Probability and Random Processes with Applications to Signal Processing

Henry Stark / John W. Woods  
Total pages
December 2011
Related Titles


For courses in Probability and Random Processes.

Probability, Statistics, and Random Processes for Engineers, 4e is a comprehensive treatment of probability and random processes that, more than any other available source, combines rigor with accessibility. Beginning with the fundamentals of probability theory and requiring only college-level calculus, the book develops all the tools needed to understand more advanced topics such as random sequences, continuous-time random processes, and statistical signal processing. The book progresses at a leisurely pace, never assuming more knowledge than contained in the material already covered. Rigor is established by developing all results from the basic axioms and carefully defining and discussing such advanced notions as stochastic convergence, stochastic integrals and resolution of stochastic processes.


  • This text combines rigor and accessibility (i.e. less mathematical than the measure theory approach but more rigorous than formula and recipe manuals) with a strong self-teaching orientation.
  • A large number of worked-out examples, MATLAB codes, and special appendices that include a review of the kind of basic math needed for solving problems in probability as well as an introduction to measure theory and its relation to probability
  • Many self-test multiple-choice exams are now available for students at the book web site: These exams were administered to senior undergraduate and graduate students at the Illinois Institute of Technology during the tenure of one of the authors who taught there from 1988-2006. The web site also includes an extensive set of small MATLAB programs that illustrate the concepts of probability.
  • Two new chapters on elementary statistics and their applications to real world problems. The first of these deals with parameter estimation and the second with hypothesis testing. The amount of added material on statistics will certainly satisfy the requirements of those curricula that offer courses that stress both probability and statistics in a one-semester course. The origin and applications of standard statistical tools such as the t-test, the Chi-square test, the F-test , order statistics, and percentiles are presented and discussed with detailed examples and end-of-chapter problems.

New to this Edition

  • A new chapter on a branch of statistics called parameter estimation with many illustrative examples;
  • A new chapter on a branch of statistics called hypothesis testing with many illustrative examples;
  • A large number of new homework problems of varying degrees of difficulty to test the student’s mastery of the principles of statistics;
  • A large number of self-test, multiple-choice, exam questions calibrated to the material in various chapters;
  • Many additional illustrative examples drawn from real-world situations where the principles of probability and statistics have useful applications;
  • A greater involvement of computers as teaching/learning aids such as (i) graphical displays of probabilistic phenomena; (ii) MATLAB programs to illustrate probabilistic concepts; (iii) homework problems requiring the use of MATLAB/ Excel to realize probability and statistical theory;
  • Numerous revised discussions–based on student feedback– meant to facilitate the understanding of difficult concepts.

Table of Contents


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 FX(x) 84

Computation of FX(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 FXY (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 fVW Directly from fXY 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

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