Probability and Random Processes
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Preface for the Second Edition xii
Preface for the First Edition xiv
1 Sets, Fields, and Events 1
1.1 Set Definitions 1
1.2 Set Operations 2
1.3 Set Algebras, Fields, and Events 5
2 Probability Space and Axioms 7
2.1 Probability Space 7
2.2 Conditional Probability 9
2.3 Independence 11
2.4 Total Probability and Bayes' Theorem 12
3 Basic Combinatorics 16
3.1 Basic Counting Principles 16
3.2 Permutations 16
3.3 Combinations 18
4 Discrete Distributions 23
4.1 Bernoulli Trials 23
4.2 Binomial Distribution 23
4.3 Multinomial Distribution 26
4.4 Geometric Distribution 26
4.5 Negative Binomial Distribution 27
4.6 Hypergeometric Distribution 28
4.7 Poisson Distribution 30
4.8 Newton-Pepys Problem and its Extensions 33
4.9 Logarithmic Distribution 40
4.9.1 Finite Law (Benford's Law) 40
4.9.2 Infinite Law 43
4.10 Summary of Discrete Distributions 44
5 Random Variables 45
5.1 Definition of Random Variables 45
5.2 Determination of Distribution and Density Functions 46
5.3 Properties of Distribution and Density Functions 50
5.4 Distribution Functions from Density Functions 51
6 Continuous Random Variables and Basic Distributions 54
6.1 Introduction 54
6.2 Uniform Distribution 54
6.3 Exponential Distribution 55
6.4 Normal or Gaussian Distribution 57
7 Other Continuous Distributions 63
7.1 Introduction 63
7.2 Triangular Distribution 63
7.3 Laplace Distribution 63
7.4 Erlang Distribution 64
7.5 Gamma Distribution 65
7.6 Weibull Distribution 66
7.7 Chi-Square Distribution 67
7.8 Chi and Other Allied Distributions 68
7.9 Student-t Density 71
7.10 Snedecor F Distribution 72
7.11 Lognormal Distribution 72
7.12 Beta Distribution 73
7.13 Cauchy Distribution 74
7.14 Pareto Distribution 75
7.15 Gibbs Distribution 75
7.16 Mixed Distributions 75
7.17 Summary of Distributions of Continuous Random Variables 76
8 Conditional Densities and Distributions 78
8.1 Conditional Distribution and Density for P{A} 0 78
8.2 Conditional Distribution and Density for P{A} = 0 80
8.3 Total Probability and Bayes' Theorem for Densities 83
9 Joint Densities and Distributions 85
9.1 Joint Discrete Distribution Functions 85
9.2 Joint Continuous Distribution Functions 86
9.3 Bivariate Gaussian Distributions 90
10 Moments and Conditional Moments 91
10.1 Expectations 91
10.2 Variance 92
10.3 Means and Variances of Some Distributions 93
10.4 Higher-Order Moments 94
10.5 Correlation and Partial Correlation Coefficients 95
10.5.1 Correlation Coefficients 95
10.5.2 Partial Correlation Coefficients 106
11 Characteristic Functions and Generating Functions 108
11.1 Characteristic Functions 108
11.2 Examples of Characteristic Functions 109
11.3 Generating Functions 111
11.4 Examples of Generating Functions 112
11.5 Moment Generating Functions 113
11.6 Cumulant Generating Functions 115
11.7 Table of Means and Variances 116
12 Functions of a Single Random Variable 118
12.1 Random Variable g(X) 118
12.2 Distribution of Y = g(X) 119
12.3 Direct Determination of Density fY(y) from fX(x) 129
12.4 Inverse Problem: Finding g(X) given fX(x) and fY(y) 132
12.5 Moments of a Function of a Random Variable 133
13 Functions of Multiple Random Variables 135
13.1 Function of Two Random Variables, Z = g(X,Y) 135
13.2 Two Functions of Two Random Variables, Z = g(X,Y), W= h(X,Y) 143
13.3 Direct Determination of Joint Density fZW(z,w) from fXY(x,y) 146
13.4 Solving Z = g(X,Y) Using an Auxiliary Random Variable 150
13.5 Multiple Functions of Random Variables 153
14 Inequalities, Convergences, and Limit Theorems 155
14.1 Degenerate Random Variables 155
14.2 Chebyshev and Allied Inequalities 155
14.3 Markov Inequality 158
14.4 Chernoff Bound 159
14.5 Cauchy-Schwartz Inequality 160
14.6 Jensen's Inequality 162
14.7 Convergence Concepts 163
14.8 Limit Theorems 165
15 Computer Methods for Generating Random Variates 169
15.1 Uniform-Distribution Random Variates 169
15.2 Histograms 170
15.3 Inverse Transformation Techniques 172
15.4 Convolution Techniques 178
15.5 Acceptance-Rejection Techniques 178
16 Elements of Matrix Algebra 181
16.1 Basic Theory of Matrices 181
16.2 Eigenvalues and Eigenvectors of Matrices 186
16.3 Vector and Matrix Differentiation 190
16.4 Block Matrices 194
17 Random Vectors and Mean-Square Estimation 196
17.1 Distributions and Densities 196
17.2 Moments of Random Vectors 200
17.3 Vector Gaussian Random Variables 204
17.4 Diagonalization of Covariance Matrices 207
17.5 Simultaneous Diagonalization of Covariance Matrices 209
17.6 Linear Estimation of Vector Variables 210
18 Estimation Theory 212
18.1 Criteria of Estimators 212
18.2 Estimation of Random Variables 213
18.3 Estimation of Parameters (Point Estimation) 218
18.4 Interval Estimation (Confidence Intervals) 225
18.5 Hypothesis Testing (Binary) 231
18.6 Bayesian Estimation 238
19 Random Processes 250
19.1 Basic Definitions 250
19.2 Stationary Random Processes 258
19.3 Ergodic Processes 269
19.4 Estimation of Parameters of Random Processes 273
19.4.1 Continuous-Time Processes 273
19.4.2 Discrete-Time Processes 280
19.5 Power Spectral Density 287
19.5.1 Continuous Time 287
19.5.2 Discrete Time 294
19.6 Adaptive Estimation 298
20 Classification of Random Processes 320
20.1 Specifications of Random Processes 320
20.1.1 Discrete-State Discrete-Time (DSDT) Process 320
20.1.2 Discrete-State Continuous-Time (DSCT) Process 320
20.1.3 Continuous-State Discrete-Time (CSDT) Process 320
20.1.4 Continuous-State Continuous-Time (CSCT) Process 320
20.2 Poisson Process 321
20.3 Binomial Process 329
20.4 Independent Increment Process 330
20.5 Random-Walk Process 333
20.6 Gaussian Process 338
20.7 Wiener Process (Brownian Motion) 340
20.8 Markov Process 342
20.9 Markov Chains 347
20.10 Birth and Death Processes 357
20.11 Renewal Processes and Generalizations 366
20.12 Martingale Process 370
20.13 Periodic Random Process 374
20.14 Aperiodic Random Process (Karhunen-Loeve Expansion) 377
21 Random Processes and Linear Systems 383
21.1 Review of Linear Systems 383
21.2 Random Processes through Linear Systems 385
21.3 Linear Filters 393
21.4 Bandpass Stationary Random Processes 401
22 Wiener and Kalman Filters 413
22.1 Review of Orthogonality Principle 413
22.2 Wiener Filtering 414
22.3 Discrete Kalman Filter 425
22.4 Continuous Kalman Filter 433
23 Probability Modeling in Traffic Engineering 437
23.1 Introduction 437
23.2 Teletraffic Models 437
23.3 Blocking Systems 438
23.4 State Probabilities for Systems with Delays 440
23.5 Waiting-Time Distribution for M/M/c/8 Systems 441
23.6 State Probabilities for M/D/c Systems 443
23.7 Waiting-Time Distribution for M/D/c/8 System 446
23.8 Comparison of M/M/c and M/D/c 448
References 451
24 Probabilistic Methods in Transmission Tomography 452
24.1 Introduction 452
24.2 Stochastic Model 453
24.3 Stochastic Estimation Algorithm 455
24.4 Prior Distribution P{M} 457
24.5 Computer Simulation 458
24.6 Results and Conclusions 460
24.7 Discussion of Results 462
References 462
APPENDICES
A A Fourier Transform Tables 463
B Cumulative Gaussian Tables 467
C Inverse Cumulative Gaussian Tables 472
D Inverse Chi-Square Tables 474
E Inverse Student-t Tables 481
F Cumulative Poisson Distribution 484
G Cumulative Binomial Distribution 488
H Computation of Roots of D(z) = 0 494
References 495
Index 498
