Probability and random processes

The second edition enhanced with new chapters, figures, and appendices to cover the new developments in applied mathematical functions This book examines the topics of applied mathematical functions to problems that engineers and researchers solve daily in the course of their work. The text covers s...

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Bibliographische Detailangaben
Hauptverfasser: Krishnan, Venkatarama, Chandra, Kavitha
Format: E-Book Buch
Sprache:Englisch
Veröffentlicht: New York Wiley 2015
John Wiley & Sons, Incorporated
Wiley-Blackwell
Ausgabe:2nd ed
Schlagworte:
Engineering
Engineering > Statistical methods
MATHEMATICS
Probabilities
Science
Science > Statistical methods
Statistical methods
Stochastic processes
ISBN:1118923138, 9781118923139, 1119011906, 9781119011903
Online-Zugang:Volltext
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Inhaltsangabe:
  • 14.4 Chernoff Bound -- 14.5 Cauchy-Schwartz Inequality -- 14.6 Jensen's Inequality -- 14.7 Convergence Concepts -- 14.8 Limit Theorems -- Chapter 15 Computer Methods for Generating Random Variates -- 15.1 Uniform-Distribution Random Variates -- 15.2 Histograms -- 15.3 Inverse Transformation Techniques -- 15.4 Convolution Techniques -- 15.5 Acceptance-Rejection Techniques -- Chapter 16 Elements of Matrix Algebra -- 16.1 Basic Theory of Matrices -- 16.2 Eigenvalues and Eigenvectors of Matrices -- 16.3 Vector and Matrix Differentiation -- 16.4 Block Matrices -- Chapter 17 Random Vectors and Mean-Square Estimation -- 17.1 Distributions and Densities -- 17.2 Moments of Random Vectors -- 17.3 Vector Gaussian Random Variables -- 17.4 Diagonalization of Covariance Matrices -- 17.5 Simultaneous Diagonalization of Covariance Matrices -- 17.6 Linear Estimation of Vector Variables -- Chapter 18 Estimation Theory -- 18.1 Criteria of Estimators -- 18.2 Estimation of Random Variables -- 18.3 Estimation of Parameters (Point Estimation) -- 18.4 Interval Estimation (Confidence Intervals) -- 18.5 Hypothesis Testing (Binary) -- 18.6 Bayesian Estimation -- Chapter 19 Random Processes -- 19.1 Basic Definitions -- 19.2 Stationary Random Processes -- 19.3 Ergodic Processes -- 19.4 Estimation of Parameters of Random Processes -- 19.4.1 Continuous-Time Processes -- 19.4.2 Discrete-Time Processes -- 19.5 Power Spectral Density -- 19.5.1 Continuous Time -- 19.5.2 Discrete Time -- 19.6 Adaptive Estimation -- Chapter 20 Classification of Random Processes -- 20.1 Specifications of Random Processes -- 20.1.1 Discrete-State Discrete-Time (DSDT) Process -- 20.1.2 Discrete-State Continuous-Time (DSCT) Process -- 20.1.3 Continuous-State Discrete-Time (CSDT) Process -- 20.1.4 Continuous-State Continuous-Time (CSCT) Process -- 20.2 Poisson Process -- 20.3 Binomial Process
  • 7.17 Summary of Distributions of Continuous Random Variables -- Chapter 8 Conditional Densities and Distributions -- 8.1 Conditional Distribution and Density for P{A}≠0 -- 8.2 Conditional Distribution and Density for P{A}=0 -- 8.3 Total Probability and Bayes' Theorem for Densities -- Chapter 9 Joint Densities and Distributions -- 9.1 Joint Discrete Distribution Functions -- 9.2 Joint Continuous Distribution Functions -- 9.3 Bivariate Gaussian Distributions -- Chapter 10 Moments and Conditional Moments -- 10.1 Expectations -- 10.2 Variance -- 10.3 Means and Variances of Some Distributions -- 10.4 Higher-Order Moments -- 10.5 Correlation and Partial Correlation Coefficients -- 10.5.1 Correlation Coefficients -- 10.5.2 Partial Correlation Coefficients -- Chapter 11 Characteristic Functions and Generating Functions -- 11.1 Characteristic Functions -- 11.2 Examples of Characteristic Functions -- 11.3 Generating Functions -- 11.4 Examples of Generating Functions -- 11.5 Moment Generating Functions -- 11.6 Cumulant Generating Functions -- 11.7 Table of Means and Variances -- Chapter 12 Functions of a Single Random Variable -- 12.1 Random Variable g(X) -- 12.2 Distribution of Y=g(X) -- 12.3 Direct Determination of Density fY(y) from fX(x) -- 12.4 Inverse Problem: Finding g(x) given fX(x) and fY(y) -- 12.5 Moments of a Function of a Random Variable -- Chapter 13 Functions of Multiple Random Variables -- 13.1 Function of Two Random Variables, Z=g(X,Y) -- 13.2 Two Functions of Two Random Variables, Z=g(X,Y), W=h(X,Y) -- 13.3 Direct Determination of Joint Density fZW(z,w) from fXY(x,y) -- 13.4 Solving Z=g(X,Y) Using an Auxiliary Random Variable -- 13.5 Multiple Functions of Random Variables -- Chapter 14 Inequalities, Convergences, and Limit Theorems -- 14.1 Degenerate Random Variables -- 14.2 Chebyshev and Allied Inequalities -- 14.3 Markov Inequality
  • 20.4 Independent Increment Process -- 20.5 Random-Walk Process -- 20.6 Gaussian Process -- 20.7 Wiener Process (Brownian Motion) -- 20.8 Markov Process -- 20.9 Markov Chains -- 20.10 Birth and Death Processes -- 20.11 Renewal Processes and Generalizations -- 20.12 Martingale Process -- 20.13 Periodic Random Process -- 20.14 Aperiodic Random Process (Karhunen-Loeve Expansion) -- Chapter 21 Random Processes and Linear Systems -- 21.1 Review of Linear Systems -- 21.2 Random Processes through Linear Systems -- 21.3 Linear Filters -- 21.4 Bandpass Stationary Random Processes -- Chapter 22 Wiener and Kalman Filters -- 22.1 Review of Orthogonality Principle -- 22.2 Wiener Filtering -- 22.3 Discrete Kalman Filter -- 22.4 Continuous Kalman Filter -- Chapter 23 Probability Modeling in Traffic Engineering -- 23.1 Introduction -- 23.2 Teletraffic Models -- 23.3 Blocking Systems -- 23.4 State Probabilities for Systems with Delays -- 23.5 Waiting-Time Distribution for M/M/c/∞ Systems -- 23.6 State Probabilities for M/D/c Systems -- 23.7 Waiting-time distribution for M/D/c/∞ System -- 23.8 Comparison of M/M/c and M/D/c -- References -- Chapter 24 Probabilistic Methods in Transmission Tomography -- 24.1 Introduction -- 24.2 Stochastic Model -- 24.3 Stochastic Estimation Algorithm -- 24.4 Prior Distribution P{M} -- 24.5 Computer Simulation -- 24.6 Results and Conclusions -- 24.7 Discussion of Results -- References -- Appendix -- Appendix A Fourier Transform Tables -- Appendix B Cumulative Gaussian Tables -- Appendix C Inverse Cumulative Gaussian Tables -- Appendix D Inverse Chi-Square Tables -- Appendix E Inverse Student-t Tables -- Appendix F Cumulative Poisson Distribution -- Appendix G Cumulative Binomial Distribution -- Appendix H Computation of Roots of D(Z) = 0 -- References -- Web References -- Index -- EULA
  • Intro -- Title Page -- Copyright Page -- Contents -- Preface for the Second Edition -- Preface for the First Edition -- Chapter 1 Sets, Fields, and Events -- 1.1 Set Definitions -- 1.2 Set Operations -- 1.3 Set Algebras, Fields, and Events -- Chapter 2 Probability Space and Axioms -- 2.1 Probability Space -- 2.2 Conditional Probability -- 2.3 Independence -- 2.4 Total Probability and Bayes´ Theorem -- Chapter 3 Basic Combinatorics -- 3.1 Basic Counting Principles -- 3.2 Permutations -- 3.3 Combinations -- Chapter 4 Discrete Distributions -- 4.1 Bernoulli Trials -- 4.2 Binomial Distribution -- 4.3 Multinomial Distribution -- 4.4 Geometric Distribution -- 4.5 Negative Binomial Distribution -- 4.6 Hypergeometric Distribution -- 4.7 Poisson Distribution -- 4.8 Newton-Pepys Problem and its Extensions -- 4.9 Logarithmic Distribution -- 4.9.1 Finite Law (Benford's Law) -- 4.9.2 Infinite Law -- 4.10 Summary of Discrete Distributions -- Chapter 5 Random Variables -- 5.1 Definition of Random Variables -- 5.2 Determination of Distribution and Density Functions -- 5.3 Properties of Distribution and Density Functions -- 5.4 Distribution Functions from Density Functions -- Chapter 6 Continuous Random Variables and Basic Distributions -- 6.1 Introduction -- 6.2 Uniform Distribution -- 6.3 Exponential Distribution -- 6.4 Normal or Gaussian Distribution -- Chapter 7 Other Continuous Distributions -- 7.1 Introduction -- 7.2 Triangular Distribution -- 7.3 Laplace Distribution -- 7.4 Erlang Distribution -- 7.5 Gamma Distribution -- 7.6 Weibull Distribution -- 7.7 Chi-Square Distribution -- 7.8 Chi and Other Allied Distributions -- 7.9 Student-t DENSITY -- 7.10 Snedecor F Distribution -- 7.11 Lognormal Distribution -- 7.12 Beta Distribution -- 7.13 Cauchy Distribution -- 7.14 Pareto Distribution -- 7.15 Gibbs Distribution -- 7.16 Mixed Distributions
  • 15.2 HISTOGRAMS -- 15.3 INVERSE TRANSFORMATION TECHNIQUES -- 15.4 CONVOLUTION TECHNIQUES -- 15.5 ACCEPTANCE-REJECTION TECHNIQUES -- 16 ELEMENTS OF MATRIX ALGEBRA -- 16.1 BASIC THEORY OF MATRICES -- 16.2 EIGENVALUES AND EIGENVECTORS OF MATRICES -- 16.3 VECTOR AND MATRIX DIFFERENTIATION -- 16.4 BLOCK MATRICES -- 17 RANDOM VECTORS AND MEAN-SQUARE ESTIMATION -- 17.1 DISTRIBUTIONS AND DENSITIES -- 17.2 MOMENTS OF RANDOM VECTORS -- 17.3 VECTOR GAUSSIAN RANDOM VARIABLES -- 17.4 DIAGONALIZATION OF COVARIANCE MATRICES -- 17.5 SIMULTANEOUS DIAGONALIZATION OF COVARIANCE MATRICES -- 17.6 LINEAR ESTIMATION OF VECTOR VARIABLES -- 18 ESTIMATION THEORY -- 18.1 CRITERIA OF ESTIMATORS -- 18.2 ESTIMATION OF RANDOM VARIABLES -- 18.3 ESTIMATION OF PARAMETERS (POINT ESTIMATION) -- 18.4 INTERVAL ESTIMATION (CONFIDENCE INTERVALS) -- 18.5 HYPOTHESIS TESTING (BINARY) -- 18.6 BAYESIAN ESTIMATION -- 19 RANDOM PROCESSES -- 19.1 BASIC DEFINITIONS -- 19.2 STATIONARY RANDOM PROCESSES -- 19.3 ERGODIC PROCESSES -- 19.4 ESTIMATION OF PARAMETERS OF RANDOM PROCESSES -- 19.5 POWER SPECTRAL DENSITY -- 19.6 ADAPTIVE ESTIMATION -- 20 CLASSIFICATION OF RANDOM PROCESSES -- 20.1 SPECIFICATIONS OF RANDOM PROCESSES -- 20.2 POISSON PROCESS -- 20.3 BINOMIAL PROCESS -- 20.4 INDEPENDENT INCREMENT PROCESS -- 20.5 RANDOM-WALK PROCESS -- 20.6 GAUSSIAN PROCESS -- 20.7 WIENER PROCESS (BROWNIAN MOTION) -- 20.8 MARKOV PROCESS -- 20.9 MARKOV CHAINS -- 20.10 BIRTH AND DEATH PROCESSES -- 20.11 RENEWAL PROCESSES AND GENERALIZATIONS -- 20.12 MARTINGALE PROCESS -- 20.13 PERIODIC RANDOM PROCESS -- 20.14 APERIODIC RANDOM PROCESS (KARHUNEN-LOEVE EXPANSION) -- 21 RANDOM PROCESSES AND LINEAR SYSTEMS -- 21.1 REVIEW OF LINEAR SYSTEMS -- 21.2 RANDOM PROCESSES THROUGH LINEAR SYSTEMS -- 21.3 LINEAR FILTERS -- 21.4 BANDPASS STATIONARY RANDOM PROCESSES -- 22 WIENER AND KALMAN FILTERS
  • 8.1 CONDITIONAL DISTRIBUTION AND DENSITY FOR P{A} -- 8.2 CONDITIONAL DISTRIBUTION AND DENSITY FOR P{A} = 0 -- 8.3 TOTAL PROBABILITY AND BAYES' THEOREM FOR DENSITIES -- 9 JOINT DENSITIES AND DISTRIBUTIONS -- 9.1 JOINT DISCRETE DISTRIBUTION FUNCTIONS -- 9.2 JOINT CONTINUOUS DISTRIBUTION FUNCTIONS -- 9.3 BIVARIATE GAUSSIAN DISTRIBUTIONS -- 10 MOMENTS AND CONDITIONAL MOMENTS -- 10.1 EXPECTATIONS -- 10.2 VARIANCE -- 10.3 MEANS AND VARIANCES OF SOME DISTRIBUTIONS -- 10.4 HIGHER-ORDER MOMENTS -- 10.5 CORRELATION AND PARTIAL CORRELATION COEFFICIENTS -- 11 CHARACTERISTIC FUNCTIONS AND GENERATING FUNCTIONS -- 11.1 CHARACTERISTIC FUNCTIONS -- 11.2 EXAMPLES OF CHARACTERISTIC FUNCTIONS -- 11.3 GENERATING FUNCTIONS -- 11.4 EXAMPLES OF GENERATING FUNCTIONS -- 11.5 MOMENT GENERATING FUNCTIONS -- 11.6 CUMULANT GENERATING FUNCTIONS -- 11.7 TABLE OF MEANS AND VARIANCES -- 12 FUNCTIONS OF A SINGLE RANDOM VARIABLE -- 12.1 RANDOM VARIABLE g(X) -- 12.2 DISTRIBUTION OF Y = g(X) -- 12.3 DIRECT DETERMINATION OF DENSITY fY(y) from fX(x) -- 12.4 INVERSE PROBLEM: FINDING g(x) GIVEN fX(x) AND fY(y) -- 12.5 MOMENTS OF A FUNCTION OF A RANDOM VARIABLE -- 13 FUNCTIONS OF MULTIPLE RANDOM VARIABLES -- 13.1 FUNCTION OF TWO RANDOM VARIABLES, Z = g(X,Y) -- 13.2 TWO FUNCTIONS OF TWO RANDOM VARIABLES, Z = g(X,Y), W = h(X,Y) -- 13.3 DIRECT DETERMINATION OF JOINT DENSITY fZW(z,w) FROM fXY(x,y) -- 13.4 SOLVING Z = g(X,Y) USING AN AUXILIARY RANDOM VARIABLE -- 13.5 MULTIPLE FUNCTIONS OF RANDOM VARIABLES -- 14 INEQUALITIES, CONVERGENCES, AND LIMIT THEOREMS -- 14.1 DEGENERATE RANDOM VARIABLES -- 14.2 CHEBYSHEV AND ALLIED INEQUALITIES -- 14.3 MARKOV INEQUALITY -- 14.4 CHERNOFF BOUND -- 14.5 CAUCHY-SCHWARTZ INEQUALITY -- 14.6 JENSEN'S INEQUALITY -- 14.7 CONVERGENCE CONCEPTS -- 14.8 LIMIT THEOREMS -- 15 COMPUTER METHODS FOR GENERATING RANDOM VARIATES -- 15.1 UNIFORM-DISTRIBUTION RANDOM VARIATES
  • Intro -- TITLE PAGE -- TABLE OF CONTENTS -- PREFACE FOR THE SECOND EDITION -- PREFACE FOR THE FIRST EDITION -- 1 SETS, FIELDS, AND EVENTS -- 1.1 SET DEFINITIONS -- 1.2 SET OPERATIONS -- 1.3 SET ALGEBRAS, FIELDS, AND EVENTS -- 2 PROBABILITY SPACE AND AXIOMS -- 2.1 PROBABILITY SPACE -- 2.2 CONDITIONAL PROBABILITY -- 2.3 INDEPENDENCE -- 2.4 TOTAL PROBABILITY AND BAYES' THEOREM -- 3 BASIC COMBINATORICS -- 3.1 BASIC COUNTING PRINCIPLES -- 3.2 PERMUTATIONS -- 3.3 COMBINATIONS -- 4 DISCRETE DISTRIBUTIONS -- 4.1 BERNOULLI TRIALS -- 4.2 BINOMIAL DISTRIBUTION -- 4.3 MULTINOMIAL DISTRIBUTION -- 4.4 GEOMETRIC DISTRIBUTION -- 4.5 NEGATIVE BINOMIAL DISTRIBUTION -- 4.6 HYPERGEOMETRIC DISTRIBUTION -- 4.7 POISSON DISTRIBUTION -- 4.8 NEWTON-PEPYS PROBLEM AND ITS EXTENSIONS -- 4.9 LOGARITHMIC DISTRIBUTION -- 4.10 SUMMARY OF DISCRETE DISTRIBUTIONS -- 5 RANDOM VARIABLES -- 5.1 DEFINITION OF RANDOM VARIABLES -- 5.2 DETERMINATION OF DISTRIBUTION AND DENSITY FUNCTIONS -- 5.3 PROPERTIES OF DISTRIBUTION AND DENSITY FUNCTIONS -- 5.4 DISTRIBUTION FUNCTIONS FROM DENSITY FUNCTIONS -- 6 CONTINUOUS RANDOM VARIABLES AND BASIC DISTRIBUTIONS -- 6.1 INTRODUCTION -- 6.2 UNIFORM DISTRIBUTION -- 6.3 EXPONENTIAL DISTRIBUTION -- 6.4 NORMAL OR GAUSSIAN DISTRIBUTION -- 7 OTHER CONTINUOUS DISTRIBUTIONS -- 7.1 INTRODUCTION -- 7.2 TRIANGULAR DISTRIBUTION -- 7.3 LAPLACE DISTRIBUTION -- 7.4 ERLANG DISTRIBUTION -- 7.5 GAMMA DISTRIBUTION -- 7.6 WEIBULL DISTRIBUTION -- 7.7 CHI-SQUARE DISTRIBUTION -- 7.8 CHI AND OTHER ALLIED DISTRIBUTIONS -- 7.9 STUDENT-t DENSITY -- 7.10 SNEDECOR F DISTRIBUTION -- 7.11 LOGNORMAL DISTRIBUTION -- 7.12 BETA DISTRIBUTION -- 7.13 CAUCHY DISTRIBUTION -- 7.14 PARETO DISTRIBUTION -- 7.15 GIBBS DISTRIBUTION -- 7.16 MIXED DISTRIBUTIONS -- 7.17 SUMMARY OF DISTRIBUTIONS OF CONTINUOUS RANDOM VARIABLES -- 8 CONDITIONAL DENSITIES AND DISTRIBUTIONS
  • 22.1 REVIEW OF ORTHOGONALITY PRINCIPLE -- 22.2 WIENER FILTERING -- 22.3 DISCRETE KALMAN FILTER1 -- 22.4 CONTINUOUS KALMAN FILTER -- 23 PROBABILITY MODELING IN TRAFFIC ENGINEERING -- 23.1 INTRODUCTION -- 23.2 TELETRAFFIC MODELS -- 23.3 BLOCKING SYSTEMS -- 23.4 STATE PROBABILITIES FOR SYSTEMS WITH DELAYS -- 23.5 WAITING-TIME DISTRIBUTION FOR M/M/c/∞ SYSTEMS -- 23.6 STATE PROBABILITIES FOR M/D/c SYSTEMS -- 23.7 WAITING-TIME DISTRIBUTION FOR M/D/c/∞ SYSTEM -- 23.8 COMPARISON OF M/M/c AND M/D/c -- REFERENCES -- 24 PROBABILISTIC METHODS IN TRANSMISSION TOMOGRAPHY -- 24.1 INTRODUCTION -- 24.2 STOCHASTIC MODEL -- 24.3 STOCHASTIC ESTIMATION ALGORITHM -- 24.4 PRIOR DISTRIBUTION P{} -- 24.5 COMPUTER SIMULATION -- 24.6 RESULTS AND CONCLUSIONS -- 24.7 DISCUSSION OF RESULTS -- REFERENCES -- APPENDIX A A FOURIER TRANSFORM TABLES -- APPENDIX B CUMULATIVE GAUSSIAN TABLES -- APPENDIX C INVERSE CUMULATIVE GAUSSIAN TABLES -- APPENDIX D INVERSE CHI-SQUARE TABLES -- APPENDIX E INVERSE STUDENT-t TABLES -- APPENDIX F CUMULATIVE POISSON DISTRIBUTION -- APPENDIX G CUMULATIVE BINOMIAL DISTRIBUTION -- APPENDIX H COMPUTATION OF ROOTS OF D(z) = 0 -- REFERENCES -- WEB REFERENCES -- INDEX -- END USER LICENSE AGREEMENT