Probability and Random Processes with Applications to Signal Processing: International Edition

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.

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 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