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## Book Description

Together with the fundamentals of probability, random processes and statistical analysis, this insightful book also presents a broad range of advanced topics and applications. There is extensive coverage of Bayesian vs. frequentist statistics, time series and spectral representation, inequalities, bound and approximation, maximum-likelihood estimation and the expectation-maximization (EM) algorithm, geometric Brownian motion and Itv process. Applications such as hidden Markov models (HMM), the Viterbi, BCJR, and BaumWelch algorithms, algorithms for machine learning, Wiener and Kalman filters, and queueing and loss networks are treated in detail. The book will be useful to students and researchers in such areas as communications, signal processing, networks, machine learning, bioinformatics, econometrics and mathematical finance. With a solutions manual, lecture slides, supplementary materials and MATLAB programs all available online, it is ideal for classroom teaching as well as a valuable reference for professionals.

1. Cover
2. Half Title
3. Title Page
5. Dedication
6. Contents
7. List of abbreviations and acronyms
8. Preface
9. Acknowledgments
10. 1. Introduction
1. 1.1 Why study probability, random processes, and statistical analysis?
2. 1.2 History and overview
3. 1.3 Discussion and further reading
11. Part I: Probability, random variables, and statistics
1. 2. Probability
1. 2.1 Randomness in the real world
2. 2.2 Axioms of probability
3. 2.3 Bernoulli trials and Bernoulli’s theorem
4. 2.4 Conditional probability, Bayes’ theorem, and statistical independence
5. 2.5 Summary of Chapter 2
6. 2.6 Discussion and further reading
7. 2.7 Problems
2. 3. Discrete random variables
1. 3.1 Random variables
2. 3.2 Discrete random variables and probability distributions
3. 3.3 Important probability distributions
4. 3.4 Summary of Chapter 3
5. 3.5 Discussion and further reading
6. 3.6 Problems
3. 4. Continuous random variables
1. 4.1 Continuous random variables
2. 4.2 Important continuous random variables and their distributions
3. 4.3 Joint and conditional probability density functions
4. 4.4 Exponential family of distributions
5. 4.5 Bayesian inference and conjugate priors
6. 4.6 Summary of Chapter 4
7. 4.7 Discussion and further reading
8. 4.8 Problems
4. 5. Functions of random variables and their distributions
1. 5.1 Function of one random variable
2. 5.2 Function of two random variables
3. 5.3 Two functions of two random variables and the Jacobian matrix
4. 5.4 Generation of random variates for Monte Carlo simulation
5. 5.5 Summary of Chapter 5
6. 5.6 Discussion and further reading
7. 5.7 Problems
5. 6. Fundamentals of statistical data analysis
1. 6.1 Sample mean and sample variance
2. 6.2 Relative frequency and histograms
3. 6.3 Graphical presentations
4. 6.4 Summary of Chapter 6
5. 6.5 Discussion and further reading
6. 6.6 Problems
6. 7. Distributions derived from the normal distribution
1. 7.1 Chi-squared distribution
2. 7.2 Student’s t-distribution
3. 7.3 Fisher’s F-distribution
4. 7.4 Log-normal distribution
5. 7.5 Rayleigh and Rice distributions
6. 7.6 Complex-valued normal variables
7. 7.7 Summary of Chapter 7
8. 7.8 Discussion and further reading
9. 7.9 Problems
12. Part II: Transform methods, bounds, and limits
1. 8. Moment-generating function and characteristic function
1. 8.1 Moment-generating function (MGF)
2. 8.2 Characteristic function (CF)
3. 8.3 Summary of Chapter 8
4. 8.4 Discussion and further reading
5. 8.5 Problems
2. 9. Generating functions and Laplace transform
1. 9.1 Generating function
2. 9.2 Laplace transform method
3. 9.3 Summary of Chapter 9
4. 9.4 Discussion and further reading
5. 9.5 Problems
3. 10. Inequalities, bounds, and large deviation approximation
1. 10.1 Inequalities frequently used in probability theory
2. 10.2 Chernoff’s bounds
3. 10.3 Large deviation theory
4. 10.4 Summary of Chapter 10
5. 10.5 Discussion and further reading
6. 10.6 Problems
4. 11. Convergence of a sequence of random variables and the limit theorems
1. 11.1 Preliminaries: convergence of a sequence of numbers or functions
2. 11.2 Types of convergence for sequences of random variables
3. 11.3 Limit theorems
4. 11.4 Summary of Chapter 11
5. 11.5 Discussion and further reading
6. 11.6 Problems
13. Part III: Random processes
1. 12. Random processes
1. 12.1 Random process
2. 12.2 Classification of random processes
3. 12.3 Stationary random process
4. 12.4 Complex-valued Gaussian process
5. 12.5 Summary of Chapter 12
6. 12.6 Discussion and further reading
7. 12.7 Problems
2. 13. Spectral representation of random processes and time series
1. 13.1 Spectral representation of random processes and time series
2. 13.2 Generalized Fourier series expansions
3. 13.3 Principal component analysis and singular value decomposition
4. 13.4 Autoregressive moving average time series and its spectrum
5. 13.5 Summary of Chapter 13
6. 13.6 Discussion and further reading
7. 13.7 Problems
3. 14. Poisson process, birth-death process, and renewal process
1. 14.1 The Poisson process
2. 14.2 Birth-death (BD) process
3. 14.3 Renewal process
4. 14.4 Summary of Chapter 14
5. 14.5 Discussion and further reading
6. 14.6 Problems
4. 15. Discrete-time Markov chains
1. 15.1 Markov processes and Markov chains
2. 15.2 Computation of state probabilities
3. 15.3 Classification of states
4. 15.4 Summary of Chapter 15
5. 15.5 Discussion and further reading
6. 15.6 Problems
5. 16. Semi-Markov processes and continuous-time Markov chains
1. 16.1 Semi-Markov process
2. 16.2 Continuous-time Markov chain (CTMC)
3. 16.3 Reversible Markov chains
4. 16.4 An application: phylogenetic tree and its Markov chain representation
5. 16.5 Summary of Chapter 16
6. 16.6 Discussion and further reading
7. 16.7 Problems
6. 17. Random walk, Brownian motion, diffusion, and Itô processes
1. 17.1 Random walk
2. 17.2 Brownian motion or Wiener process
3. 17.3 Diffusion processes and diffusion equations
4. 17.4 Stochastic differential equations and Itô process
5. 17.5 Summary of Chapter 17
6. 17.6 Discussion and further reading
7. 17.7 Problems
14. Part IV: Statistical inference
1. 18. Estimation and decision theory
1. 18.1 Parameter estimation
2. 18.2 Hypothesis testing and statistical decision theory
3. 18.3 Bayesian estimation and decision theory
4. 18.4 Summary of Chapter 18
5. 18.5 Discussion and further reading
6. 18.6 Problems
2. 19 Estimation algorithms
1. 19.1 Classical numerical methods for estimation
2. 19.2 Expectation-maximization algorithm for maximum-likelihood estimation
3. 19.3 Summary of Chapter 19
4. 19.4 Discussion and further reading
5. 19.5 Problems
15. Part V: Applications and advanced topics
1. 20. Hidden Markov models and applications
1. 20.1 Introduction
2. 20.2 Formulation of a hidden Markov model
3. 20.3 Evaluation of a hidden Markov model
4. 20.4 Estimation algorithms for state sequence
5. 20.5 The BCJR algorithm
6. 20.6 Maximum-likelihood estimation of model parameters
7. 20.7 Application: parameter estimation of mixture distributions
8. 20.8 Summary of Chapter 20
9. 20.9 Discussion and further reading
10. 20.10 Problems
2. 21. Probabilistic models in machine learning
1. 21.1 Introduction
2. 21.2 Maximum a posteriori probability (MAP) recognition
3. 21.3 Clustering
4. 21.4 Sequence recognition
5. 21.5 Bayesian networks
6. 21.6 Factor graphs
7. 21.7 Markov chain Monte Carlo (MCMC) methods
8. 21.8 Summary of Chapter 21
9. 21.9 Discussion and further reading
10. 21.10 Problems
3. 22. Filtering and prediction of random processes
1. 22.1 Conditional expectation, minimum mean square error estimation and regression analysis
2. 22.2 Linear smoothing and prediction: Wiener filter theory
3. 22.3 Kalman filter
4. 22.4 Summary of Chapter 22
5. 22.5 Discussion and further reading
6. 22.6 Problems
4. 23. Queueing and loss models
1. 23.1 Introduction
2. 23.2 Little’s formula: L = λ W
3. 23.3 Queueing models
4. 23.4 Loss models
5. 23.5 Summary of Chapter 23
6. 23.6 Discussion and further reading
7. 23.7 Problems
16. References
17. Index