Intuitive Probability And Random Processes Using Matlab

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ISBN: 9780387241579 | 0387241574
Cover: Hardcover
Copyright: 12/30/2005

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This book covers:- The use of MATLAB examples to provide motivation for the theory to come.- The incorporation of MATLAB code to allow students to understand how the theory is applied in practice.- Numerous computer exercises to familiarize the student with MATLAB and how it is used to solve real problems.- The incorporation of "real-world" problems from various disciplines in each chapter to illlustrate the application of the chapter concepts.- Discussion of discrete random variables first, followed by continuous random variables to minimize confusion.

Preface

vii

Introduction

(12)

What Is Probability?

(2)

Types of Probability Problems

(1)

Probabilistic Modeling

(3)

Analysis versus Computer Simulation

(1)

Some Notes to the Reader

(5)

References

(1)

Problems

(3)

Computer Simulation

(24)

Introduction

(1)

Summary

(1)

Why Use Computer Simulation?

(3)

Computer Simulation of Random Phenomena

(1)

Determining Characteristics of Random Variables

(6)

Real-World Example -- Digital Communications

(7)

References

(1)

Problems

(5)

Brief Introduction to MATLAB

(6)

Basic Probability

(36)

Introduction

(1)

Summary

(1)

Review of Set Theory

(5)

Assigning and Determining Probabilities

(5)

Properties of the Probability Function

(4)

Probabilities for Continuous Sample Spaces

(2)

Probabilities for Finite Sample Spaces -- Equally Likely Outcomes

(1)

Combinatorics

(7)

Binomial Probability Law

(2)

Real-World Example -- Quality Control

(9)

References

(1)

Problems

(7)

Conditional Probability

(32)

Introduction

(1)

Summary

(1)

Joint Events and the Conditional Probability

(9)

Statistically Independent Events

(3)

Bayes' Theorem

(3)

Multiple Experiments

(8)

Real-World Example -- Cluster Recognition

(8)

References

100

(1)

Problems

100

(5)

Discrete Random Variables

105

(28)

Introduction

105

(1)

Summary

105

(1)

Definition of Discrete Random Variable

106

(2)

Probability of Discrete Random Variables

108

(3)

Important Probability Mass Functions

111

(2)

Approximation of Binomial PMF by Poisson PMF

113

(2)

Transformation of Discrete Random Variables

115

(2)

Cumulative Distribution Function

117

(5)

Computer Simulation

122

(2)

Real-World Example -- Servicing Customers

124

(9)

References

128

(1)

Problems

128

(5)

Expected Values for Discrete Random Variables

133

(34)

Introduction

133

(1)

Summary

133

(1)

Determining Averages from the PMF

134

(3)

Expected Values of Some Important Random Variables

137

(3)

Expected Value for a Function of a Random Variable

140

(3)

Variance and Moments of a Random Variable

143

(4)

Characteristic Functions

147

(6)

Estimating Means and Variances

153

(2)

Real-World Example -- Data Compression

155

(8)

References

157

(1)

Problems

158

(5)

Derivation of E[g(X)] Formula

163

(2)

MATLAB Code Used to Estimate Mean and Variance

165

(2)

Multiple Discrete Random Variables

167

(48)

Introduction

167

(1)

Summary

168

(1)

Jointly Distributed Random Variables

169

(5)

Marginal PMFs and CDFs

174

(4)

Independence of Multiple Random Variables

178

(3)

Transformations of Multiple Random Variables

181

(5)

Expected Values

186

(3)

Joint Moments

189

(3)

Prediction of a Random Variable Outcome

192

(6)

Joint Characteristic Functions

198

(2)

Computer Simulation of Random Vectors

200

(2)

Real-World Example Assessing Health Risks

202

(11)

References

204

(1)

Problems

204

(9)

Derivation of the Cauchy-Schwarz Inequality

213

(2)

Conditional Probability Mass Functions

215

(32)

Introduction

215

(1)

Summary

216

(1)

Conditional Probability Mass Function

217

(3)

Joint, Conditional, and Marginal PMFs

220

(5)

Simplifying Probability Calculations using Conditioning

225

(4)

Mean of the Conditional PMF

229

(6)

Computer Simulation Based on Conditioning

235

(2)

Real-World Example -- Modeling Human Learning

237

(10)

References

240

(1)

Problems

240

(7)

Discrete N-Dimensional Random Variables

247

(38)

Introduction

247

(1)

Summary

247

(1)

Random Vectors and Probability Mass Functions

248

(3)

Transformations

251

(4)

Expected Values

255

(10)

Joint Moments and the Characteristic Function

265

(1)

Conditional Probability Mass Functions

266

(3)

Computer Simulation of Random Vectors

269

(3)

Real-World Example -- Image Coding

272

(13)

References

277

(1)

Problems

277

(8)

Continuous Random Variables

285

(58)

Introduction

285

(1)

Summary

286

(1)

Definition of a Continuous Random Variable

287

(6)

The PDF and Its Properties

293

(2)

Important PDFs

295

(8)

Cumulative Distribution Functions

303

(8)

Transformations

311

(6)

Mixed Random Variables

317

(7)

Computer Simulation

324

(4)

Real-World Example -- Setting Clipping Levels

328

(11)

References

331

(1)

Problems

331

(8)

Derivation of PDF of a Transformed Continuous Random Variable

339

(2)

MATLAB Subprograms to Compute Q and Inverse Q Functions

341

(2)

Expected Values for Continuous Random Variables

343

(34)

Introduction

343

(1)

Summary

343

(1)

Determining the Expected Value

344

(5)

Expected Values for Important PDFs

349

(2)

Expected Value for a Function of a Random Variable

351

(4)

Variance and Moments

355

(4)

Characteristic Functions

359

(2)

Probability, Moments, and the Chebyshev Inequality

361

(2)

Estimating the Mean and Variance

363

(1)

Real-World Example -- Critical Software Testing

364

(11)

References

367

(1)

Problems

367

(8)

Partial Proof of Expected Value of Function of Continuous Random Variable

375

(2)

Multiple Continuous Random Variables

377

(56)

Introduction

377

(1)

Summary

378

(1)

Jointly Distributed Random Variables

379

(8)

Marginal PDFs and the Joint CDF

387

(5)

Independence of Multiple Random Variables

392

(2)

Transformations

394

(10)

Expected Values

404

(8)

Joint Moments

412

(1)

Prediction of Random Variable Outcome

412

(2)

Joint Characteristic Functions

414

(1)

Computer Simulation

415

(4)

Real-World Example - Optical Character Recognition

419

(14)

References

423

(1)

Problems

423

(10)

Conditional Probability Density Functions

433

(24)

Introduction

433

(1)

Summary

433

(1)

Conditional PDF

434

(6)

Joint, Conditional, and Marginal PDFs

440

(4)

Simplifying Probability Calculations Using Conditioning

444

(2)

Mean of Conditional PDF

446

(1)

Computer Simulation of Jointly Continuous Random Variables

447

(2)

Real-World Example -- Retirement Planning

449

(8)

References

452

(1)

Problems

452

(5)

Continuous N-Dimensional Random Variables

457

(28)

Introduction

457

(1)

Summary

457

(1)

Random Vectors and PDFs

458

(5)

Transformations

463

(2)

Expected Values

465

(2)

Joint Moments and the Characteristic Function

467

(4)

Conditional PDFs

471

(1)

Prediction of a Random Variable Outcome

471

(4)

Computer Simulation of Gaussian Random Vectors

475

(1)

Real-World Example -- Signal Detection

476

(9)

References

479

(1)

Problems

479

(6)

Probability and Moment Approximations Using Limit Theorems

485

(30)

Introduction

485

(1)

Summary

486

(1)

Convergence and Approximation of a Sum

486

(1)

Law of Large Numbers

487

(5)

Central Limit Theorem

492

(11)

Real-World Example -- Opinion Polling

503

(8)

References

506

(1)

Problems

507

(4)

MATLAB Program to Compute Repeated Convolution of PDFs

511

(2)

Proof of Central Limit Theorem

513

(2)

Basic Random Processes

515

(32)

Introduction

515

(1)

Summary

516

(1)

What Is a Random Process?

517

(3)

Types of Random Processes

520

(3)

The Important, Property of Stationarity

523

(5)

Some More Examples

528

(5)

Joint Moments

533

(5)

Real-World Example - Statistical Data Analysis

538

(9)

References

542

(1)

Problems

542

(5)

Wide Sense Stationary Random Processes

547

(50)

Introduction

547

(1)

Summary

548

(1)

Definition of WSS Random Process

549

(3)

Autocorrelation Sequence

552

(10)

Ergodicity and Temporal Averages

562

(5)

The Power Spectral Density

567

(9)

Estimation of the ACS and PSD

576

(4)

Continuous-Time WSS Random Processes

580

(6)

Real-World Example - Random Vibration Testing

586

(11)

References

589

(1)

Problems

590

(7)

Linear Systems and Wide Sense Stationary Random Processes

597

(44)

Introduction

597

(1)

Summary

598

(1)

Random Process at Output of Linear System

598

(9)

Interpretation of the PSD

607

(2)

Wiener Filtering

609

(14)

Continuous-Time Definitions and Formulas

623

(3)

Real-World Example -- Speech Synthesis

626

(11)

References

630

(1)

Problems

631

(6)

Solution for Infinite Length Predictor

637

(4)

Multiple Wide Sense Stationary Random Processes

641

(32)

Introduction

641

(1)

Summary

642

(1)

Jointly Distributed WSS Random Processes

642

(5)

The Cross-Power Spectral Density

647

(5)

Transformations of Multiple Random Processes

652

(5)

Continuous-Time Definitions and Formulas

657

(4)

Cross-Correlation Sequence Estimation

661

(2)

Real-World Example -- Brain Physiology Research

663

(10)

References

667

(1)

Problems

667

(6)

Gaussian Random Processes

673

(38)

Introduction

673

(2)

Summary

675

(1)

Definition of the Gaussian Random Process

676

(5)

Linear Transformations

681

(2)

Nonlinear Transformations

683

(3)

Continuous-Time Definitions and Formulas

686

(3)

Special Continuous-Time Gaussian Random Processes

689

(7)

Computer Simulation

696

(2)

Real-World Example -- Estimating Fish Populations

698

(11)

References

701

(1)

Problems

702

(7)

MATLAB Listing for Figure 20.2

709

(2)

Poisson Random Processes

711

(28)

Introduction

711

(2)

Summary

713

(1)

Derivation of Poisson Counting Random Process

714

(4)

Interarrival Times

718

(3)

Arrival Times

721

(2)

Compound Poisson Random Process

723

(4)

Computer Simulation

727

(1)

Real-World Example -- Automobile Traffic Signal Planning

728

(9)

References

732

(1)

Problems

732

(5)

Joint PDF for Interarrival Times

737

(2)

Markov Chains

739

(38)

Introduction

739

(5)

Summary

744

(1)

Definitions

744

(4)

Computation of State Probabilities

748

(8)

Ergodic Markov Chains

756

(3)

Further Steady-State Characteristics

759

(3)

K-State Markov Chains

762

(2)

Computer Simulation

764

(1)

Real-World Example -- Strange Markov Chain Dynamics

765

(10)

References

767

(1)

Problems

767

(8)

Solving for the Stationary PMF

775

(2)

A. Glossary of Symbols and Abbrevations

777

(6)

B. Assorted Math Facts and Formulas

783

(6)

B.1 Proof by Induction

783

(1)

B.2 Trigonometry

784

(1)

B.3 Limits

784

(1)

B.4 Sums

785

(1)

B.5 Calculus

786

(3)

C. Linear and Matrix Algebra

789

(6)

C.1 Definitions

789

(2)

C.2 Special Matrices

791

(1)

C.3 Matrix Manipulation and Formulas

792

(1)

C.4 Some Properties of PD (PSD) Matrices

793

(1)

C.5 Eigendecomposition of Matrices

793

(2)

D. Summary of Signals, Linear Transforms, and Linear Systems

795

(14)

D.1 Discrete-Time Signals

795

(1)

D.2 Linear Transforms

796

(4)

D.3 Discrete-Time Linear Systems

800

(4)

D.4 Continuous-Time Signals

804

(1)

D.5 Linear Transforms

805

(2)

D.6 Continuous-Time Linear Systems

807

(2)

E. Answers to Selected Problems

809

(14)

Index

823

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Intuitive Probability And Random Processes Using Matlab

Summary

Author Biography

Table of Contents

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