continuous Bivariate Random Variables 163 5.5 Determining Probabilities from a Joint CDF 165 5.6 Conditional Distributions 168 5.6.1 Conditional PMF for Discrete Bivariate Random Variables 168 5.6.2 Conditional PDF for Continuous Bivariate Random Variables 169 5.6.3 Conditional Means and Variances 170 5.6.4 Simple Rule for Independence 171 5.7 Covariance and Correlation Coefficient 172 5.8 Multivariate Random Variables 176 5.9 Multinomial Distributions 177 5.10 Chapter Summary 179 5.11 Problems 179 Section 5.3 Discrete Bivariate Random Variables 179 Section 5.4 Continuous Bivariate Random Variables 180 Section 5.6 Conditional Distributions 182 Section 5.7 Covariance and Correlation Coefficient 183 Section 5.9 Multinomial Distributions 183 CHAPTER 6 Functions of Random Variables 185 6.1 Introduction 185 6.2 Functions of One Random Variable 185 6.2.1 Linear Functions 185

Contents 6.2.2 Power Functions 187 6.3 Expectation of a Function of One Random Variable 188 6.3.1 Moments of a Linear Function 188 6.3.2 Expected Value of a Conditional Expectation 189 6.4 Sums of Independent Random Variables 189 6.4.1 Moments of the Sum of Random Variables 196 6.4.2 Sum of Discrete Random Variables 197 6.4.3 Sum of Independent Binomial Random Variables 200 6.4.4 Sum of Independent Poisson Random Variables.. 201 6.4.5 The Spare Parts Problem 201 6.5 Minimum of Two Independent Random Variables 204 6.6 Maximum of Two Independent Random Variables 205 6.7 Comparison of the Interconnection Models 207 6.8 Two Functions of Two Random Variables 209 6.8.1 Application of the Transformation Method 210 6.9 Laws of Large Numbers 212 6.10 The Central Limit Theorem 214 6.11 Order Statistics 215 6.12 Chapter Summary 219 6.13 Problems 219 Section 6.2 Functions of One Random Variable 219 Section 6.4 Sums of Random Variables 220 Sections 6.4 and 6.5 Maximum and Minimum of Independent Random Variables... 221 Section 6.8 Two Functions of Two Random Variables... 222 Section 6.10 The Central Limit Theorem 222 Section 6.11 Order Statistics 223 CHAPTER 7 Transform Methods 225 7.1 Introduction 225 7.2 The Characteristic Function 225 7.2.1 Moment-Generating Property of the Characteristic Function 226 7.2.2 Sums of Independent Random Variables 227 7.2.3 The Characteristic Functions of Some Well-Known Distributions 228 7.3 The s-transform 231 7.3.1 Moment-Generating Property of the s-transform 231 7.3.2 The s-transform of the PDF of the Sum of Independent Random Variables 232 7.3.3 The s-transforms of Some Well-Known PDFs 232

Confer 7.4 The z-transform 236 7.4.1 Moment-Generating Property of the z-transform 239 7.4.2 The z-transform of the PMF of the Sum of Independent Random Variables 240 7.4.3 The z-transform of Some Well-Known PMFs 240 7.5 Random Sum of Random Variables 242 7.6 Chapter Summary 246 7.7 Problems 247 Section 7.2 Characteristic Functions 247 Section 7.3 s-transforms 247 Section 7.4 z-transforms 249 Section 7.5 Random Sum of Random Variables 250 CHAPTER 8 Introduction to Descriptive Statistics 253 8.1 Introduction 253 8.2 Descriptive Statistics 255 8.3 Measures of Central Tendency 255 8.3.1 Mean 256 8.3.2 Median 256 8.3.3 Mode 257 8.4 Measures of Dispersion 257 8.4.1 Range 257 8.4.2 Quartiles and Percentiles 258 8.4.3 Variance 259 8.4.4 Standard Deviation 259 8.5 Graphical and Tabular Displays 261 8.5.1 Dot Plots 261 8.5.2 Frequency Distribution 262 8.5.3 Histograms 263 8.5.4 Frequency Polygons 263 8.5.5 Bar Graphs 264 8.5.6 Pie Chart 265 8.5.7 Box and Whiskers Plot 266 8.6 Shape of Frequency Distributions: Skewness 269 8.7 Shape of Frequency Distributions: Peakedness 271 8.8 Chapter Summary 272 8.9 Problems 273 Section 8.3 Measures of Central Tendency 273 Section 8.4 Measures of Dispersion 273 Section 8.6 Graphical Displays 274 Section 8.7 Shape of Frequency Distribution 274

Contents CHAPTER 9 Introduction to Inferential Statistics 275 9.1 Introduction 275 9.2 Sampling Theory 276 9.2.1 The Sample Mean 277 9.2.2 The Sample Variance 279 9.2.3 Sampling Distributions 280 9.3 Estimation Theory 281 9.3.1 Point Estimate, Interval Estimate, and Confidence Interval 283 9.3.2 Maximum Likelihood Estimation 285 9.3.3 Minimum Mean Squared Error Estimation 289 9.4 Hypothesis Testing 291 9.4.1 Hypothesis Test Procedure 291 9.4.2 Type I and Type II Errors 292 9.4.3 One-Tailed and Two-Tailed Tests 293 9.5 Regression Analysis 298 9.6 Chapter Summary 301 9.7 Problems 302 Section 9.2 Sampling Theory 302 Section 9.3 Estimation Theory 303 Section 9.4 Hypothesis Testing 303 Section 9.5 Regression Analysis 304 CHAPTER 10 Introduction to Random Processes 307 10.1 Introduction 307 10.2 Classification of Random Processes 308 10.3 Characterizing a Random Process 309 10.3.1 Mean and Autocorrelation Function 309 10.3.2 The Autocovariance Function 310 10.4 Crosscorrelation and Crosscovariance Functions 311 10.4.1 Review of Some Trigonometric Identities 312 10.5 Stationary Random Processes 314 10.5.1 Strict-Sense Stationary Processes 314 10.5.2 Wide-Sense Stationary Processes 315 10.6 Ergodic Random Processes 321 10.7 Power Spectral Density 323 10.7.1 White Noise 328 10.8 Discrete-Time Random Processes 329 10.8.1 Mean, Autocorrelation Function and Autocovariance Function 329 10.8.2 Power Spectral Density of a Random Sequence... 330 10.8.3 Sampling of Continuous-Time Processes 331

Conter 10.9 Chapter Summary 333 10.10 Problems 334 Section 10.3 Mean, Autocorrelation Function and Autocovariance Function 334 Section 10.4 Crosscorrelation and Crosscovariance Functions 335 Section 10.5 Wide-Sense Stationary Processes 336 Section 10.6 Ergodic Random Processes 339 Section 10.7 Power Spectral Density 339 Section 10.8 Discrete-Time Random Processes 342 CHAPTER 11 Linear Systems with Random Inputs 34-5 11.1 Introduction 345 11.2 Overview of Linear Systems with Deterministic Inputs...345 11.3 Linear Systems with Continuous-Time Random Inputs...347 11.4 Linear Systems with Discrete-Time Random Inputs 352 11.5 Autoregressive Moving Average Process 354 11.5.1 Moving Average Process 355 11.5.2 Autoregressive Process 357 11.5.3 ARMA Process 360 11.6 Chapter Summary 361 11.7 Problems 361 Section 11.2 Linear Systems with Deterministic Input 361 Section 11.3 Linear Systems with Continuous Random Input 362 Section 11.4 Linear Systems with Discrete Random Input 365 Section 11.5 Autoregressive Moving Average Processes 367 CHAPTER 12 Special Random Processes 369 12.1 Introduction 369 12.2 The Bernoulli Process 369 12.3 Random Walk Process 371 12.3.1 Symmetric Simple Random Walk 372 12.3.2 Gambler's Ruin 373 12.4 The Gaussian Process 375 12.4.1 White Gaussian Noise Process 377 12.5 Poisson Process 378 12.5.1 Counting Processes 378 12.5.2 Independent Increment Processes 379 12.5.3 Stationary Increments 379

12.5.4 Definitions of a Poisson Process 380 12.5.5 Interarrival Times for the Poisson Process 381 12.5.6 Conditional and Joint PMFs for Poisson Processes 382 12.5.7 Compound Poisson Process 383 12.5.8 Combinations of Independent Poisson Processes 385 12.5.9 Competing Independent Poisson Processes 386 12.5.10 Subdivision of a Poisson Process and the Filtered Poisson Process 387 12.5.11 Random Incidence 388 12.6 Markov Processes 391 12.7 Discrete-Time Markov Chains 393 12.7.1 State Transition Probability Matrix 393 12.7.2 The n-step State Transition Probability 393 12.7.3 State Transition Diagrams 395 12.7.4 Classification of States 396 12.7.5 Limiting-State Probabilities 399 12.7.6 Doubly Stochastic Matrix 402 12.8 Continuous-Time Markov Chains 404 12.8.1 Birth and Death Processes 406 12.9 Gambler's Ruin as a Markov Chain 409 12.10 Chapter Summary 411 12.11 Problems 411 Section 12.2 Bernoulli Process 411 Section 12.3 Random Walk 413 Section 12.4 Gaussian Process 414 Section 12.5 Poisson Process 415 Section 12.7 Discrete-Time Markov Chains 418 Section 12.8 Continuous-Time Markov Chains 423 APPENDIX 427 BIBLIOGRAPHY 429 INDEX 431