STOCHASTIC PROBABILITY THEORY PROCESSES. Universities Press. Y Mallikarjuna Reddy EDITION

Transcription

1 PROBABILITY THEORY STOCHASTIC PROCESSES FOURTH EDITION Y Mallikarjuna Reddy Department of Electronics and Communication Engineering Vasireddy Venkatadri Institute of Technology, Guntur, A.R < Universities Press

2 Preface I Introduction to Probability 1.1 Introduction Set Theory Terminology Set Operations Laws of Sets Sample Spaces Events The Relative Frequency and Axioms of Probability Probability Introduced through Relative Frequency Probability Introduced through Axioms Classical Definition of Probability Mathematical Model of Experiments Examples of Experiments Joint and Conditional Probability Joint Probability Conditional Probability Properties of Conditional Probability Total Probability Theorem Bayes' Theorem Independent Events Multiplication Theorem of Probability Properties of Independent Events Combined Sample Space Independent Experiments Permutations and Combinations Bernoulli Trials 27 Additional Problems 29 More SolvedExamples 52 Questions 70 Problems Multiple-Choice Questions 75 78

3 2 The Random Variable Introduction Random Variable Classifications ofrandom Variables Probability Distribution Function Expression for Distribution Function Probability Density Function Expression for Density Function Properties of Probability Distribution Functions Properties of Probability Density Functions Probability Mass Function Examples of Distribution and Density Functions Gaussian Density Function Uniform Density Function Exponential Probability Density Function Rayleigh Probability Density Function Binomial Probability Density Function Poisson's Probability Density Function Conditional Distribution Function Properties of Conditional Distribution Function Conditional Density Function Properties of Conditional Density Functions Distribution Function for a Conditional Event 103 Additional Problems 105 More Solved Examples 114 Questions 136 Problems Multiple-Choice Questions Operations on a Single Random Variable Introduction Mathematical Expectation Expected Value ofa Random Variable Expected Value ofa Function of a Random Variable Conditional Expectation of a Random Variable Properties of Expectation 145

4 3.4 Moments Moments about the Origin Moments about the Mean 3.5 Variance Physical Significance of Variance and Standard Deviation Skew and Coefficient of Skewness Properties of Variance Relationship between Central Moments and Moments about Origin 3.6 Functions for Moments Characteristic Function Properties of the Characteristic Function Moment Generating Function Properties of the Moment Generating Function 3.7 Inequalities Chebychev's Inequality Markov Inequality Chernoffs Inequality and Bound 3.8 Transformations ofa Random Variable Monotonic Transformation of a Continuous Random Variable Non-Monotonic Transformation of a Continuous Random Variable Transformation of a Discrete Random Variable Additional Problems More Solved Examples Questions Problems Multiple-Choice Questions Multiple Random Variables 4.1 Introduction 4.2 Joint probability Distribution Functions Properties of Joint Distribution Functions 4.3 Joint Probability Density Function Properties of Joint Density Function 4.4 Conditional Distribution and Density Functions Point Conditioning Internal Conditioning

5 4.5 Statistical Independence ofrandom Variables Sum ofrandom Variables Two Random Variables Multiple Random Variables Central Limit Theorem Probability Mass Function 248 Additional Problems 251 More Solved Examples 273 Questions 287 Problems Multiple-Choice Questions Operations on Multiple Random Variables Introduction Function ofjoint Random Variables Joint Moments Joint Moments about the Origin Correlation Properties of Correlation Joint Central Moments Covariance Correlation Coefficient Properties of Covariance Joint Characteristic Function Properties of Joint Characteristic Functions Joint Moment Generating Function Properties of Joint Moment Generating Functions Gaussian Random Variables Two Random Variables ^Random Variables Properties ofgaussian Random Variables Transformation ofrandom Variables Linear Transformation of Gaussian Random Variables Conditional Gaussian Density Functions 322 Additional Problems 323 More Solved Examples 344

6 Questions 360 Problems Multiple-Choice Questions Random Processes Introduction Definition Classification of Random Processes Continuous Random Processes Discrete Random Processes Continuous Random Sequencees Discrete Random Sequencees Distribution and Density Functions of Random Processes Joint Distribution Functions of a Random Process Joint Density Functions of a Random Process Independent Random Processes Statistical Properties ofrandom Processes Mean Autocorrelation Cross Correlation Stationary Processes First-order Stationary Processes Second-Order Stationary Processes Wide-Sense Stationary Processes (WSS) Jointly Wide-Sense Stationary Processes Strict-Sense Stationary Processes (SSS) Time Averages of a Random Process Time Average Function Time Autocorrelation Function Time Cross Correlation Function Ergodic Theorem and Ergodic Processes Ergodic Processes Jointly Ergodic Processes Mean Ergodic Processes Autocorrelation Ergodic Processes Cross Correlation Ergodic Processes 383

7 6.10 Properties of Autocorrelation Functions Properties of Cross Correlation Functions Covariance Functions for Random Processes Autocovariance Function Cross Covariance Function Gaussian Random Processes Poisson Random Processes 396 Additional Problems 398 More Solved Examples 402 Questions 423 Problems Multiple-Choice Questions Random Processes: Spectral Characteristics Spectrum. 7.1 Introduction Power Density Spectrum Average Power of the Random Process Properties of the Power Density Spectrum Bandwidth ofthe Power Density Spectrum Cross Power Density Spectrum Average Cross Power 444 Properties of Cross Power Density 446 Additional Problems 451 More Solved Examples 458 Questions 411 Problems Multiple-Choice Questions Linear Systems with Random Processes Introduction Linear and Time Invariant Systems Linear System Response of a Linear System Linear Time Invariant Systems (LTI) 489

8 8.2.4 Transfer Function ofan LTI System Causal Systems Stable Systems Ideal Systems Response of Linear Systems to Random Signals System Response Mean Value of Output Response Mean Square Value of Output Response 4% Autocorrelation Function of Output Response Cross Correlation Function of Response Spectral Characteristics of System Response Power Density Spectrum of Response Spectrum Bandwidth Types of Random Processes Lowpass Random Processes Bandpass Random Processes Band Limited Random Processes Narrow Band Random Processes Properties ofband Limited Random Processes Noise Introduction Classification ofnoise External Noise Internal Noise White Noise or White Gaussian Noise Power Spectrum of White Noise Band Limited White Noise ResistorNoise Voltage Equivalent Noise Resistor Resistor Noise Spectral Density Available Noise Power Equivalent Noise Temperature Noise through Two Port Networks Signal-to-Noise Ratio Available Power Gain Equivalent Noise Bandwidth Equivalent (Effective) Input Noise Temperature Noise Figure 526

9 Noise Figure in Terms of Available Power Gain Noise Figure in Terms of Input Noise Temperature Noise Figure in Terms of Signal-to-Noise Ratio Noise Figure in Terms ofnetwork Transfer Function Spot Noise Figure Average Operating Noise Figure Output Noise Power and System Noise Power Noise in CascadeAmplifiers Antenna Noise Temperature Narrow Band Noise In Phase and Quadrature Components of a Narrow Band Noise Properties of a Narrow Band Noise Ideal Narrow Band White Noise 536 Additional Problems 538 More Solved Examples 555 Questions 572 Problems Multiple-Choice Questions More Solved Questions on All Chapters 585 Appendix A: Indefinite Integrals, Definite Integrals and Finite Series 653 Appendix B: Fourier Transform Pairs 655 Bibliography 656 Index 657