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Probability, Random Processes, and Statistical Analysis

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.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  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?
      1. 1.1.1 Communications, information, and control systems
      2. 1.1.2 Signal processing
      3. 1.1.3 Machine learning
      4. 1.1.4 Biostatistics, bioinformatics, and related fields
      5. 1.1.5 Econometrics and mathematical finance
      6. 1.1.6 Queueing and loss systems
      7. 1.1.7 Other application domains
    2. 1.2 History and overview
      1. 1.2.1 Classical probability theory
      2. 1.2.2 Modern probability theory
      3. 1.2.3 Random processes
      4. 1.2.4 Statistical analysis and inference
    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
        1. 2.1.1 Repeated experiments and statistical regularity
        2. 2.1.2 Random experiments and relative frequencies
      2. 2.2 Axioms of probability
        1. 2.2.1 Sample space
        2. 2.2.2 Event
        3. 2.2.3 Probability measure
        4. 2.2.4 Properties of probability measure
      3. 2.3 Bernoulli trials and Bernoulli’s theorem
      4. 2.4 Conditional probability, Bayes’ theorem, and statistical independence
        1. 2.4.1 Joint probability and conditional probability
        2. 2.4.2 Bayes’ theorem
        3. 2.4.3 Statistical independence of events
      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
        1. 3.1.1 Distribution function
        2. 3.1.2 Two random variables and joint distribution function
      2. 3.2 Discrete random variables and probability distributions
        1. 3.2.1 Joint and conditional probability distributions
        2. 3.2.2 Moments, central moments, and variance
        3. 3.2.3 Covariance and correlation coefficient
      3. 3.3 Important probability distributions
        1. 3.3.1 Bernoulli distribution and binomial distribution
        2. 3.3.2 Geometric distribution
        3. 3.3.3 Poisson distribution
        4. 3.3.4 Negative binomial (or Pascal) distribution
        5. 3.3.5 Zipf’s law and zeta distribution
      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
        1. 4.1.1 Distribution function and probability density function
        2. 4.1.2 Expectation, moments, central moments, and variance
      2. 4.2 Important continuous random variables and their distributions
        1. 4.2.1 Uniform distribution
        2. 4.2.2 Exponential distribution
        3. 4.2.3 Gamma distribution
        4. 4.2.4 Normal (or Gaussian) distribution
        5. 4.2.5 Weibull distributions
        6. 4.2.6 Pareto distribution
      3. 4.3 Joint and conditional probability density functions
        1. 4.3.1 Bivariate normal (or Gaussian) distribution
        2. 4.3.2 Multivariate normal (or Gaussian) distribution
      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
        1. 5.4.1 Random number generator (RNG)
        2. 5.4.2 Generation of variates from general distributions
        3. 5.4.3 Generation of normal (or Gaussian) variates
      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
        1. 6.3.1 Histogram on probability paper
        2. 6.3.2 Log-survivor function curve
        3. 6.3.3 Hazard function and mean residual life curves
        4. 6.3.4 Dot diagram and correlation coefficient
      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
        1. 7.5.1 Rayleigh distribution
        2. 7.5.2 Rice distribution
      6. 7.6 Complex-valued normal variables
        1. 7.6.1 Complex-valued Gaussian variables and their properties
        2. 7.6.2 Multivariate Gaussian 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)
        1. 8.1.1 Moment-generating function of one random variable
        2. 8.1.2 Moment-generating function of sum of independent random variables
        3. 8.1.3 Joint moment-generating function of multivariate random variables
      2. 8.2 Characteristic function (CF)
        1. 8.2.1 Characteristic function of one random variable
        2. 8.2.2 Sum of independent random variables and convolution
        3. 8.2.3 Moment generation from characteristic function
        4. 8.2.4 Joint characteristic function of multivariate random variables
        5. 8.2.5 Application of the characteristic function: the central limit theorem (CLT)
        6. 8.2.6 Characteristic function of multivariate complex-valued normal variables
      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
        1. 9.1.1 Probability-generating function (PGF)
        2. 9.1.2 Sum of independent variables and convolutions
        3. 9.1.3 Sum of a random number of random variables
        4. 9.1.4 Inverse transform of generating functions
      2. 9.2 Laplace transform method
        1. 9.2.1 Laplace transform and moment generation
        2. 9.2.2 Inverse Laplace transform
      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
        1. 10.1.1 Cauchy–Schwarz inequality
        2. 10.1.2 Jensen’s inequality
        3. 10.1.3 Shannon’s lemma and log-sum inequality
        4. 10.1.4 Markov’s inequality
        5. 10.1.5 Chebyshev’s inequality
        6. 10.1.6 Kolmogorov’s inequalities for martingales and submartingales
      2. 10.2 Chernoff’s bounds
        1. 10.2.1 Chernoff’s bound for a single random variable
        2. 10.2.2 Chernoff’s bound for a sum of i.i.d. random variables
      3. 10.3 Large deviation theory
        1. 10.3.1 Large deviation approximation
        2. 10.3.2 Large deviation rate function
      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
        1. 11.1.1 Sequence of numbers
        2. 11.1.2 Sequence of functions
      2. 11.2 Types of convergence for sequences of random variables
        1. 11.2.1 Convergence in distribution
        2. 11.2.2 Convergence in probability
        3. 11.2.3 Almost sure convergence
        4. 11.2.4 Convergence in the rth mean
        5. 11.2.5 Relations between the modes of convergence
      3. 11.3 Limit theorems
        1. 11.3.1 Infinite sequence of events
        2. 11.3.2 Weak law of large numbers (WLLN)
        3. 11.3.3 Strong laws of large numbers (SLLN)
        4. 11.3.4 The central limit theorem (CLT) revisited
      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
        1. 12.2.1 Discrete-time versus continuous-time processes
        2. 12.2.2 Discrete-state versus continuous-state processes
        3. 12.2.3 Stationary versus nonstationary processes
        4. 12.2.4 Independent versus dependent processes
        5. 12.2.5 Markov chains and Markov processes
        6. 12.2.6 Point processes and renewal processes
        7. 12.2.7 Real-valued versus complex-valued processes
        8. 12.2.8 One-dimensional versus vector processes
      3. 12.3 Stationary random process
        1. 12.3.1 Strict stationarity versus wide-sense stationarity
        2. 12.3.2 Gaussian process
        3. 12.3.3 Ergodic processes and ergodic theorems
      4. 12.4 Complex-valued Gaussian process
        1. 12.4.1 Complex-valued Gaussian random variables
        2. 12.4.2 Complex-valued Gaussian process
        3. 12.4.3 Hilbert transform and analytic signal
        4. 12.4.4 Complex envelope
      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
        1. 13.1.1 Fourier series
        2. 13.1.2 Fourier transform
        3. 13.1.3 Analysis of periodic wide-sense stationary random process
        4. 13.1.4 Power spectrum
        5. 13.1.5 Power spectrum and periodogram of time series
      2. 13.2 Generalized Fourier series expansions
        1. 13.2.1 Review of matrix analysis
        2. 13.2.2 Karhunen-Loève expansion and its applications
      3. 13.3 Principal component analysis and singular value decomposition
        1. 13.3.1 Principal component analysis (PCA)
        2. 13.3.2 Singular value decomposition (SVD)
        3. 13.3.3 Matrix decomposition methods for Web information retrieval
      4. 13.4 Autoregressive moving average time series and its spectrum
        1. 13.4.1 Autoregressive (AR) time series
        2. 13.4.2 Moving average (MA) time series
        3. 13.4.3 Autoregressive moving average (ARMA) time series
        4. 13.4.4 Autoregressive integrated moving average (ARIMA)time series
      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
        1. 14.1.1 Derivation of the Poisson process
        2. 14.1.2 Properties of the Poisson process
      2. 14.2 Birth-death (BD) process
      3. 14.3 Renewal process
        1. 14.3.1 Renewal function and renewal equation
        2. 14.3.2 Residual life in a 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
        1. 15.2.1 Generating function method
        2. 15.2.2 Spectral expansion method
      3. 15.3 Classification of states
        1. 15.3.1 Recurrent and transient states
        2. 15.3.2 Criteria for state classification
        3. 15.3.3 Communicating states and an irreducible Markov chain
        4. 15.3.4 Stationary distribution of an aperiodic irreducible chain
      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)
        1. 16.2.1 Infinitesimal generator and embedded Markov chain of a continuous-time Markov chain
        2. 16.2.2 Kolmogorov’s forward and backward equations
        3. 16.2.3 Spectral expansion of the infinitesimal generator
      3. 16.3 Reversible Markov chains
        1. 16.3.1 Reversible discrete-time Markov chain
        2. 16.3.2 Reversible continuous-time Markov chain
      4. 16.4 An application: phylogenetic tree and its Markov chain representation
        1. 16.4.1 Trees and phylogenetic trees
        2. 16.4.2 Markov process on a tree
        3. 16.4.3 Continuous-time Markov chain model
      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
        1. 17.1.1 Simple random walk
        2. 17.1.2 Gambler’s ruin
      2. 17.2 Brownian motion or Wiener process
        1. 17.2.1 Wiener process as a limit process
        2. 17.2.2 Properties of the Wiener process
        3. 17.2.3 White noise
      3. 17.3 Diffusion processes and diffusion equations
        1. 17.3.1 Fokker-Planck equation for Brownian motion with drift
        2. 17.3.2 Einstein’s diffusion equation for Brownian motion
        3. 17.3.3 Forward and backward equations for general diffusion processes
        4. 17.3.4 Ornstein–Uhlenbeck process: Gauss–Markov process
      4. 17.4 Stochastic differential equations and Itô process
        1. 17.4.1 Itô’s formula
        2. 17.4.2 Geometric Brownian motion (GBM)
        3. 17.4.3 The Black–Scholes model: an application of an 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
        1. 18.1.1 Exponential family of distributions revisited
        2. 18.1.2 Maximum-likelihood estimation
        3. 18.1.3 Cramér-Rao lower bound
        4. 18.1.4 Interval or region estimation
      2. 18.2 Hypothesis testing and statistical decision theory
        1. 18.2.1 Hypothesis testing
        2. 18.2.2 Neyman-Pearson criterion and likelihood ratio test
        3. 18.2.3 Receiver operating characteristic (ROC)
        4. 18.2.4 Receiver operating characteristic application example: radar signal detection
      3. 18.3 Bayesian estimation and decision theory
        1. 18.3.1 The Bayes risk
        2. 18.3.2 The conditional risk
        3. 18.3.3 Maximum a posteriori probability (MAP) estimation
      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
        1. 19.1.1 Method of moments
        2. 19.1.2 Minimum chi-square estimation method
        3. 19.1.3 Minimum Kullback-Leibler divergence method
        4. 19.1.4 Newton-Raphson algorithm for maximum-likelihood estimation
      2. 19.2 Expectation-maximization algorithm for maximum-likelihood estimation
        1. 19.2.1 Expectation-maximization algorithm for transformed data
        2. 19.2.2 Expectation-maximization algorithm for missing data
      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
        1. 20.2.1 Discrete-time Markov chain and hidden Markov model
        2. 20.2.2 Examples of hidden Markov models
      3. 20.3 Evaluation of a hidden Markov model
        1. 20.3.1 Evaluation of a likelihood function
        2. 20.3.2 Forward recursion algorithm
        3. 20.3.3 Backward algorithm and forward-backward algorithm
      4. 20.4 Estimation algorithms for state sequence
        1. 20.4.1 Forward algorithm for maximum a posteriori probability state sequence estimation
        2. 20.4.2 The Viterbi algorithm
      5. 20.5 The BCJR algorithm
      6. 20.6 Maximum-likelihood estimation of model parameters
        1. 20.6.1 Forward-backward algorithm for a transition-based hidden Markov model
        2. 20.6.2 The Baum-Welch algorithm
      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
        1. 21.3.1 K-means algorithm
        2. 21.3.2 EM algorithm for clustering
        3. 21.3.3 The k-nearest neighbor rule
      4. 21.4 Sequence recognition
        1. 21.4.1 Speech recognition
        2. 21.4.2 Biological sequence analysis
      5. 21.5 Bayesian networks
      6. 21.6 Factor graphs
        1. 21.6.1 Sum-product algorithm
        2. 21.6.2 Applications of factor graphs to Bayesian networks
      7. 21.7 Markov chain Monte Carlo (MCMC) methods
        1. 21.7.1 Metropolis-Hastings algorithm
        2. 21.7.2 Metropolis-Hastings algorithm for continuous variables
        3. 21.7.3 Block Metropolis–Hastings algorithm
        4. 21.7.4 The Gibbs sampler
        5. 21.7.5 Simulated annealing
      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
        1. 22.1.1 Minimum mean square error estimation
        2. 22.1.2 Linear minimum mean square error estimation
        3. 22.1.3 Conditional expectation and minimum mean square error estimation
        4. 22.1.4 Regression analysis
      2. 22.2 Linear smoothing and prediction: Wiener filter theory
        1. 22.2.1 Linear time-invariant system with random input
        2. 22.2.2 Optimal smoothing and prediction of a stationary process
        3. 22.2.3 Noncausal (or physically unrealizable) filter
        4. 22.2.4 Prediction of a random signal in the absence of noise
        5. 22.2.5 Prediction of a random signal in the presence of noise
      3. 22.3 Kalman filter
        1. 22.3.1 State space model
        2. 22.3.2 Derivation of the 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
        1. 23.3.1 M/M/1: the simplest queueing model
        2. 23.3.2 M/M/∞ and M/G/∞: infinite servers
        3. 23.3.3 M/M/m: multiple server queue
        4. 23.3.4 M(K)/M/m and G(K)/M/m: multiple access models
        5. 23.3.5 M/G/1 queueing model
        6. 23.3.6 M/G/1 with processor sharing (PS)
      4. 23.4 Loss models
        1. 23.4.1 M/M/m(0) and M/G/m(0): Erlang loss models
        2. 23.4.2 M(K)/M/m(0) and G(K)/G/m(0): Engset loss models
        3. 23.4.3 Loss network model and generalized loss station
      5. 23.5 Summary of Chapter 23
      6. 23.6 Discussion and further reading
      7. 23.7 Problems
  16. References
  17. Index