機率、統計與隨機過程（第四版）（英文版）

本書從工程套用的角度，全面闡述機率、統計與隨機過程的基本理論及其套用。

全書共11章，首先簡單介紹機率論，然後各章分別討論隨機變數、隨機變數的函式、均值與矩、隨機矢量、統計（包括參數估計和假設檢驗）、隨機序列、隨機過程基礎知識和深入探討，最後討論了統計信號處理中的相關套用。書中給出了大量電子和信息系統相關實例，每章給出了豐富的習題。

基本介紹

• 書名：機率、統計與隨機過程（第四版）（英文版）
• 作者：亨利 斯塔克、約翰 伍茲 
• ISBN：9787121176685
• 出版社：電子工業出版社
• 出版時間：2012-08-01

圖書內容

本書從工程套用的角度，全面闡述機率、統計與隨機過程的基本理論及其套用。

全書共11章，首先簡單介紹機率論，然後各章分別討論隨機變數、隨機變數的函式、均值與矩、隨機矢量、統計（包括參數估計和假設檢驗）、隨機序列、隨機過程基礎知識和深入探討，最後討論了統計信號處理中的相關套用。書中給出了大量電子和信息系統相關實例，每章給出了豐富的習題。

目錄

Preface 11

Chapter1 Introduction to Probability 13

1.1 Introduction: Why Study Probability? 13

1.2 The Different Kinds of Probability 14

Probability as Intuition 14

Probability as the Ratio of Favorable to Total Outcomes (Classical Theory) 15

Probability as a Measure of Frequency of Occurrence 16

Probability Based on an Axiomatic Theory 17

1.3 Misuses, Miscalculations, and Paradoxes in Probability 19

1.4 Sets, Fields, and Events 20

Examples of Sample Spaces 20

1.5 Axiomatic Definition of Probability 27

1.6 Joint, Conditional, and Total Probabilities; Independence 32

Compound Experiments 35

1.7 Bayes’ Theorem and Applications 47

1.8 Combinatorics 50

Occupancy Problems 54

Extensions and Applications 58

1.9 Bernoulli Trials—Binomial and Multinomial Probability Laws 60

Multinomial Probability Law 66

1.10 Asymptotic Behavior of the Binomial Law: The Poisson Law 69

1.11 Normal Approximation to the Binomial Law 75

Summary 77

Problems 78

References 89

Chapter2 Random Variables 91

2.1 Introduction 91

2.2 Definition of a Random Variable 92

2.3 Cumulative Distribution Function 95

Properties of FX(x) 96

Computation of FX(x) 97

2.4 Probability Density Function (pdf) 100

Four Other Common Density Functions 107

More Advanced Density Functions 109

2.5 Continuous, Discrete, and Mixed Random Variables 112

Some Common Discrete Random Variables 114

2.6 Conditional and Joint Distributions and Densities 119

Properties of Joint CDF FXY (x, y) 130

2.7 Failure Rates 149

Summary 153

Problems 153

References 161

Additional Reading 161

Chapter3 Functions of Random Variables 163

3.1 Introduction 163

Functions of a Random Variable (FRV): Several Views 166

3.2 Solving Problems of the Type Y = g(X) 167

General Formula of Determining the pdf of Y = g(X) 178

3.3 Solving Problems of the Type Z = g(X, Y ) 183

3.4 Solving Problems of the Type V = g(X, Y ), W = h(X, Y ) 205

Fundamental Problem 205

Obtaining fVW Directly from fXY 208

3.5 Additional Examples 212

Summary 217

Problems 218

References 226

Additional Reading 226

Chapter4 Expectation and Moments 227

4.1 Expected Value of a Random Variable 227

On the Validity of Equation 4.1-8 230

4.2 Conditional Expectations 244

Conditional Expectation as a Random Variable 251

4.3 Moments of Random Variables 254

Joint Moments 258

Properties of Uncorrelated Random Variables 260

Jointly Gaussian Random Variables 263

4.4 Chebyshev and Schwarz Inequalities 267

Markov Inequality 269

The Schwarz Inequality 270

4.5 Moment-Generating Functions 273

4.6 Chernoff Bound 276

4.7 Characteristic Functions 278

Joint Characteristic Functions 285

The Central Limit Theorem 288

4.8 Additional Examples 293

Summary 295

Problems 296

References 305

Additional Reading 306

Chapter5 Random Vectors 307

5.1 Joint Distribution and Densities 307

5.2 Multiple Transformation of Random Variables 311

5.3 Ordered Random Variables 314

5.4 Expectation Vectors and Covariance Matrices 323

5.5 Properties of Covariance Matrices 326

Whitening Transformation 330

5.6 The Multidimensional Gaussian (Normal) Law 331

5.7 Characteristic Functions of Random Vectors 340

Properties of CF of Random Vectors 342

The Characteristic Function of the Gaussian (Normal) Law 343

Summary 344

Problems 345

References 351

Additional Reading 351

Chapter6 Statistics: Part 1 Parameter Estimation 352

6.1 Introduction 352

Independent, Identically, Observations 353

Estimation of Probabilities 355

6.2 Estimators 358

6.3 Estimation of the Mean 360

Properties of the Mean-Estimator Function (MEF) 361

Procedure for Getting a δ-confidence Interval on the Mean of a Normal

Random Variable When σX Is Known 364

Confidence Interval for the Mean of a Normal Distribution When σX Is Not

Known 364

Procedure for Getting a δ-Confidence Interval Based on n Observations on

the Mean of a Normal Random Variable when σX Is Not Known 367

Interpretation of the Confidence Interval 367

6.4 Estimation of the Variance and Covariance 367

Confidence Interval for the Variance of a Normal Random

variable 369

Estimating the Standard Deviation Directly 371

Estimating the covariance 372

6.5 Simultaneous Estimation of Mean and Variance 373

6.6 Estimation of Non-Gaussian Parameters from Large Samples 375

6.7 Maximum Likelihood Estimators 377

6.8 Ordering, more on Percentiles, Parametric Versus Nonparametric Statistics 381

The Median of a Population Versus Its Mean 383

Parametric versus Nonparametric Statistics 384

Confidence Interval on the Percentile 385

Confidence Interval for the Median When n Is Large 387

6.9 Estimation of Vector Means and Covariance Matrices 388

Estimation of μ 389

Estimation of the covariance K 390

6.10 Linear Estimation of Vector Parameters 392

Summary 396

Problems 396

References 400

Additional Reading 401

Chapter7 Statistics: Part 2 Hypothesis Testing 402

7.1 Bayesian Decision Theory 403

7.2 Likelihood Ratio Test 408

7.3 Composite Hypotheses 414

Generalized Likelihood Ratio Test (GLRT) 415

How Do We Test for the Equality of Means of Two Populations? 420

Testing for the Equality of Variances for Normal Populations:

The F-test 424

TestingWhether the Variance of a Normal Population Has a Predetermined

Value: 428

7.4 Goodness of Fit 429

7.5 Ordering, Percentiles, and Rank 435

How Ordering is Useful in Estimating Percentiles and the Median 437

Confidence Interval for the Median When n Is Large 440

Distribution-free Hypothesis Testing: Testing If Two Population are the

Same Using Runs 441

Ranking Test for Sameness of Two Populations 444

Summary 445

Problems 445

References 451

Chapter8 Random Sequences 453

8.1 Basic Concepts 454

Infinite-length Bernoulli Trials 459

Continuity of Probability Measure 464

Statistical Specification of a Random Sequence 466

8.2 Basic Principles of Discrete-Time Linear Systems 483

8.3 Random Sequences and Linear Systems 489

8.4 WSS Random Sequences 498

Power Spectral Density 501

Interpretation of the psd 502

Synthesis of Random Sequences and Discrete-Time Simulation 505

Decimation 508

Interpolation 509

8.5 Markov Random Sequences 512

ARMA Models 515

Markov Chains 516

8.6 Vector Random Sequences and State Equations 523

8.7 Convergence of Random Sequences 525

8.8 Laws of Large Numbers 533

Summary 538

Problems 538

References 553

Chapter9 Random Processes 555

9.1 Basic Definitions 556

9.2 Some Important Random Processes 560

Asynchronous Binary Signaling 560

Poisson Counting Process 562

Alternative Derivation of Poisson Process 567

Random Telegraph Signal 569

Digital Modulation Using Phase-Shift Keying 570

Wiener Process or Brownian Motion 572

Markov Random Processes 575

Birth–Death Markov Chains 579

Chapman–Kolmogorov Equations 583

Random Process Generated from Random Sequences 584

9.3 Continuous-Time Linear Systems with Random Inputs 584

White Noise 589

9.4 Some Useful Classifications of Random Processes 590

Stationarity 591

9.5 Wide-Sense Stationary Processes and LSI Systems 593

Wide-Sense Stationary Case 594

Power Spectral Density 596

An Interpretation of the psd 598

More on White Noise 602

Stationary Processes and Differential Equations 608

9.6 Periodic and Cyclostationary Processes 612

9.7 Vector Processes and State Equations 618

State Equations 620

Summary 623

Problems 623

References 645

Chapters 10 and 11 are available as Web chapters on the companion

Web site at www.pearsoninternationaleditions.com/stark.

Chapter10 Advanced Topics in Random Processes 647

10.1 Mean-Square (m.s.) Calculus 647

Stochastic Continuity and Derivatives [10-1] 647

Further Results on m.s. Convergence [10-1] 657

10.2 Mean-Square Stochastic Integrals 662

10.3 Mean-Square Stochastic Differential Equations 665

10.4 Ergodicity [10-3] 670

10.5 Karhunen–Lo`eve Expansion [10-5] 677

10.6 Representation of Bandlimited and Periodic Processes 683

Bandlimited Processes 683

Bandpass Random Processes 686

WSS Periodic Processes 689

Fourier Series for WSS Processes 692

Summary 694

Appendix: Integral Equations 694

Existence Theorem 695

Problems 698

References 711

Chapter11 Applications to Statistical Signal Processing 712

11.1 Estimation of Random Variables and Vectors 712

More on the Conditional Mean 718

Orthogonality and Linear Estimation 720

Some Properties of the Operator ˆE 728

11.2 Innovation Sequences and Kalman Filtering 730

Predicting Gaussian Random Sequences 734

Kalman Predictor and Filter 736

Error-Covariance Equations 741

11.3 Wiener Filters for Random Sequences 745

Unrealizable Case (Smoothing) 746

Causal Wiener Filter 748

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Contents 9

11.4 Expectation-Maximization Algorithm 750

Log-likelihood for the Linear Transformation 752

Summary of the E-M algorithm 754

E-M Algorithm for Exponential Probability

Functions 755

Application to Emission Tomography 756

Log-likelihood Function of Complete Data 758

E-step 759

M-step 760

11.5 Hidden Markov Models (HMM) 761

Specification of an HMM 763

Application to Speech Processing 765

Efficient Computation of P[E|M] with a Recursive

Algorithm 766

Viterbi Algorithm and the Most Likely State Sequence

for the Observations 768

11.6 Spectral Estimation 771

The Periodogram 772

Bartlett’s Procedure---Averaging Periodograms 774

Parametric Spectral Estimate 779

Maximum Entropy Spectral Density 781

11.7 Simulated Annealing 784

Gibbs Sampler 785

Noncausal Gauss–Markov Models 786

Compound Markov Models 790

Gibbs Line Sequence 791

Summary 795

Problems 795

References 800

Appendix A Review of Relevant Mathematics A-1

A.1 Basic Mathematics A-1

Sequences A-1

Convergence A-2

Summations A-3

Z-Transform A-3

A.2 Continuous Mathematics A-4

Definite and Indefinite Integrals A-5

Differentiation of Integrals A-6

Integration by Parts A-7

Completing the Square A-7

Double Integration A-8

Functions A-8

A.3 Residue Method for Inverse Fourier Transformation A-10

Fact A-11

Inverse Fourier Transform for psd of Random Sequence A-13

A.4 Mathematical Induction A-17

References A-17

Appendix B Gamma and Delta Functions B-1

B.1 Gamma Function B-1

B.2 Incomplete Gamma Function B-2

B.3 Dirac Delta Function B-2

References B-5

Appendix C Functional Transformations and Jacobians C-1

C.1 Introduction C-1

C.2 Jacobians for n = 2 C-2

C.3 Jacobian for General n C-4

Appendix D Measure and Probability D-1

D.1 Introduction and Basic Ideas D-1

Measurable Mappings and Functions D-3

D.2 Application of Measure Theory to Probability D-3

Distribution Measure D-4

Appendix E Sampled Analog Waveforms and Discrete-time Signals E-1

Appendix F Independence of Sample Mean and Variance for Normal Random Variables F-1

Appendix G Tables of Cumulative Distribution Functions: the Normal, Student t, Chi-square, and F G-1

Index I-1