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APPENDIX D: Table of Continuous Random Variables and Properties
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APPENDIX D: Table of Continuous Random Variables and Properties
by Lonnie C. Ludeman
Random Processes: Filtering, Estimation, and Detection
Cover Page
Title Page
Copyright
Dedication
CONTENTS
PREFACE
Chapter 1: Experiments and Probability
1.1 DEFINITION OF AN EXPERIMENT
1.2 COMBINED EXPERIMENTS
1.3 CONDITIONAL PROBABILITY
1.4 RANDOM POINTS
1.5 SUMMARY
Chapter 2: Random Variables
2.1 DEFINITION OF A RANDOM VARIABLE
2.2 COMMON CONTINUOUS RANDOM VARIABLES
2.3 COMMON DISCRETE RANDOM VARIABLES
2.4 TRANSFORMATIONS OF ONE RANDOM VARIABLE
2.5 COMPUTATION OF EXPECTED VALUES
2.6 TWO RANDOM VARIABLES
2.7 TWO FUNCTIONS OF TWO RANDOM VARIABLES
2.8 ONE FUNCTION OF TWO RANDOM VARIABLES
2.9 COMPUTATION OF E [ h ( X , Y )]
2.10 MULTIPLE RANDOM VARIABLES
2.11 M FUNCTIONS OF N RANDOM VARIABLES
Chapter 3: Estimation of Random Variables
3.1 ESTIMATION OF VARIABLES
3.2 LINEAR MMSE ESTIMATION
3.3 NONLINEAR MMSE ESTIMATION
3.4 PROPERTIES OF ESTIMATORS OF RANDOM VARIABLES
3.5 BAYES ESTIMATION
3.6 ESTIMATION OF NONRANDOM PARAMETERS
3.7 SUMMARY
Chapter 4: Random Processes
4.1 DEFINITION OF A RANDOM PROCESS
4.2 CHARACTERIZATIONS OF A RANDOM PROCESS
4.3 STATIONARITY OF RANDOM PROCESSES
4.4 EXAMPLES OF RANDOM PROCESSES
4.5 DEFINITE INTEGRALS OF RANDOM PROCESSES
4.6 JOINT CHARACTERIZATIONS OF RANDOM PROCESSES
4.7 GAUSSIAN RANDOM PROCESSES
4.8 WHITE RANDOM PROCESSES
4.9 ARMA RANDOM PROCESSES
4.10 PERIODIC RANDOM PROCESSES
4.11 SAMPLING OF CONTINUOUS RANDOM PROCESSES
4.12 ERGODIC RANDOM PROCESSES
Chapter 5: Linear Systems: Random Processes
5.1 INTRODUCTION
5.2 CLASSIFICATION OF SYSTEMS
5.3 CONTINUOUS LINEAR TIME-INVARIANT SYSTEMS (RANDOM INPUTS)
5.4 CONTINUOUS TIME-VARYING SYSTEMS WITH RANDOM INPUT
5.5 DISCRETE TIME-INVARIANT LINEAR SYSTEMS WITH RANDOM INPUTS
5.6 DISCRETE TIME-VARYING LINEAR SYSTEMS WITH RANDOM INPUTS
5.7 LINEAR SYSTEM IDENTIFICATION
5.8 DERIVATIVES OF RANDOM PROCESSES
5.9 MULTI-INPUT, MULTI-OUTPUT LINEAR SYSTEMS
5.10 TRANSIENTS IN LINEAR SYSTEMS
5.11 SUMMARY
Chapter 6: Nonlinear Systems: Random Processes
6.1 INTRODUCTION
6.2 CLASSIFICATION OF NONLINEAR SYSTEMS
6.3 RANDOM OUTPUTS FOR INSTANTANEOUS NONLINEAR SYSTEMS
6.4 CHARACTERIZATIONS FOR BILINEAR SYSTEMS
6.5 CHARACTERIZATIONS FOR TRILINEAR SYSTEMS
6.6 CHARACTERIZATIONS FOR VOLTERRA NONLINEAR SYSTEMS
6.7 HIGHER-ORDER CHARACTERIZATIONS
6.8 SUMMARY
Chapter 7: Optimum Linear Filters: The Wiener Approach
7.1 OPTIMUM FILTER FORMULATION
7.2 BASIC PROBLEMS
7.3 THE WIENER FILTER
7.4 THE DISCRETE WIENER FILTER
7.5 OPTIMAL LINEAR SYSTEM OF PARAMETRIC FORM
7.6 SUMMARY
Chapter 8: Optimum Linear Systems: The Kalman Approach
8.1 INTRODUCTION
8.2 DISCRETE TIME SYSTEMS
8.3 BASIC ESTIMATION PROBLEM
8.4 OPTIMAL FILTERED ESTIMATE
8.5 OPTIMAL PREDICTION
8.6 OPTIMAL SMOOTHING
8.7 STEADY STATE EQUIVALENCE OF THE KALMAN AND WIENER FILTERS
8.8 SUMMARY
Chapter 9: Detection Theory: Discrete Observation
9.1 BASIC DETECTION PROBLEM
9.2 MAXIMUM A POSTERIORI DECISION RULE
9.3 MINIMUM PROBABILITY OF ERROR CLASSIFIER
9.4 BAYES DECISION RULE
9.5 SPECIAL CASES FOR THE MULTIPLE-CLASS PROBLEM (BAYES)
9.6 NEYMAN-PEARSON CLASSIFIER
9.7 GENERAL CALCULATION OF PROBABILITY OF ERROR
9.8 GENERAL GAUSSIAN PROBLEM
9.9 COMPOSITE HYPOTHESES
9.10 SUMMARY
Chapter 10: Detection Theory: Continuous Observation
10.1 CONTINUOUS OBSERVATIONS
10.2 DETECTION OF KNOWN SIGNALS IN WHITE GAUSSIAN NOISE
10.3 DETECTION OF KNOWN SIGNALS IN NONWHITE GAUSSIAN NOISE (ANWGN)
10.4 DETECTION OF KNOWN SIGNALS IN COMBINATION OF WHITE AND NONWHITE GAUSSIAN NOISE (AW&NWGN)
10.5 OPTIMUM CLASSIFIER FOR GENERAL GAUSSIAN PROCESSES (TWO-CLASS DETECTION)
10.6 DETECTION OF KNOWN SIGNALS WITH RANDOM PARAMETERS IN ADDITIVE WHITE GAUSSIAN NOISE
10.7 SUMMARY
APPENDIX A: The Bilateral Laplace Transform
APPENDIX B: Table of Binomial Probabilities
APPENDIX C: Table of Discrete Random Variables and Properties
APPENDIX D: Table of Continuous Random Variables and Properties
APPENDIX E: Table for Gaussian Cumulative Distribution Function
INDEX
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APPENDIX C: Table of Discrete Random Variables and Properties
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APPENDIX E: Table for Gaussian Cumulative Distribution Function
APPENDIX D
Table of Continuous Random Variables and Properties
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