Probability and Stochastic Processes

Roy Yates received the B.S.E. degree in 1983 from Princeton and the S.M. and Ph.D. degrees in 1986 and 1990 from MIT, all in Electrical Engineering. Since 1990, he has been with the Wireless Information Networks Laboratory (WINLAB) and the ECE department at Rutgers University. Presently, he is an Associate Director of WINLAB and a Professor in the ECE Dept. He is a co-author (with David Goodman) of the text Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers, published by John Wiley and Sons. He is a co-recipient (with Christopher Rose and Sennur Ulukus) of the 2003 IEEE Marconi Prize Paper Award in Wireless Communications. His research interests include power control, interference suppression and spectrum regulation for wireless systems.

• Preface vii1 Random Experiments, Models, and Probabilities 1Getting Started with Probability 11.1 Applying Set Theory to Probability 21.2 Probability Axioms 71.3 Conditional Probability 101.4 Partitions and the Law of Total Probability 131.5 Bayes’ Theorem 171.6 Independence 181.7 Matlab 22Problems 242 Sequential Random Experiments 312.1 Tree Diagrams 312.2 Counting Methods 352.3 Independent Trials 432.4 Matlab 46Problems 483 Discrete Random Variables 533.1 Definitions 533.2 Probability Mass Function 563.3 Families of Discrete Random Variables 593.4 Cumulative Distribution Function (CDF) 653.5 Averages and Expected Value 693.6 Functions of a Random Variable 743.7 Expected Value of a Derived Random Variable 773.8 Variance and Standard Deviation 803.9 Matlab 86Problems 934 Continuous Random Variables 1034.1 Continuous Sample Space 1034.2 The Cumulative Distribution Function 1054.3 Probability Density Function 1084.4 Expected Values 1134.5 Families of Continuous Random Variables 1164.6 Gaussian Random Variables 1224.7 Delta Functions, Mixed Random Variables 1284.8 Matlab 134Problems 1365 Multiple Random Variables 1455.1 Joint Cumulative Distribution Function 1465.2 Joint Probability Mass Function 1495.3 Marginal PMF 1525.4 Joint Probability Density Function 1545.5 Marginal PDF 1595.6 Independent Random Variables 1615.7 Expected Value of a Function of Two Random Variables 1645.8 Covariance, Correlation and Independence 1675.9 Bivariate Gaussian Random Variables 1745.10 Multivariate Probability Models 1785.11 Matlab 183Problems 1886 Probability Models of Derived Random Variables 1996.1 PMF of a Function of Two Discrete Random Variables 2006.2 Functions Yielding Continuous Random Variables 2016.3 Functions Yielding Discrete or Mixed Random Variables 2076.4 Continuous Functions of Two Continuous Random Variables 2116.5 PDF of the Sum of Two Random Variables 2146.6 Matlab 216Problems 2177 Conditional Probability Models 2257.1 Conditioning a Random Variable by an Event 2257.2 Conditional Expected Value Given an Event 2317.3 Conditioning Two Random Variables by an Event 2337.4 Conditioning by a Random Variable 2377.5 Conditional Expected Value Given a Random Variable 2417.6 Bivariate Gaussian Random Variables: Conditional PDFs 2457.7 Matlab 248Problems 2498 Random Vectors 2578.1 Vector Notation 2578.2 Independent Random Variables and Random Vectors 2608.3 Functions of Random Vectors 2618.4 Expected Value Vector and Correlation Matrix 2658.5 Gaussian Random Vectors 2708.6 Matlab 277Problems 2799 Sums of Random Variables 2859.1 Expected Values of Sums 2859.2 Moment Generating Functions 2899.3 MGF of the Sum of Independent Random Variables 2939.4 Characteristic Function and Probability Generating Function 2979.5 Matlab 301Problems 30410 Hypothesis Testing 30710.1 Significance Testing 30810.2 Binary Hypothesis Testing 31110.3 Multiple Hypothesis Test 32410.4 Matlab 327Problems 32911 Estimation of a Random Variable 33911.1 Minimum Mean Square Error Estimation 33911.2 Linear Estimation of X given Y 34411.3 MAP and ML Estimation 34911.4 Linear Estimation of Random Variables from Random Vectors 35311.5 Matlab 360Problems 36212 Some Probabilistic Inequalities and Bounds 36912.1 Markov Inequality 36912.2 Chebyshev’s Inequality 37312.3 Chernoff Bound 37412.4 Central Limit Theorem 37612.5 Sample Mean and Variance 38012.6 Laws of Large Numbers (LLN) 382Problems 38413 Stochastic Processes and Markov Chains 39113.1 Definitions and Examples 39113.2 Random Variables from Random Processes 39713.3 Independent, Identically Distributed Random Sequences 39913.4 The Poisson Process 40013.5 Properties of the Poisson Process 40413.6 The Brownian Motion Process 40713.7 Markov Process 40913.8 Discrete-Time Markov Chains 41013.9 Higher Transition Probabilities: Chapman–Kolmogorov Equations 41413.10 Long-Run Behavior of Markov Chains 41913.11 Classification of States of Chains 42213.12 Markov Chains with Countably Infinite States 42613.13 Ergodic and Reducible Chains 42913.14 Birth Process and Death Process 43313.15 Queuing Models – Poisson Queues 43513.16 Matlab 441Problems 44714 Stationary Processes and Random Signal Processing 45714.1 Expected Value and Correlation 45714.2 Stationary Processes 46014.3 Wide Sense Stationary Processes 46314.4 Cross-Correlation 46614.5 Gaussian Processes 46914.6 Linear Filtering of Continuous-Time Stochastic Processes 47114.7 Linear Filtering of a Random Sequence 47514.8 Discrete-Time Linear Filtering: Vectors and Matrices 48114.9 Power Spectral Density of a Continuous-Time Process 48514.10 Power Spectral Density of a Random Sequence 49014.11 Cross Power Spectral Density 49414.12 Frequency Domain Filter Relationships 49614.13 Matlab 501Problems 510Appendix A The Sample Mean 517A. 1 Sample Mean: Expected Value and Variance 517A. 2 Deviation of a Random Variable from the Expected Value 519A. 3 Laws of Large Numbers 523A. 4 Point Estimates of Model Parameters 525A. 5 Confidence Intervals 531A. 6 Matlab 538Appendix B Families of Random Variables 541B. 1 Discrete Random Variables 541B. 2 Continuous Random Variables 543Appendix C A Few Math Facts 547References 553Index 555