PROBABILITY AND STOCHASTIC PROCESSES A Friendly Introduction for Electrical and Computer Engineers

Transcription

1 PROBABILITY AND STOCHASTIC PROCESSES A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates Rutgers, The State University ofnew Jersey David J. Goodman Rutgers, The State University ofnew Jersey JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Singapore Toronto

2 CHAPTER 1 EXPERIMENTS, MODELS, AND PROBABILITIES / Getting Started with Probability SetTheory Applying Set Theory to Probability Probability Axioms Some Consequences of the Axioms Conditional Probability Independence Sequential Experiments and Tree Diagrams Counting Methods Independent Trials 31 Summary 36 Problems 36 CHAPTER 2 DISCRETE RANDOM VARIABLES Definitions Probability Mass Function Some Useful Discrete Random Variables Cumulative Distribution Function (CDF) Averages Functions of a Random Variable Expected Value of a Derived Random Variable 2.8 Variance and Standard Deviation Conditional Probability Mass Function 74 Summary 79 Problems CHAPTER 3 MULTIPLE DISCRETE RANDOM VARIABLES 3.1 Joint Probability Mass Function Marginal PMF Functions of Two Random Variables xm

3 XIV 3.4 Expectations Conditioning a Joint PMF by an Event Conditional PMF Independent Random Variables More Than Two Discrete Random Variables 108 Summary 777 Problems 772 CHAPTER 4 CONTINUOUS RANDOM VARIABLES 119 Continuous Sample Space The Cumulative Distribution Function Probability Density Function Expected Values Some Useful Continuous Random Variables Gaussian Random Variables Delta Functions, Mixed Random Variables Probability Models of Derived Random Variables Conditioning a Continuous Random Variable 755 Summary 759 Problems 759 CHAPTER 5 MULTIPLE CONTINUOUS RANDOM VARIABLES Joint Cumulative Distribution Function Joint Probability Density Function Marginal PDF Functions of Two Random Variables Expected Values Conditioning a Joint PDF by an Event Conditional PDF Independent Random Variables Jointly Gaussian Random Variables More Than Two Continuous Random Variables 797 Summary 795 Problems 796 CHAPTER 6 STOCHASTIC PROCESSES 201 Definitions 207

4 XV 6.1 Stochastic Process Examples Types of Stochastic Processes Random Variables from Random Processes Independent, Identically Distributed Random Sequences The Poisson Process The Brownian Motion Process Expected Value and Correlation Stationary Processes Wide Sense Stationary Random Processes 223 Summary 225 Problems 226 CHAPTER 7 SUMS OF RANDOM VARIABLES Expectations of Sums PDF of the Sum oftwo Random Variables Moment Generating Function MGF of the Sum of Independent Random Variables Sums of Independent Gaussian Random Variables Random Sums of Independent Random Variables Central Limit Theorem Applications of the Central Limit Theorem 252 Summary 255 Problems 256 CHAPTER 8 THE SAMPLE MEAN Expected Value and Variance Useful Inequalities Sample Mean of Large Numbers Laws of Large Numbers 269 Summary 275 Problems 276 CHAPTER 9 STATISTICAL INFERENCE Significance Testing Binary Hypothesis Testing Multiple Hypothesis Test Estimation of a Random Variable 295

5 XVI 9.5 Linear Estimation of X given Y MAP and ML Estimation Estimation of Model Parameters 310 Summary 316 Problems 317 CHAPTER 10 RANDOM SIGNAL PROCESSING Linear Filtering of a Random Process Power Spectral Density Cross Correlations Gaussian Processes White Gaussian Noise Processes Digital Signal Processing 340 Summary 341 Problems 342 CHAPTER 11 RENEWAL PROCESSES AND MARKOV CHAINS Renewal Processes Poisson Process Renewal-Reward Processes Discrete Time Markov Chains Discrete Time Markov Chain Dynamics Limiting State Probabilities State Classification Limit Theorems For Discrete Time Markov Chains Periodic States and Multiple Communicating Classes Continuous Time Markov Chains Birth-Death Processes and Queueing Systems 386 Summary 391 Problems 392 APPENDIX A COMMON RANDOM VARIABLES 397 A. 1 Discrete Random Variables 397 A.2 Continuous Random Variables 399 APPENDIX B QUIZ SOLUTIONS 403

6 XV11 Quiz Solutions Quiz Solutions Quiz Solutions REFERENCES 449 INDEX 450