Probability, Statistics, and Random Processes for Engineers

Häftad, Engelska, 2012

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For courses in Probability and Random Processes. Probability, Statistics, and Random Processes for Engineers, 4e is a useful text for electrical and computer engineers. This book 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.

Produktinformation

Utgivningsdatum2012-03-02
Mått176 x 50 x 230 mm
Vikt1 320 g
FormatHäftad
SpråkEngelska
Antal sidor864
Upplaga4
FörlagPearson Education
ISBN9780132311236

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Preface 1 Introduction to Probability 11.1 Introduction: Why Study Probability? 11.2 The Different Kinds of Probability 2Probability as Intuition 2Probability as the Ratio of Favorable to Total Outcomes (Classical Theory) 3Probability as a Measure of Frequency of Occurrence 4Probability Based on an Axiomatic Theory 51.3 Misuses, Miscalculations, and Paradoxes in Probability 71.4 Sets, Fields, and Events 8Examples of Sample Spaces 81.5 Axiomatic Definition of Probability 151.6 Joint, Conditional, and Total Probabilities; Independence 20Compound Experiments 231.7 Bayes’ Theorem and Applications 351.8 Combinatorics 38Occupancy Problems 42Extensions and Applications 461.9 Bernoulli Trials–Binomial and Multinomial Probability Laws 48Multinomial Probability Law 541.10 Asymptotic Behavior of the Binomial Law: The Poisson Law 571.11 Normal Approximation to the Binomial Law 63Summary 65Problems 66References 772 Random Variables 792.1 Introduction 792.2 Definition of a Random Variable 802.3 Cumulative Distribution Function 83Properties of F X(x) 84Computation of F X(x) 852.4 Probability Density Function (pdf) 88Four Other Common Density Functions 95More Advanced Density Functions 972.5 Continuous, Discrete, and Mixed Random Variables 100Some Common Discrete Random Variables 1022.6 Conditional and Joint Distributions and Densities 107Properties of Joint CDF F XY (x, y) 1182.7 Failure Rates 137Summary 141Problems 141References 149Additional Reading 1493 Functions of Random Variables 1513.1 Introduction 151Functions of a Random Variable (FRV): Several Views 1543.2 Solving Problems of the Type Y = g(X) 155General Formula of Determining the pdf of Y = g(X) 1663.3 Solving Problems of the Type Z = g(X, Y ) 1713.4 Solving Problems of the Type V = g(X, Y ), W = h(X, Y ) 193Fundamental Problem 193Obtaining f VW Directly from f XY 1963.5 Additional Examples 200Summary 205Problems 206References 214Additional Reading 2144 Expectation and Moments 2154.1 Expected Value of a Random Variable 215On the Validity of Equation 4.1-8 2184.2 Conditional Expectations 232Conditional Expectation as a Random Variable 2394.3 Moments of Random Variables 242Joint Moments 246Properties of Uncorrelated Random Variables 248Jointly Gaussian Random Variables 2514.4 Chebyshev and Schwarz Inequalities 255Markov Inequality 257The Schwarz Inequality 2584.5 Moment-Generating Functions 2614.6 Chernoff Bound 2644.7 Characteristic Functions 266Joint Characteristic Functions 273The Central Limit Theorem 2764.8 Additional Examples 281Summary 283Problems 284References 293Additional Reading 2945 Random Vectors 2955.1 Joint Distribution and Densities 2955.2 Multiple Transformation of Random Variables 2995.3 Ordered Random Variables 302Distribution of area random variables 3055.4 Expectation Vectors and Covariance Matrices 3115.5 Properties of Covariance Matrices 314Whitening Transformation 3185.6 The Multidimensional Gaussian (Normal) Law 3195.7 Characteristic Functions of Random Vectors 328Properties of CF of Random Vectors 330The Characteristic Function of the Gaussian (Normal) Law 331Summary 332Problems 333References 339Additional Reading 3396 Statistics: Part 1 Parameter Estimation 3406.1 Introduction 340Independent, Identically Distributed (i.i.d.) Observations 341Estimation of Probabilities 3436.2 Estimators 3466.3 Estimation of the Mean 348Properties of the Mean-Estimator Function (MEF) 349Procedure for Getting a δ-confidence Interval on the Mean of a NormalRandom Variable When σ X Is Known 352Confidence Interval for the Mean of a Normal Distribution When σX Is NotKnown 352Procedure for Getting a δ-Confidence Interval Based on n Observations onthe Mean of a Normal Random Variable when σ X Is Not Known 355Interpretation of the Confidence Interval 3556.4 Estimation of the Variance and Covariance 355Confidence Interval for the Variance of a Normal Randomvariable 357Estimating the Standard Deviation Directly 359Estimating the covariance 3606.5 Simultaneous Estimation of Mean and Variance 3616.6 Estimation of Non-Gaussian Parameters from Large Samples 3636.7 Maximum Likelihood Estimators 3656.8 Ordering, more on Percentiles, Parametric Versus Nonparametric Statistics 369The Median of a Population Versus Its Mean 371Parametric versus Nonparametric Statistics 372Confidence Interval on the Percentile 373Confidence Interval for the Median When n Is Large 3756.9 Estimation of Vector Means and Covariance Matrices 376Estimation of μ 377Estimation of the covariance K 3786.10 Linear Estimation of Vector Parameters 380Summary 384Problems 384References 388Additional Reading 3897 Statistics: Part 2 Hypothesis Testing 3907.1 Bayesian Decision Theory 3917.2 Likelihood Ratio Test 3967.3 Composite Hypotheses 402Generalized Likelihood Ratio Test (GLRT) 403How Do We Test for the Equality of Means of Two Populations? 408Testing for the Equality of Variances for Normal Populations:The F-test 412Testing Whether the Variance of a Normal Population Has aPredetermined Value: 4167.4 Goodness of Fit 4177.5 Ordering, Percentiles, and Rank 423How Ordering is Useful in Estimating Percentiles and the Median 425Confidence Interval for the Median When n Is Large 428Distribution-free Hypothesis Testing: Testing If Two Population are theSame Using Runs 429Ranking Test for Sameness of Two Populations 432Summary 433Problems 433References 4398 Random Sequences 4418.1 Basic Concepts 442Infinite-length Bernoulli Trials 447Continuity of Probability Measure 452Statistical Specification of a Random Sequence 4548.2 Basic Principles of Discrete-Time Linear Systems 4718.3 Random Sequences and Linear Systems 4778.4 WSS Random Sequences 486Power Spectral Density 489Interpretation of the psd 490Synthesis of Random Sequences and Discrete-Time Simulation 493Decimation 496Interpolation 4978.5 Markov Random Sequences 500ARMA Models 503Markov Chains 5048.6 Vector Random Sequences and State Equations 5118.7 Convergence of Random Sequences 5138.8 Laws of Large Numbers 521Summary 526Problems 526References 5419 Random Processes 5439.1 Basic Definitions 5449.2 Some Important Random Processes 548Asynchronous Binary Signaling 548Poisson Counting Process 550Alternative Derivation of Poisson Process 555Random Telegraph Signal 557Digital Modulation Using Phase-Shift Keying 558Wiener Process or Brownian Motion 560Markov Random Processes 563Birth—Death Markov Chains 567Chapman—Kolmogorov Equations 571Random Process Generated from Random Sequences 5729.3 Continuous-Time Linear Systems with Random Inputs 572White Noise 5779.4 Some Useful Classifications of Random Processes 578Stationarity 5799.5 Wide-Sense Stationary Processes and LSI Systems 581Wide-Sense Stationary Case 582Power Spectral Density 584An Interpretation of the psd 586More on White Noise 590Stationary Processes and Differential Equations 5969.6 Periodic and Cyclostationary Processes 6009.7 Vector Processes and State Equations 606State Equations 608Summary 611Problems 611References 633Chapters 10 and 11 are available as Web chapters on the companionWeb site at http://www.pearsonhighered.com/stark.10 Advanced Topics in Random Processes 63510.1 Mean-Square (m.s.) Calculus 635Stochastic Continuity and Derivatives [10-1] 635Further Results on m.s. Convergence [10-1] 64510.2 Mean-Square Stochastic Integrals 65010.3 Mean-Square Stochastic Differential Equations 65310.4 Ergodicity [10-3] 65810.5 Karhunen—Lo`eve Expansion [10-5] 66510.6 Representation of Bandlimited and Periodic Processes 671Bandlimited Processes 671Bandpass Random Processes 674WSS Periodic Processes 677Fourier Series for WSS Processes 680Summary 682Appendix: Integral Equations 682Existence Theorem 683Problems 686References 69911 Applications to Statistical Signal Processing 70011.1 Estimation of Random Variables and Vectors 700More on the Conditional Mean 706Orthogonality and Linear Estimation 708Some Properties of the Operator ˆE 71611.2 Innovation Sequences and Kalman Filtering 718Predicting Gaussian Random Sequences 722Kalman Predictor and Filter 724Error-Covariance Equations 72911.3 Wiener Filters for Random Sequences 733Unrealizable Case (Smoothing) 734Causal Wiener Filter 73611.4 Expectation-Maximization Algorithm 738Log-likelihood for the Linear Transformation 740Summary of the E-M algorithm 742E-M Algorithm for Exponential ProbabilityFunctions 743Application to Emission Tomography 744Log-likelihood Function of Complete Data 746E-step 747M-step 74811.5 Hidden Markov Models (HMM) 749Specification of an HMM 751Application to Speech Processing 753Efficient Computation of P[E | M] with a RecursiveAlgorithm 754Viterbi Algorithm and the Most Likely State Sequencefor the Observations 75611.6 Spectral Estimation 759The Periodogram 760Bartlett’s Procedure---Averaging Periodograms 762Parametric Spectral Estimate 767Maximum Entropy Spectral Density 76911.7 Simulated Annealing 772Gibbs Sampler 773Noncausal Gauss—Markov Models 774Compound Markov Models 778Gibbs Line Sequence 779Summary 783Problems 783References 788Appendix A Review of Relevant Mathematics A-1A.1 Basic Mathematics A-1Sequences A-1Convergence A-2Summations A-3Z-Transform A-3A.2 Continuous Mathematics A-4Definite and Indefinite Integrals A-5Differentiation of Integrals A-6Integration by Parts A-7Completing the Square A-7Double Integration A-8Functions A-8A.3 Residue Method for Inverse Fourier Transformation A-10Fact A-11Inverse Fourier Transform for psd of Random Sequence A-13A.4 Mathematical Induction A-17References A-17Appendix B Gamma and Delta Functions B-1B.1 Gamma Function B-1B.2 Incomplete Gamma Function B-2B.3 Dirac Delta Function B-2References B-5Appendix C Functional Transformations and Jacobians C-1C.1 Introduction C-1C.2 Jacobians for n = 2 C-2C.3 Jacobian for General n C-4Appendix D Measure and Probability D-1D.1 Introduction and Basic Ideas D-1Measurable Mappings and Functions D-3D.2 Application of Measure Theory to Probability D-3Distribution Measure D-4Appendix E Sampled Analog Waveforms and Discrete-time Signals E-1Appendix F Independence of Sample Mean and Variance for NormalRandom Variables F-1Appendix G Tables of Cumulative Distribution Functions: the Normal,Student t, Chi-square, and F G-1Index I-1