Fundamentals of Applied Probability and Random Processes

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1 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

2 Contents ACKNOWLEDGMENT PREFACE TO THE SECOND EDITION PREFACE TO FIRST EDITION xiv xvi xix CHAPTER 1 Basic Probability Concepts Introduction Sample Space and Events Definitions of Probability Axiomatic Definition Relative-Frequency Definition Classical Definition Applications of Probability Information Theory Reliability Engineering Quality Control Channel Noise System Simulation Elementary Set Theory Set Operations Number of Subsets of a Set Venn Diagram Set Identities Duality Principle Properties of Probability Conditional Probability Total Probability and the Bayes' Theorem Tree Diagram Independent Events Combined Experiments 29

3 1.10 Basic Combinatorial Analysis Permutations Circular Arrangement Applications of Permutations in Probability Combinations The Binomial Theorem Stirling's Formula The Fundamental Counting Rule Applications of Combinations in Probability Reliability Applications Chapter Summary 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 Introduction Definition of a Random Variable Events Defined by Random Variables Distribution Functions Discrete Random Variables Obtaining the PMF from the CDF Continuous Random Variables Chapter Summary 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 Introduction Expectation Expectation of Nonnegative Random Variables Moments of Random Variables and the Variance Conditional Expectations The Markov Inequality The Chebyshev Inequality 97

4 Contents 3.8 Chapter Summary 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 Introduction The Bernoulli Trial and Bernoulli Distribution Binomial Distribution Geometric Distribution CDF of the Geometric Distribution Modified Geometric Distribution "Forgetfulness" Property of the Geometric Distribution Pascal Distribution Distinction Between Binomial and Pascal Distributions Hypergeometric Distribution Poisson Distribution Poisson Approximation of the Binomial Distribution Exponential Distribution "Forgetfulness" Property of the Exponential Distribution Relationship between the Exponential and Poisson Distributions Erlang Distribution Uniform Distribution The Discrete Uniform Distribution Normal Distribution Normal Approximation of the Binomial Distribution The Error Function The Q-Function The Hazard Function Truncated Probability Distributions Truncated Binomial Distribution Truncated Geometric Distribution 145 0

5 SSSiiSMefSSSftS' J -fe v Truncated Poisson Distribution Truncated Normal Distribution Chapter Summary 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 Introduction Joint CDFs of Bivariate Random Variables Properties of the Joint CDF Discrete Bivariate Random Variables Continuous Bivariate Random Variables Determining Probabilities from a Joint CDF Conditional Distributions Conditional PMF for Discrete Bivariate Random Variables Conditional PDF for Continuous Bivariate Random Variables Conditional Means and Variances Simple Rule for Independence Covariance and Correlation Coefficient Multivariate Random Variables Multinomial Distributions Chapter Summary 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 Introduction Functions of One Random Variable Linear Functions 185

6 Contents Power Functions Expectation of a Function of One Random Variable Moments of a Linear Function Expected Value of a Conditional Expectation Sums of Independent Random Variables Moments of the Sum of Random Variables Sum of Discrete Random Variables Sum of Independent Binomial Random Variables Sum of Independent Poisson Random Variables The Spare Parts Problem Minimum of Two Independent Random Variables Maximum of Two Independent Random Variables Comparison of the Interconnection Models Two Functions of Two Random Variables Application of the Transformation Method Laws of Large Numbers The Central Limit Theorem Order Statistics Chapter Summary 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 Section 6.8 Two Functions of Two Random Variables Section 6.10 The Central Limit Theorem 222 Section 6.11 Order Statistics 223 CHAPTER 7 Transform Methods Introduction The Characteristic Function Moment-Generating Property of the Characteristic Function Sums of Independent Random Variables The Characteristic Functions of Some Well-Known Distributions The s-transform Moment-Generating Property of the s-transform The s-transform of the PDF of the Sum of Independent Random Variables The s-transforms of Some Well-Known PDFs 232

7 Confer 7.4 The z-transform Moment-Generating Property of the z-transform The z-transform of the PMF of the Sum of Independent Random Variables The z-transform of Some Well-Known PMFs Random Sum of Random Variables Chapter Summary 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 Introduction Descriptive Statistics Measures of Central Tendency Mean Median Mode Measures of Dispersion Range Quartiles and Percentiles Variance Standard Deviation Graphical and Tabular Displays Dot Plots Frequency Distribution Histograms Frequency Polygons Bar Graphs Pie Chart Box and Whiskers Plot Shape of Frequency Distributions: Skewness Shape of Frequency Distributions: Peakedness Chapter Summary 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

8 Contents CHAPTER 9 Introduction to Inferential Statistics Introduction Sampling Theory The Sample Mean The Sample Variance Sampling Distributions Estimation Theory Point Estimate, Interval Estimate, and Confidence Interval Maximum Likelihood Estimation Minimum Mean Squared Error Estimation Hypothesis Testing Hypothesis Test Procedure Type I and Type II Errors One-Tailed and Two-Tailed Tests Regression Analysis Chapter Summary 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 Introduction Classification of Random Processes Characterizing a Random Process Mean and Autocorrelation Function The Autocovariance Function Crosscorrelation and Crosscovariance Functions Review of Some Trigonometric Identities Stationary Random Processes Strict-Sense Stationary Processes Wide-Sense Stationary Processes Ergodic Random Processes Power Spectral Density White Noise Discrete-Time Random Processes Mean, Autocorrelation Function and Autocovariance Function Power Spectral Density of a Random Sequence Sampling of Continuous-Time Processes 331

9 Conter 10.9 Chapter Summary 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 Introduction Overview of Linear Systems with Deterministic Inputs Linear Systems with Continuous-Time Random Inputs Linear Systems with Discrete-Time Random Inputs Autoregressive Moving Average Process Moving Average Process Autoregressive Process ARMA Process Chapter Summary 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 Introduction The Bernoulli Process Random Walk Process Symmetric Simple Random Walk Gambler's Ruin The Gaussian Process White Gaussian Noise Process Poisson Process Counting Processes Independent Increment Processes Stationary Increments 379

10 Definitions of a Poisson Process Interarrival Times for the Poisson Process Conditional and Joint PMFs for Poisson Processes Compound Poisson Process Combinations of Independent Poisson Processes Competing Independent Poisson Processes Subdivision of a Poisson Process and the Filtered Poisson Process Random Incidence Markov Processes Discrete-Time Markov Chains State Transition Probability Matrix The n-step State Transition Probability State Transition Diagrams Classification of States Limiting-State Probabilities Doubly Stochastic Matrix Continuous-Time Markov Chains Birth and Death Processes Gambler's Ruin as a Markov Chain Chapter Summary 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

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