Probability and Stochastic Processes

Transcription

1 Probability and Stochastic Processes A Friendly Introduction Electrical and Computer Engineers Third Edition Roy D. Yates Rutgers, The State University of New Jersey David J. Goodman New York University Wiley

2 Contents Features of this Text i Preface vii 1 Experiments, Models, and Probabilities 1 Getting Started with Probability Set Theory Applying Set Theory to Probability Probability Axioms Conditional Probability Partitions and the Law of Total Probability Independence Matlab 27 Problems 29 2 Sequential Experiments Tree Diagrams Counting Methods Independent Trials Reliability Analysis Matlab 55 Problems 57 3 Discrete Random Variables Definitions Probability Mass Function Families of Discrete Random Variables Cumulative Distribution Function (CDF) Averages and Expected Value Functions of a Random Variable Expected Value of a Derived Random Variable Variance and Standard Deviation Matlab 99 Problems 106 xiii

3 x/v CONTENTS 4 Continuous Random Variables Continuous Sample Space The Cumulative Distribution Function Probability Density Function Expected Values Families of Continuous Random Variables Gaussian Random Variables Delta Functions, Mixed Random Variables Matlab 152 Problems Multiple Random Variables Joint Cumulative Distribution Function Joint Probability Mass Function Marginal PMF Joint Probability Density Function Marginal PDF Independent Random Variables Expected Value of a Function of Two Random Variables Covariance, Correlation and Independence Bivariate Gaussian Random Variables Multivariate Probability Models Matlab 201 Problems Probability Models of Derived Random Variables PMF of a Function of Two Discrete Random Variables Functions Yielding Continuous Random Variables Functions Yielding Discrete or Mixed Random Variables Continuous Functions of Two Continuous Random Variables PDF of the Sum of Two Random Variables Matlab 234 Problems Conditional Probability Models Conditioning a Random Variable by an Event Conditional Expected Value Given an Event 248

4 CONTENTS xv 7.3 Conditioning Two Random Variables by an Event Conditioning by a Random Variable Conditional Expected Value Given a Random Variable Bivariate Gaussian Random Variables: Conditional PDFs Matlab 268 Problems Random Vectors Vector Notation Independent Random Variables and Random Vectors Functions of Random Vectors Expected Value Vector and Correlation Matrix Gaussian Random Vectors Matlab 298 Problems Sums of Random Variables Expected Values of Sums Moment Generating Functions MGF of the Sum of Independent Random Variables Random Sums of Independent Random Variables Central Limit Theorem Matlab 328 Problems The Sample Mean Sample Mean: Expected Value and Variance Deviation of a Random Variable from the Expected Value Laws of Large Numbers Point Estimates of Model Parameters Confidence Intervals Matlab 358 Problems Hypothesis Testing Significance Testing Binary Hypothesis Testing Multiple Hypothesis Test 384

5 xvi CONTENTS 11.4 MATLAB 387 Problems Estimation of a Random Variable Minimum Mean Square Error Estimation Linear Estimation of X given Y MAP and ML Estimation Linear Estimation of Random, Variables from Random Vectors 4H 12.5 Matlab 421 Problems Stochastic Processes 4^ Definitions and Examples Random Variables from Random Processes Independent. Identically Distributed Random Sequences The Poisson Process Properties of the Poisson Process The Brownian Motion Process Expected Value and Correlation 44$ 13.8 Stationary Processes Wide Sense Stationary Stochastic Processes Cross-Correlation Gaussian Processes 4^ Matlab 464 Problems 4^ Appendix A Families of R.andom Variables,?7 A.l Discrete Random Variables 4-77 A.2 Continuous Random Variables.$7$ Appendix B A Feiv Math Facts 4$.$ References 4$9 Index 491