Probability and Random Variables for Electrical Engineering. Probability: Measurement of Uncertainty

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Author(s): Muammer Catak, Tofigh Allahviranloo, Witold Pedrycz
Series: Studies in Systems, Decision and Control, 390
Publisher: Springer
Year: 2022

Language: English

Preface
Overview
Chapter Descriptions
Contents
Part I Concepts of Probability Theory
1 Introduction
1.1 Set Theory
1.2 Set Operations
1.3 Sample Space and Events
1.4 Probability Axioms
1.5 Probability of the Equally Likely Outcomes
1.6 Conditional Probability and Bayes' Theory
1.7 Problems
2 Continuous Random Variables
2.1 The Probability Distribution Function (PDF)
2.2 The Probability Density Function (PDF)
2.3 Expected Value and Variance
2.4 Common Continuous Probability Distribution Functions
2.4.1 Uniform Distribution
2.4.2 Normal (Gaussian) Distribution
2.4.3 Exponential Distribution
2.4.4 Gamma Distribution
2.5 Problems
3 Discrete Random Variables
3.1 The Probability Distribution Function (PDF)
3.2 The Probability Density Function (PDF)
3.3 Expected Value and Variance
3.4 Common Discrete Probability Distribution Functions
3.4.1 Uniform Distribution
3.4.2 Bernoulli Distribution
3.4.3 Binomial Distribution
3.4.4 Geometric Distribution
3.4.5 The Memoryless Property of the Geometric Distribution
3.4.6 Poisson Distribution
3.5 The Mixed Probability Distributions
3.6 Problems
4 Multiple Random Variables
4.1 The Joint Probability Distribution Function
4.2 The Joint Probability Density Function
4.3 Marginal Probability Functions
4.3.1 Marginal Probability Distribution Function
4.3.2 Marginal Probability Density Function
4.4 Conditional Probability and Statistical Independence
4.5 Functions of Random Variables
4.5.1 Y=aX+b, a>0, Continuous Random Variables
4.5.2 Y=aX+b, a<0, Continuous Random Variables
4.5.3 Y=aX+b, a>0, Discrete Random Variables
4.5.4 Y=X2
4.5.5 Sum of Two Statistically Independent Continuous Random Variables, Z=X+Y
4.5.6 Sum of Two Statistically Independent Discrete Random Variables, Z=X+Y
4.5.7 Z=XY
4.5.8 Z=XY
4.5.9 Central Limit Theorem
4.6 Problems
5 Statistical Analysis of Random Variables
5.1 Statistical Analysis of One Random Variable
5.1.1 Expected Value and Mean
5.1.2 Variance and Standard Deviation
5.2 Moment Generating Functions
5.2.1 Maclaurin Series
5.2.2 Characteristic Function
5.3 Statistical Analysis of Multiple Random Variables
5.3.1 Normalized Joint Moments
5.3.2 Joint Moments Around the Expected Values
5.3.3 Expected Operations of Functions of Random Variables
5.4 Problems
Part II Random Processes
6 Random Processes
6.1 Random Processes
6.1.1 Probability Functions Associated with a Random Process
6.1.2 Classification of Random Processes
6.2 Correlation Functions
6.2.1 Autocorrelation Function, RXX(t1,t2)
6.2.2 Cross-Correlation Function, RXY(t1,t2)
6.3 Covariance Functions
6.3.1 Autocovariance Function, CXX(t1,t2)
6.3.2 Cross-covariance Function, CXX(t1,t2)
6.4 Gaussian Random Process
6.5 Poisson Random Process
6.5.1 Autocorrelation Function of a Poisson Process
6.5.2 Covariance Function of a Poisson Process
6.6 Problems
7 Spectral Analysis of Random Processes
7.1 The Power Density Spectrum
7.1.1 Continuous Time Random Processes
7.2 Properties of the Power Density Spectrum
7.3 Discrete Time Random Processes
7.4 Problems
8 Linear Systems with Random Inputs
8.1 System Properties
8.1.1 Linear Systems
8.1.2 Time-Invariant Systems
8.1.3 Stable Systems
8.1.4 Causal Systems
8.1.5 Linear Time-Invariant Systems
8.2 Systems with Random Inputs
8.2.1 Statistical Analysis of Random Outputs
8.3 Power Density Spectrum of Random Outputs
8.4 Noisy Inputs
8.4.1 White Noise
8.5 Discrete Time Systems with Random Inputs
8.6 Problems
9 Random Samples
9.1 Random Sample Sequences
9.2 Random Sample Matrix
9.2.1 Correlation Matrices
9.2.2 Covariance Matrices
9.2.3 Sample Correlation Coefficient
9.3 Confidence Interval
9.4 Linear Transformation of Gaussian Random Variables
9.5 Problems
Appendix A Fourier Transform
A.1 Continuous Time Fourier Transform, CTFT
A.1.1 Properties of CTFT
A.1.1.1 Linearity
A.1.1.2 Shifting in Time
A.1.1.3 Shifting in Frequency
A.1.1.4 Scaling
A.1.1.5 Differentiation in Time Domain
A.1.1.6 Differentiation in Frequency Domain
A.1.1.7 Convolution in Time Domain
A.1.1.8 Multiplication in Time Domain or Convolution in Frequency Domain
A.2 Some Commonly Used CTFT Pairs
A.3 Discrete Time Fourier Transform, DTFT
A.3.1 Properties of DTFT
A.3.1.1 Linearity
A.3.1.2 Shifting in Time
A.3.1.3 Shifting in Frequency
A.3.1.4 Differentiation in Frequency Domain
A.3.1.5 Convolution Sum in Time Domain
A.3.1.6 Multiplication in Time Domain or Convolution in Frequency Domain
A.4 Some Commonly Used DTFT Pairs
Appendix B Probability Distribution Functions, Summary
B.1 Continuous Time Distributions
B.1.1 Uniform Distribution
B.1.2 Normal (Gaussian) Distribution
B.1.3 Exponential Distribution
B.1.4 Gamma Distribution
B.2 Discrete Time Distributions
B.2.1 Uniform Distribution
B.2.2 Bernoulli Distribution
B.2.3 Binomial Distribution
B.2.4 Geometric Distribution
B.2.5 Poisson Distribution
Appendix Bibliography
Index