Random Signal Analysis in Engineering Systems

Preface1 Set Theory and Probability 1.1 Introduction 1.2 Set Theory 1.3 Probability Theory 1.4 Conditional Probability and Statistical Independence Problems2 Random Variables and Vectors 2.1 Random Variables 2.2 Gaussian Random Variable 2.3 Other Random Variables 2.4 Random Vectors 2.5 Conditional Distribution and Density Functions Problems3 Functions of Random Variables 3.1 One-to-One Transformation of Random Variables 3.2 One-to-One Transformation of Random Vectors 3.3 Non-One-to-One Transformation 3.4 Sum of Random Variables 3.5 Special Transformations Problems4 Statistical Averages 4.1 Expected Value 4.2 Characteristic Function 4.3 Multiple Random Variables 4.4 TV-Dimensional Gaussian 4.5 Binary Communication Scheme 4.6 Inequalities 4.7 Central Limit Theorem Problems5 Simulation 5.1 Theoretical Histogram 5.2 Pseudorandom Numbers 5.3 Empirical Histogram 5.4 Goodness of Fit 5.5 Simulation Confidence Statements 5.6 Correlated Random Variables Problems6 Statistical Inference 6.1 Sampling and Estimation 6.2 Maximum Likelihood Estimation 6.3 Sequential Estimation 6.4 Random Parameter Estimation Problems7 Random Processes 7.1 Statistical Description 7.2 Statistical Averages 7.3 Spectral Density 7.4 Linear Systems with Random Inputs ProblemsAppendix A: Evaluation of Gaussian Probabilities A.l Q Function Evaluation A.2 Inverse Q FunctionAppendix B: Sum of N Uniform Random VariablesAppendix C: Moments of Random VariablesAppendix D: Matrix NotationAppendix E: Mathematical Quantities E.l Trigonometric Identities E.2 Indefinte Integrals E.3 Definite Integrals E.4 Series E.5 Fourier TransformsBibliographyAnswers to Selected ProblemsIndex