Probability, Random Processes, and Statistical Analysis: Applications to Communications, Signal Processing, Queueing Theory and Mathematical Finance

Contents......Page 8
Abbreviations and Acronyms......Page 19
Organization of the book......Page 24
Suggested course plans......Page 28
Matlab exercises and programs......Page 30
Acknowledgments......Page 31
1.1 Why study probability, random processes, and statistical analysis?......Page 34
1.1.2 Signal processing......Page 35
1.1.3 Machine learning......Page 36
1.1.5 Econometrics and mathematical finance......Page 37
1.1.7 Other application domains......Page 39
1.2.1 Classical probability theory......Page 40
1.2.2 Modern probability theory......Page 42
1.2.3.1 Poisson process to Markov process......Page 43
1.2.3.2 Brownian motion to Itô process......Page 44
1.2.4 Statistical analysis and inference......Page 45
1.2.4.1 Frequentist statistics versus Bayesian statistics......Page 46
1.3 Discussion and further reading......Page 47
Part I Probability, random variables, and statistics......Page 48
2.1.1 Repeated experiments and statistical regularity......Page 50
2.2 Axioms of probability......Page 51
2.2.2 Event......Page 52
2.2.3 Probability measure......Page 53
2.2.4 Properties of probability measure......Page 54
2.3 Bernoulli trials and Bernoulli's theorem......Page 59
2.4.1 Joint probability and conditional probability......Page 63
2.4.2 Bayes' theorem......Page 64
2.4.2.1 Frequentist probabilities and Bayesian probabilities......Page 66
2.4.3 Statistical independence of events......Page 67
2.5 Summary of Chapter 2......Page 69
2.7 Problems......Page 70
3.1 Random variables......Page 75
3.1.1 Distribution function......Page 76
3.1.2 Two random variables and joint distribution function......Page 77
3.2 Discrete random variables and probability distributions......Page 78
3.2.1 Joint and conditional probability distributions......Page 79
3.2.2 Moments, central moments, and variance......Page 83
3.2.3 Covariance and correlation coefficient......Page 84
3.3 Important probability distributions......Page 86
3.3.1 Bernoulli distribution and binomial distribution......Page 87
3.3.2 Geometric distribution......Page 88
3.3.3 Poisson distribution......Page 89
3.3.4 Negative binomial (or Pascal) distribution......Page 92
3.3.4.1 Shifted negative binomial distribution......Page 94
3.3.5 Zipf's law and zeta distribution......Page 95
3.3.5.1 Euler and Riemann zeta functions......Page 97
3.4 Summary of Chapter 3......Page 98
3.6 Problems......Page 99
4.1.1 Distribution function and probability density function......Page 105
4.1.2 Expectation, moments, central moments, and variance......Page 106
4.2.1 Uniform distribution......Page 108
4.2.2 Exponential distribution......Page 109
4.2.3 Gamma distribution......Page 111
4.2.4 Normal (or Gaussian) distribution......Page 113
4.2.4.1 Moments of the unit normal distribution......Page 115
4.2.4.2 The normal approximation to the binomial distribution and the DeMoivre–Laplace limit theorem4......Page 116
4.2.5 Weibull distributions......Page 119
4.2.6 Pareto distribution......Page 121
4.3 Joint and conditional probability density functions......Page 123
4.3.1 Bivariate normal (or Gaussian) distribution......Page 125
4.3.2 Multivariate normal (or Gaussian) distribution......Page 127
4.4 Exponential family of distributions......Page 128
4.5 Bayesian inference and conjugate priors......Page 130
4.6 Summary of Chapter 4......Page 136
4.8 Problems......Page 137
5.1 Function of one random variable......Page 145
5.2 Function of two random variables......Page 148
5.3 Two functions of two random variables and the Jacobian matrix......Page 152
5.4 Generation of random variates for Monte Carlo simulation3......Page 156
5.4.1 Random number generator (RNG)......Page 157
5.4.2.1 Transform method......Page 158
5.4.2.2 Acceptance–rejection method......Page 159
5.4.2.3 Composition methods......Page 162
5.4.3.2 Box–Muller method......Page 163
5.6 Discussion and further reading......Page 164
5.7 Problems......Page 165
6.1 Sample mean and sample variance......Page 171
6.2 Relative frequency and histograms......Page 173
6.3 Graphical presentations......Page 174
6.3.1.1 Testing the normal distribution hypothesis......Page 175
6.3.1.2 Testing the log-normal distribution hypothesis......Page 176
6.3.2 Log-survivor function curve......Page 177
6.3.2.1 Testing the Pareto distribution hypothesis......Page 179
6.3.3 Hazard function and mean residual life curves......Page 181
6.3.4 Dot diagram and correlation coefficient......Page 182
6.5 Discussion and further reading......Page 185
6.6 Problems......Page 186
7.1 Chi-squared distribution......Page 190
7.2 Student's t-distribution......Page 194
7.3 Fisher's F-distribution......Page 196
7.4 Log-normal distribution......Page 198
7.5 Rayleigh and Rice distributions......Page 200
7.5.1 Rayleigh distribution......Page 201
7.5.1.1 Rayleigh distribution and the Weibull distribution......Page 202
7.5.2 Rice distribution......Page 203
7.6.1 Complex-valued Gaussian variables and their properties......Page 205
7.6.2 Multivariate Gaussian variables......Page 206
7.7 Summary of Chapter 7......Page 209
7.8 Discussion and further reading......Page 210
7.9 Problems......Page 211
Part II Transform methods, bounds, and limits......Page 216
8.1.1 Moment-generating function of one random variable......Page 218
8.1.2 Moment-generating function of sum of independent random variables......Page 222
8.1.3 Joint moment-generating function of multivariate random variables......Page 223
8.2.1 Characteristic function of one random variable......Page 225
8.2.2 Sum of independent random variables and convolution......Page 229
8.2.3 Moment generation from characteristic function......Page 231
8.2.4 Joint characteristic function of multivariate random variables......Page 232
8.2.5 Application of the characteristic function: the central limit theorem (CLT)......Page 234
8.2.6 Characteristic function of multivariate complex-valued normal variables......Page 235
8.3 Summary of Chapter 8......Page 237
8.4 Discussion and further reading......Page 238
8.5 Problems......Page 239
9.1 Generating function......Page 244
9.1.1 Probability-generating function (PGF)......Page 245
9.1.1.1 Generating function of the complementary distribution......Page 246
9.1.1.2 Expectation and factorial moments......Page 247
9.1.2 Sum of independent variables and convolutions......Page 248
9.1.3 Sum of a random number of random variables......Page 250
9.1.4 Inverse transform of generating functions......Page 251
9.1.4.1 Partial-fraction expansion method......Page 253
9.1.4.2 Asymptotic formula in partial-fraction expansion......Page 257
9.1.4.3 Recursion method......Page 258
9.2.1 Laplace transform and moment generation......Page 259
9.2.2 Inverse Laplace transform......Page 262
9.3 Summary of Chapter 9......Page 267
9.5 Problems......Page 268
10.1.1 Cauchy–Schwarz inequality......Page 274
10.1.2 Jensen's inequality......Page 278
10.1.3 Shannon's lemma and log-sum inequality......Page 279
10.1.4 Markov's inequality......Page 281
10.1.5 Chebyshev's inequality......Page 282
10.1.6 Kolmogorov's inequalities for martingales and submartingales......Page 283
10.2.1 Chernoff's bound for a single random variable......Page 286
10.2.2 Chernoff's bound for a sum of i.i.d. random variables......Page 287
10.2.2.1 The role of entropy function......Page 289
10.3.1 Large deviation approximation......Page 290
10.3.2 Large deviation rate function......Page 296
10.4 Summary of Chapter 10......Page 300
10.5 Discussion and further reading......Page 301
10.6 Problems......Page 302
11.1.1 Sequence of numbers......Page 310
11.1.2 Sequence of functions......Page 311
11.2.1 Convergence in distribution......Page 313
11.2.2 Convergence in probability......Page 315
11.2.3 Almost sure convergence......Page 318
11.2.4 Convergence in the rth mean......Page 321
11.2.5 Relations between the modes of convergence......Page 325
11.3 Limit theorems......Page 326
11.3.1 Infinite sequence of events......Page 327
11.3.2 Weak law of large numbers (WLLN)......Page 331
11.3.3 Strong laws of large numbers (SLLN)......Page 333
11.3.4 The central limit theorem (CLT) revisited......Page 336
11.4 Summary of Chapter 11......Page 339
11.5 Discussion and further reading......Page 340
11.6 Problems......Page 341
Part III Random processes......Page 346
12.1 Random process......Page 348
12.2.1 Discrete-time versus continuous-time processes......Page 349
12.2.3 Stationary versus nonstationary processes......Page 350
12.2.5.1 Discrete-time Markov chain (DTMC)......Page 351
12.2.5.2 Continuous-time Markov chain (CTMC)......Page 352
12.2.5.4 Discrete-Time, Continuous-Space (DTCS) Markov Process......Page 353
12.2.7 Real-valued versus complex-valued processes......Page 354
12.3 Stationary random process......Page 355
12.3.1 Strict stationarity versus wide-sense stationarity......Page 356
12.3.2 Gaussian process......Page 359
12.3.3 Ergodic processes and ergodic theorems......Page 360
12.4.1 Complex-valued Gaussian random variables......Page 362
12.4.2 Complex-valued Gaussian process......Page 363
12.4.2.2 Wide-sense stationarity and strict-sense stationarity of complex Gaussian processes......Page 365
12.4.3 Hilbert transform and analytic signal......Page 366
12.4.4 Complex envelope......Page 371
12.5 Summary of Chapter 12......Page 372
12.7 Problems......Page 373
13.1.1 Fourier series......Page 376
13.1.2 Fourier transform......Page 378
13.1.3 Analysis of periodic wide-sense stationary random process......Page 379
13.1.4 Power spectrum......Page 381
13.1.5 Power spectrum and periodogram of time series......Page 383
13.1.5.1 Power spectrum and the Wiener–Khinchin formula......Page 384
13.1.5.2 Periodogram of a time series......Page 385
13.2.1 Review of matrix analysis......Page 390
13.2.2 Karhunen–Loève expansion and its applications......Page 396
13.3.1.1 Correlation across the m variates......Page 405
13.3.1.3 Correlation across n data points......Page 410
13.3.2 Singular value decomposition (SVD)......Page 413
13.3.2.1 Computation of pseudo-inverse......Page 416
13.3.3 Matrix decomposition methods for Web information retrieval......Page 417
13.4.1 Autoregressive (AR) time series......Page 418
13.4.2 Moving average (MA) time series......Page 422
13.4.3 Autoregressive moving average (ARMA) time series......Page 423
13.4.4 Autoregressive integrated moving average (ARIMA) time series......Page 425
13.5 Summary of Chapter 13......Page 426
13.6 Discussion and further reading......Page 427
13.7 Problems......Page 428
14.1.1 Derivation of the Poisson process......Page 433
14.1.2.1 Memoryless property of Poisson process......Page 436
14.1.2.3 Decomposition of a Poisson process......Page 437
14.1.2.4 Uniformity of Poisson arrivals......Page 438
14.1.2.5 Infinitesimal generator of Poisson process......Page 439
14.2 Birth–death (BD) process......Page 440
14.3 Renewal process......Page 443
14.3.1 Renewal function and renewal equation......Page 444
14.3.2 Residual life in a renewal process......Page 446
14.4 Summary of Chapter 14......Page 450
14.5 Discussion and further reading......Page 451
14.6 Problems......Page 452
15.1 Markov processes and Markov chains......Page 457
15.2 Computation of state probabilities......Page 462
15.2.1 Generating function method......Page 463
15.2.2 Spectral expansion method......Page 466
15.2.2.1 Multiplicity of eigenvalues......Page 469
15.3 Classification of states......Page 471
15.3.1 Recurrent and transient states......Page 472
15.3.2 Criteria for state classification......Page 473
15.3.3 Communicating states and an irreducible Markov chain......Page 478
15.3.4 Stationary distribution of an aperiodic irreducible chain......Page 481
15.4 Summary of Chapter 15......Page 483
15.5 Discussion and further reading......Page 484
15.6 Problems......Page 485
16.1 Semi-Markov process......Page 488
16.2.1 Infinitesimal generator and embedded Markov chain of a continuous-time Markov chain......Page 492
16.2.2 Kolmogorov's forward and backward equations......Page 495
16.2.3 Spectral expansion of the infinitesimal generator......Page 497
16.3.1 Reversible discrete-time Markov chain......Page 500
16.3.2 Reversible continuous-time Markov chain......Page 502
16.4 An application: phylogenetic tree and its Markov chain representation......Page 503
16.4.1 Trees and phylogenetic trees......Page 504
16.4.2 Markov process on a tree......Page 506
16.4.3 Continuous-time Markov chain model......Page 507
16.5 Summary of Chapter 16......Page 509
16.6 Discussion and further reading......Page 510
16.7 Problems......Page 511
17.1.1 Simple random walk......Page 516
17.1.2.1 Opponent with infinite capital......Page 519
17.1.2.2 Opponent with finite capital......Page 521
17.2.1 Wiener process as a limit process......Page 522
17.2.2 Properties of the Wiener process......Page 524
17.2.3 White noise......Page 526
17.2.3.1 Thermal noise in a band-limited system......Page 528
17.3.1 Fokker–Planck equation for Brownian motion with drift......Page 529
17.3.2 Einstein's diffusion equation for Brownian motion......Page 532
17.3.3 Forward and backward equations for general diffusion processes......Page 533
17.3.4 Ornstein–Uhlenbeck process: Gauss–Markov process......Page 535
17.3.4.1 Analysis of one-dimensional Ornstein–Uhlenbeck process......Page 536
17.4 Stochastic differential equations and Itô process......Page 539
17.4.1 Itô's formula......Page 540
17.4.2 Geometric Brownian motion (GBM)......Page 541
17.4.3 The Black–Scholes model: an application of an Itô process......Page 544
17.5 Summary of Chapter 17......Page 547
17.7 Problems......Page 549
Part IV Statistical inference......Page 554
18.1 Parameter estimation......Page 556
18.1.1 Exponential family of distributions revisited......Page 560
18.1.2 Maximum-likelihood estimation......Page 561
18.1.2.1 Maximum-likelihood estimate for the exponential family distribution......Page 564
18.1.3 Cramér–Rao lower bound......Page 565
18.1.3.1 Asymptotic unbiasedness, efficiency, and normality of the maximum-likelihood estimate......Page 568
18.1.4 Interval or region estimation......Page 569
18.2 Hypothesis testing and statistical decision theory......Page 571
18.2.1 Hypothesis testing......Page 572
18.2.2 Neyman–Pearson criterion and likelihood ratio test......Page 573
18.2.3.1 Properties of the receiver operating characterstic of a Neyman–Pearson test......Page 575
18.2.4 Receiver operating characteristic application example: radar signal detection......Page 576
18.3 Bayesian estimation and decision theory......Page 577
18.3.1 The Bayes risk......Page 578
18.3.2 The conditional risk......Page 579
18.3.3 Maximum a posteriori probability (MAP) estimation......Page 580
18.6 Problems......Page 582
19.1.1 Method of moments......Page 587
19.1.2 Minimum chi-square estimation method......Page 588
19.1.3 Minimum Kullback–Leibler divergence method......Page 589
19.1.4 Newton–Raphson algorithm for maximum-likelihood estimation......Page 591
19.2.1 Expectation-maximization algorithm for transformed data......Page 592
19.2.1.2 Derivation of the E-step and the M-step......Page 594
19.2.1.3 Geometrical interpretation of the EM algorithm......Page 595
19.2.2 Expectation-maximization algorithm for missing data......Page 597
19.2.2.2 Monte Carlo EM algorithm......Page 598
19.4 Discussion and further reading......Page 599
19.5 Problems......Page 600
Part V Applications and advanced topics......Page 604
20.2.1.1 Discrete-time Markov chain (DTMC)......Page 606
20.2.1.2 Hidden Markov model......Page 608
20.2.2 Examples of hidden Markov models......Page 611
20.2.2.1 Discrete memoryless channel......Page 613
20.3.1 Evaluation of a likelihood function......Page 616
20.3.2 Forward recursion algorithm......Page 618
20.3.3 Backward algorithm and forward-backward algorithm......Page 619
20.4.1 Forward algorithm for maximum a posteriori probability state sequence estimation......Page 621
20.4.1.1 The logarithmic conversion of probabilities......Page 622
20.4.2 The Viterbi algorithm......Page 623
20.5 The BCJR algorithm......Page 625
20.6.1 Forward–backward algorithm for a transition-based hidden Markov model......Page 627
20.6.1.1 The M-step of the EM algorithm for a transition-based hidden Morkov model......Page 629
20.6.2 The Baum–Welch algorithm......Page 631
20.7 Application: parameter estimation of mixture distributions......Page 634
20.8 Summary of Chapter 20......Page 637
20.9 Discussion and further reading......Page 638
20.10 Problems......Page 639
21.1 Introduction......Page 648
21.2 Maximum a posteriori probability (MAP) recognition......Page 650
21.3.1 K-means algorithm......Page 652
21.3.2 EM algorithm for clustering......Page 653
21.4.1 Speech recognition......Page 654
21.4.1.2 Connected-word recognition......Page 655
21.4.2 Biological sequence analysis......Page 656
21.5 Bayesian networks......Page 657
21.6 Factor graphs......Page 660
21.6.1 Sum-product algorithm......Page 661
21.6.2 Applications of factor graphs to Bayesian networks......Page 664
21.7 Markov chain Monte Carlo (MCMC) methods......Page 665
21.7.1 Metropolis–Hastings algorithm......Page 666
21.7.1.1 Choices of proposal distributions......Page 669
21.7.2 Metropolis–Hastings algorithm for continuous variables......Page 670
21.7.3 Block Metropolis–Hastings algorithm......Page 671
21.7.4 The Gibbs sampler......Page 672
21.7.5 Simulated annealing......Page 674
21.9 Discussion and further reading......Page 675
21.10 Problems......Page 676
22.1 Conditional expectation, minimum mean square error estimation and regression analysis......Page 678
22.1.1 Minimum mean square error estimation......Page 679
22.1.2 Linear minimum mean square error estimation......Page 680
22.1.2.1 Unbiased linear minimum mean square error estimate......Page 681
22.1.3 Conditional expectation and minimum mean square error estimation......Page 682
22.1.4 Regression analysis......Page 684
22.1.4.1 Linear regression......Page 685
22.1.4.3 Statistical analysis of regression6......Page 686
22.2 Linear smoothing and prediction: Wiener filter theory......Page 689
22.2.1 Linear time-invariant system with random input......Page 690
22.2.2 Optimal smoothing and prediction of a stationary process......Page 692
22.2.3 Noncausal (or physically unrealizable) filter......Page 695
22.2.4 Prediction of a random signal in the absence of noise......Page 696
22.2.5 Prediction of a random signal in the presence of noise......Page 703
22.3 Kalman filter......Page 712
22.3.1 State space model......Page 713
22.3.2 Derivation of the Kalman filter......Page 715
22.3.2.1 Kalman filter estimate......Page 716
22.4 Summary of Chapter 22......Page 722
22.6 Problems......Page 723
23.2 Little's formula: L = λW......Page 728
23.3.1 M/M/1: the simplest queueing model......Page 730
23.3.1.1 Waiting and system times......Page 733
23.3.2 M/M/∞and M/G/∞: infinite servers......Page 736
23.3.2.1 M/ G/∞model and formula (23.46)......Page 737
23.3.3 M/M/m: multiple server queue......Page 738
23.3.4 M(K)/M/m and G(K)/M/m: multiple access models......Page 740
23.3.4.2 G(K )/ M/1: The machine repairman model (m =1).......Page 741
23.3.4.3 Waiting time distribution in M(K )/M/m.......Page 742
23.3.5 M/G/1 queueing model......Page 743
23.3.5.1 Queue distribution in M/G/1......Page 745
23.3.5.2 Pollaczek–Khintchine mean value formula......Page 747
23.3.5.3 Waiting time distribution in M/G/1......Page 748
23.3.6 M/G/1 with processor sharing (PS)......Page 749
23.4 Loss models......Page 750
23.4.1 M/M/m(0) and M/G/m(0): Erlang loss models......Page 751
23.4.1.1 Generalized Erlang loss model: M/ G/ m(0)......Page 752
23.4.1.3 Carried load......Page 753
23.4.2 M(K)/M/m(0) and G(K)/G/m(0): Engset loss models......Page 754
23.4.2.1 Probability distribution seen by arrivals......Page 756
23.4.3 Loss network model and generalized loss station......Page 757
23.4.3.1 Open loss network......Page 758
23.4.3.2 Generalized loss station (GLS)......Page 759
23.5 Summary of Chapter 23......Page 762
23.6 Discussion and further reading......Page 763
23.7 Problems......Page 764
References......Page 773
Index......Page 792