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CONTENTS-

1 Introduction


2 Probability
2.1 Introduction
2.2 Spinning pointers and flipping coins
2.3 Probability spaces
2.4 Discrete probability spaces
2.5 Continuous probability spaces
2.6 Independence
2.7 Elementary conditional probability
2.8 Problems


3 Random variables, vectors, and processes
3.1 Introduction
3.2 Random variables
3.3 Distributions of random variables
3.4 Random vectors and random processes
3.5 Distributions of random vectors
3.6 Independent random variables
3.7 Conditional distributions
3.8 Statistical detection and classification
3.9 Additive noise
3.10 Binary detection in Gaussian noise
3.11 Statistical estimation
3.12 Characteristic functions
3.13 Gaussian random vectors
3.14 Simple random processes
3.15 Directly given random processes
3.16 Discrete time Markov processes
3.17 ⋆Nonelementary conditional probability
3.18 Problems


4 Expectation and averages
4.1 Averages
4.2 Expectation
4.3 Functions of random variables
4.4 Functions of several random variables
4.5 Properties of expectation
4.6 Examples
4.7 Conditional expectation
4.8 ⋆Jointly Gaussian vectors
4.9 Expectation as estimation
4.10 ⋆Implications for linear estimation
4.11 Correlation and linear estimation
4.12 Correlation and covariance functions
4.13 ⋆The central limit theorem
4.14 Sample averages
4.15 Convergence of random variables
4.16 Weak law of large numbers
4.17 ⋆Strong law of large numbers
4.18 Stationarity
4.19 Asymptotically uncorrelated processes
4.20 Problems


5 Second-order theory
5.1 Linear filtering of random processes
5.2 Linear systems I/O relations
5.3 Power spectral densities
5.4 Linearly filtered uncorrelated processes
5.5 Linear modulation
5.6 White noise
5.7 ⋆Time averages
5.8 ⋆Mean square calculus
5.9 ⋆Linear estimation and filtering
5.10 Problems


6 A menagerie of processes
6.1 Discrete time linear models
6.2 Sums of iid random variables
6.3 Independent stationary increment processes
6.4 ⋆Second-order moments of isi processes
6.5 Specification of continuous time isi processes
6.6 Moving-average and autoregressive processes
6.7 The discrete time Gauss–Markov process
6.8 Gaussian random processes
6.9 The Poisson counting process
6.10 Compound processes
6.11 Composite random processes
6.12 ⋆Exponential modulation
6.13 ⋆Thermal noise
6.14 Ergodicity
6.15 Random fields
6.16 Problems


Appendix A Preliminaries
A.1 Set theory
A.2 Examples of proofs
A.3 Mappings and functions
A.4 Linear algebra
A.5 Linear system fundamentals
A.6 Problems


Appendix B Sums and integrals
B.1 Summation
B.2 ⋆Double sums
B.3 Integration
B.4 ⋆The Lebesgue integral


Appendix C Common univariate distributions


Appendix D Supplementary reading


References


Index