### Table of content:

Cover

Half Title

Title Page

Copyright

Dedication

Contents

Chapter dependencies

Preface

1. Introduction to probability

1.1 Sample spaces, outcomes, and events

1.2 Review of set notation

1.3 Probability models

1.4 Axioms and properties of probability

1.5 Conditional probability

1.6 Independence

1.7 Combinatorics and probability

Notes

Problems

Exam preparation

2. Introduction to discrete random variables

2.1 Probabilities involving random variables

2.2 Discrete random variables

2.3 Multiple random variables

2.4 Expectation

Notes

Problems

Exam preparation

3. More about discrete random variables

3.1 Probability generating functions

3.2 The binomial random variable

3.3 The weak law of large numbers

3.4 Conditional probability

3.5 Conditional expectation

Notes

Problems

Exam preparation

4. Continuous random variables

4.1 Densities and probabilities

4.2 Expectation of a single random variable

4.3 Transform methods

4.4 Expectation of multiple random variables

4.5 Probability bounds

Notes

Problems

Exam preparation

5. Cumulative distribution functions and their applications

5.1 Continuous random variables

5.2 Discrete random variables

5.3 Mixed random variables

5.4 Functions of random variables and their cdfs

5.5 Properties of cdfs

5.6 The central limit theorem

5.7 Reliability

Notes

Problems

Exam preparation

6. Statistics

6.1 Parameter estimators and their properties

6.2 Histograms

6.3 Confidence intervals for the mean – known variance

6.4 Confidence intervals for the mean – unknown variance

6.5 Confidence intervals for Gaussian data

6.6 Hypothesis tests for the mean

6.7 Regression and curve fitting

6.8 Monte Carlo estimation

Notes

Problems

Exam preparation

7. Bivariate random variables

7.1 Joint and marginal probabilities

7.2 Jointly continuous random variables

7.3 Conditional probability and expectation

7.4 The bivariate normal

7.5 Extension to three or more random variables

Notes

Problems

Exam preparation

8. Introduction to random vectors

8.1 Review of matrix operations

8.2 Random vectors and random matrices

8.3 Transformations of random vectors

8.4 Linear estimation of random vectors (Wiener filters)

8.5 Estimation of covariance matrices

8.6 Nonlinear estimation of random vectors

Notes

Problems

Exam preparation

9. Gaussian random vectors

9.1 Introduction

9.2 Definition of the multivariate Gaussian

9.3 Characteristic function

9.4 Density function

9.5 Conditional expectation and conditional probability

9.6 Complex random variables and vectors

Notes

Problems

Exam preparation

10. Introduction to random processes

10.1 Definition and examples

10.2 Characterization of random processes

10.3 Strict-sense and wide-sense stationary processes

10.4 WSS processes through LTI systems

10.5 Power spectral densities for WSS processes

10.6 Characterization of correlation functions

10.7 The matched filter

10.8 The Wiener filter

10.9 The Wiener–Khinchin theorem

10.10 Mean-square ergodic theorem for WSS processes

10.11 Power spectral densities for non-WSS processes

Notes

Problems

Exam preparation

11. Advanced concepts in random processes

11.1 The Poisson process

11.2 Renewal processes

11.3 The Wiener process

11.4 Specification of random processes

Notes

Problems

Exam preparation

12. Introduction to Markov chains

12.1 Preliminary results

12.2 Discrete-time Markov chains

12.3 Recurrent and transient states

12.4 Limiting n-step transition probabilities

12.5 Continuous-time Markov chains

Notes

Problems

Exam preparation

13. Mean convergence and applications

13.1 Convergence in mean of order p

13.2 Normed vector spaces of random variables

13.3 The Karhunen–Loève expansion

13.4 The Wiener integral (again)

13.5 Projections, orthogonality principle, projection theorem

13.6 Conditional expectation and probability

13.7 The spectral representation

Notes

Problems

Exam preparation

14. Other modes of convergence

14.1 Convergence in probability

14.2 Convergence in distribution

14.3 Almost-sure convergence

Notes

Problems

Exam preparation

15. Self similarity and long-range dependence

15.1 Self similarity in continuous time

15.2 Self similarity in discrete time

15.3 Asymptotic second-order self similarity

15.4 Long-range dependence

15.5 ARMA processes

15.6 ARIMA processes

Problems

Exam preparation

Bibliography

Index

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