## Table of Contents

**Preface **

**1 Introduction to Probability 1**

1.1 Introduction: Why Study Probability? 1

1.2 The Different Kinds of Probability 2

Probability as Intuition 2

Probability as the Ratio of Favorable to Total Outcomes (Classical Theory) 3

Probability as a Measure of Frequency of Occurrence 4

Probability Based on an Axiomatic Theory 5

1.3 Misuses, Miscalculations, and Paradoxes in Probability 7

1.4 Sets, Fields, and Events 8

Examples of Sample Spaces 8

1.5 Axiomatic Definition of Probability 15

1.6 Joint, Conditional, and Total Probabilities; Independence 20

Compound Experiments 23

1.7 Bayes’ Theorem and Applications 35

1.8 Combinatorics 38

Occupancy Problems 42

Extensions and Applications 46

1.9 Bernoulli Trials–Binomial and Multinomial Probability Laws 48

Multinomial Probability Law 54

1.10 Asymptotic Behavior of the Binomial Law: The Poisson Law 57

1.11 Normal Approximation to the Binomial Law 63

Summary 65

Problems 66

References 77

** **

**2 Random Variables 79**

2.1 Introduction 79

2.2 Definition of a Random Variable 80

2.3 Cumulative Distribution Function 83

Properties of *F**X*(*x*) 84

Computation of *F**X*(*x*) 85

2.4 Probability Density Function (pdf) 88

Four Other Common Density Functions 95

More Advanced Density Functions 97

2.5 Continuous, Discrete, and Mixed Random Variables 100

Some Common Discrete Random Variables 102

2.6 Conditional and Joint Distributions and Densities 107

Properties of Joint CDF *F**XY *(*x, y*) 118

2.7 Failure Rates 137

Summary 141

Problems 141

References 149

Additional Reading 149

** **

**3 Functions of Random Variables 151**

3.1 Introduction 151

Functions of a Random Variable (FRV): Several Views 154

3.2 Solving Problems of the Type *Y *= *g*(*X*) 155

General Formula of Determining the pdf of *Y *= *g*(*X*) 166

3.3 Solving Problems of the Type *Z *= *g*(*X, Y *) 171

3.4 Solving Problems of the Type *V *= *g*(*X, Y *), *W *= *h*(*X, Y *) 193

Fundamental Problem 193

Obtaining *f**VW *Directly from *f**XY *196

3.5 Additional Examples 200

Summary 205

Problems 206

References 214

Additional Reading 214

** **

**4 Expectation and Moments 215**

4.1 Expected Value of a Random Variable 215

On the Validity of Equation 4.1-8 218

4.2 Conditional Expectations 232

Conditional Expectation as a Random Variable 239

4.3 Moments of Random Variables 242

Joint Moments 246

Properties of Uncorrelated Random Variables 248

Jointly Gaussian Random Variables 251

4.4 Chebyshev and Schwarz Inequalities 255

Markov Inequality 257

The Schwarz Inequality 258

4.5 Moment-Generating Functions 261

4.6 Chernoff Bound 264

4.7 Characteristic Functions 266

Joint Characteristic Functions 273

The Central Limit Theorem 276

4.8 Additional Examples 281

Summary 283

Problems 284

References 293

Additional Reading 294

** **

**5 Random Vectors 295**

5.1 Joint Distribution and Densities 295

5.2 Multiple Transformation of Random Variables 299

5.3 Ordered Random Variables 302

Distribution of area random variables 305

5.4 Expectation Vectors and Covariance Matrices 311

5.5 Properties of Covariance Matrices 314

Whitening Transformation 318

5.6 The Multidimensional Gaussian (Normal) Law 319

5.7 Characteristic Functions of Random Vectors 328

Properties of CF of Random Vectors 330

The Characteristic Function of the Gaussian (Normal) Law 331

Summary 332

Problems 333

References 339

Additional Reading 339

** **

**6 Statistics: Part 1 Parameter Estimation 340**

6.1 Introduction 340

Independent, Identically Distributed (i.i.d.) Observations 341

Estimation of Probabilities 343

6.2 Estimators 346

6.3 Estimation of the Mean 348

Properties of the Mean-Estimator Function (MEF) 349

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

Random Variable When *σ**X *Is Known 352

Confidence Interval for the Mean of a Normal Distribution When σ*X *Is Not

Known 352

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

the Mean of a Normal Random Variable when *σ**X *Is Not Known 355

Interpretation of the Confidence Interval 355

6.4 Estimation of the Variance and Covariance 355

Confidence Interval for the Variance of a Normal Random

variable 357

Estimating the Standard Deviation Directly 359

Estimating the covariance 360

6.5 Simultaneous Estimation of Mean and Variance 361

6.6 Estimation of Non-Gaussian Parameters from Large Samples 363

6.7 Maximum Likelihood Estimators 365

6.8 Ordering, more on Percentiles, Parametric Versus Nonparametric Statistics 369

The Median of a Population Versus Its Mean 371

Parametric versus Nonparametric Statistics 372

Confidence Interval on the Percentile 373

Confidence Interval for the Median When *n *Is Large 375

6.9 Estimation of Vector Means and Covariance Matrices 376

Estimation of *μ *377

Estimation of the covariance *K *378

6.10 Linear Estimation of Vector Parameters 380

Summary 384

Problems 384

References 388

Additional Reading 389

** **

**7 Statistics: Part 2 Hypothesis Testing 390**

7.1 Bayesian Decision Theory 391

7.2 Likelihood Ratio Test 396

7.3 Composite Hypotheses 402

Generalized Likelihood Ratio Test (GLRT) 403

How Do We Test for the Equality of Means of Two Populations? 408

Testing for the Equality of Variances for Normal Populations:

The F-test 412

Testing Whether the Variance of a Normal Population Has a

Predetermined Value: 416

7.4 Goodness of Fit 417

7.5 Ordering, Percentiles, and Rank 423

How Ordering is Useful in Estimating Percentiles and the Median 425

Confidence Interval for the Median When *n *Is Large 428

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

Same Using *Runs *429

Ranking Test for Sameness of Two Populations 432

Summary 433

Problems 433

References 439

** **

**8 Random Sequences 441**

8.1 Basic Concepts 442

Infinite-length Bernoulli Trials 447

Continuity of Probability Measure 452

Statistical Specification of a Random Sequence 454

8.2 Basic Principles of Discrete-Time Linear Systems 471

8.3 Random Sequences and Linear Systems 477

8.4 WSS Random Sequences 486

Power Spectral Density 489

Interpretation of the psd 490

Synthesis of Random Sequences and Discrete-Time Simulation 493

Decimation 496

Interpolation 497

8.5 Markov Random Sequences 500

ARMA Models 503

Markov Chains 504

8.6 Vector Random Sequences and State Equations 511

8.7 Convergence of Random Sequences 513

8.8 Laws of Large Numbers 521

Summary 526

Problems 526

References 541

** **

**9 Random Processes 543**

9.1 Basic Definitions 544

9.2 Some Important Random Processes 548

Asynchronous Binary Signaling 548

Poisson Counting Process 550

Alternative Derivation of Poisson Process 555

Random Telegraph Signal 557

Digital Modulation Using Phase-Shift Keying 558

Wiener Process or Brownian Motion 560

Markov Random Processes 563

Birth—Death Markov Chains 567

Chapman—Kolmogorov Equations 571

Random Process Generated from Random Sequences 572

9.3 Continuous-Time Linear Systems with Random Inputs 572

White Noise 577

9.4 Some Useful Classifications of Random Processes 578

Stationarity 579

9.5 Wide-Sense Stationary Processes and LSI Systems 581

Wide-Sense Stationary Case 582

Power Spectral Density 584

An Interpretation of the psd 586

More on White Noise 590

Stationary Processes and Differential Equations 596

9.6 Periodic and Cyclostationary Processes 600

9.7 Vector Processes and State Equations 606

State Equations 608

Summary 611

Problems 611

References 633

* *

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

*Web site at http://www.pearsonhighered.com/stark.*

**10 Advanced Topics in Random Processes 635**

10.1 Mean-Square (m.s.) Calculus 635

Stochastic Continuity and Derivatives [10-1] 635

Further Results on m.s. Convergence [10-1] 645

10.2 Mean-Square Stochastic Integrals 650

10.3 Mean-Square Stochastic Differential Equations 653

10.4 Ergodicity [10-3] 658

10.5 Karhunen—Lo`eve Expansion [10-5] 665

10.6 Representation of Bandlimited and Periodic Processes 671

Bandlimited Processes 671

Bandpass Random Processes 674

WSS Periodic Processes 677

Fourier Series for WSS Processes 680

Summary 682

Appendix: Integral Equations 682

Existence Theorem 683

Problems 686

References 699

** **

**11 Applications to Statistical Signal Processing 700**

11.1 Estimation of Random Variables and Vectors 700

More on the Conditional Mean 706

Orthogonality and Linear Estimation 708

Some Properties of the Operator ˆ*E *716

11.2 Innovation Sequences and Kalman Filtering 718

Predicting Gaussian Random Sequences 722

Kalman Predictor and Filter 724

Error-Covariance Equations 729

11.3 Wiener Filters for Random Sequences 733

Unrealizable Case (Smoothing) 734

Causal Wiener Filter 736

11.4 Expectation-Maximization Algorithm 738

Log-likelihood for the Linear Transformation 740

Summary of the E-M algorithm 742

E-M Algorithm for Exponential Probability

Functions 743

Application to Emission Tomography 744

Log-likelihood Function of Complete Data 746

E-step 747

M-step 748

11.5 Hidden Markov Models (HMM) 749

Specification of an HMM 751

Application to Speech Processing 753

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

Algorithm 754

Viterbi Algorithm and the Most Likely State Sequence

for the Observations 756

11.6 Spectral Estimation 759

The Periodogram 760

Bartlett’s Procedure---Averaging Periodograms 762

Parametric Spectral Estimate 767

Maximum Entropy Spectral Density 769

11.7 Simulated Annealing 772

Gibbs Sampler 773

Noncausal Gauss—Markov Models 774

Compound Markov Models 778

Gibbs Line Sequence 779

Summary 783

Problems 783

References 788

** **

**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**