Random Processes: Filtering, Estimation, and Detection

CONTENTS

Preface

1 Experiments and Probability

1.1 Definition of an Experiment

1.1.1 The Sample Space

1.1.2 The Borel Field

1.1.3 The Probability Measure

1.2 Combined Experiments

1.2.1 Cartesian Product of Two Experiments

1.2.2 Cartesian Product of n Experiments

1.2.3 Counting Experiments

1.2.4 Selection Combined Experiment

1.3 Conditional Probability

1.3.1 Total Probability Theorem

1.3.2 Bayes’s Theorem

1.4 Random Points

1.4.1 Uniform Random Points in an Interval

1.4.2 Nonuniform Random Points in an Interval

1.5 Summary

Problems

References

2 Random Variables

2.1 Definition of a Random Variable

2.1.1 Cumulative Distribution Function (CDF)

2.1.2 Probability Density Function (PDF)

2.1.3 Partial Characterizations

2.1.4 Conditional Cumulative Distribution Functions

2.1.5 Characteristic Function

2.1.6 Higher-Order Moments for Gaussian Random Variables

2.2 Common Continuous Random Variables

2.3 Common Discrete Random Variables

2.4 Transformations of One Random Variable

2.4.1 Transformation of One Random Variable

2.4.2 Cumulative Distribution Function

2.5 Computation of Expected Values

2.6 Two Random Variables

2.6.1 Joint Cumulative Distribution Function

2.6.2 Joint Probability Density Function

2.6.3 Partial Characterizations

2.6.4 Jointly Normal Random Variables

2.7 Two Functions of Two Random Variables

2.7.1 Probability Density Function (Discrete Random Variables)

2.7.2 Probability Density Function (Continuous Random Variables and Continuous Functions)

2.7.3 Distribution Function (Continuous, ...