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Probability, Statistics, and Random Processes for Electrical Engineering.pdf

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Contents
Preface ix
CHAPTER 1 Probability Models in Electrical
and Computer Engineering 1
1.1 Mathematical Models as Tools in Analysis and Design 2
1.2 Deterministic Models 4
1.3 Probability Models 4
1.4 A Detailed Example: A Packet Voice Transmission System 9
1.5 Other Examples 11
1.6 Overview of Book 16
Summary 17
Problems 18
CHAPTER 2 Basic Concepts of Probability Theory 21
2.1 Specifying Random Experiments 21
2.2 The Axioms of Probability 30
2.3 Computing Probabilities Using Counting Methods 41
2.4 Conditional Probability 47
2.5 Independence of Events 53
2.6 Sequential Experiments 59
2.7 Synthesizing Randomness: Random Number Generators 67
2.8 Fine Points: Event Classes 70
2.9 Fine Points: Probabilities of Sequences of Events 75
Summary 79
Problems 80
CHAPTER 3 Discrete Random Variables 96
3.1 The Notion of a Random Variable 96
3.2 Discrete Random Variables and Probability Mass Function 99
3.3 Expected Value and Moments of Discrete Random Variable 104
3.4 Conditional Probability Mass Function 111
3.5 Important Discrete Random Variables 115
3.6 Generation of Discrete Random Variables 127
Summary 129
Problems 130
*
*
*
*
vi Contents
CHAPTER 4 One Random Variable 141
4.1 The Cumulative Distribution Function 141
4.2 The Probability Density Function 148
4.3 The Expected Value of X 155
4.4 Important Continuous Random Variables 163
4.5 Functions of a Random Variable 174
4.6 The Markov and Chebyshev Inequalities 181
4.7 Transform Methods 184
4.8 Basic Reliability Calculations 189
4.9 Computer Methods for Generating Random Variables 194
4.10 Entropy 202
Summary 213
Problems 215
CHAPTER 5 Pairs of Random Variables 233
5.1 Two Random Variables 233
5.2 Pairs of Discrete Random Variables 236
5.3 The Joint cdf of X and Y 242
5.4 The Joint pdf of Two Continuous Random Variables 248
5.5 Independence of Two Random Variables 254
5.6 Joint Moments and Expected Values of a Function of Two Random
Variables 257
5.7 Conditional Probability and Conditional Expectation 261
5.8 Functions of Two Random Variables 271
5.9 Pairs of Jointly Gaussian Random Variables 278
5.10 Generating Independent Gaussian Random Variables 284
Summary 286
Problems 288
CHAPTER 6 Vector Random Variables 303
6.1 Vector Random Variables 303
6.2 Functions of Several Random Variables 309
6.3 Expected Values of Vector Random Variables 318
6.4 Jointly Gaussian Random Vectors 325
6.5 Estimation of Random Variables 332
6.6 Generating Correlated Vector Random Variables 342
Summary 346
Problems 348
*
Contents vii
CHAPTER 7 Sums of Random Variables and Long-Term Averages 359
7.1 Sums of Random Variables 360
7.2 The Sample Mean and the Laws of Large Numbers 365
Weak Law of Large Numbers 367
Strong Law of Large Numbers 368
7.3 The Central Limit Theorem 369
Central Limit Theorem 370
7.4 Convergence of Sequences of Random Variables 378
7.5 Long-Term Arrival Rates and Associated Averages 387
7.6 Calculating Distribution’s Using the Discrete Fourier
Transform 392
Summary 400
Problems 402
CHAPTER 8 Statistics 411
8.1 Samples and Sampling Distributions 411
8.2 Parameter Estimation 415
8.3 Maximum Likelihood Estimation 419
8.4 Confidence Intervals 430
8.5 Hypothesis Testing 441
8.6 Bayesian Decision Methods 455
8.7 Testing the Fit of a Distribution to Data 462
Summary 469
Problems 471
CHAPTER 9 Random Processes 487
9.1 Definition of a Random Process 488
9.2 Specifying a Random Process 491
9.3 Discrete-Time Processes: Sum Process, Binomial Counting Process,
and Random Walk 498
9.4 Poisson and Associated Random Processes 507
9.5 Gaussian Random Processes, Wiener Process
and Brownian Motion 514
9.6 Stationary Random Processes 518
9.7 Continuity, Derivatives, and Integrals of Random Processes 529
9.8 Time Averages of Random Processes and Ergodic Theorems 540
9.9 Fourier Series and Karhunen-Loeve Expansion 544
9.10 Generating Random Processes 550
Summary 554
Problems 557
*
*
*
viii Contents
CHAPTER 10 Analysis and Processing of Random Signals 577
10.1 Power Spectral Density 577
10.2 Response of Linear Systems to Random Signals 587
10.3 Bandlimited Random Processes 597
10.4 Optimum Linear Systems 605
10.5 The Kalman Filter 617
10.6 Estimating the Power Spectral Density 622
10.7 Numerical Techniques for Processing Random Signals 628
Summary 633
Problems 635
CHAPTER 11 Markov Chains 647
11.1 Markov Processes 647
11.2 Discrete-Time Markov Chains 650
11.3 Classes of States, Recurrence Properties, and Limiting
Probabilities 660
11.4 Continuous-Time Markov Chains 673
11.5 Time-Reversed Markov Chains 686
11.6 Numerical Techniques for Markov Chains 692
Summary 700
Problems 702
CHAPTER 12 Introduction to Queueing Theory 713
12.1 The Elements of a Queueing System 714
12.2 Little’s Formula 715
12.3 The M/M/1 Queue 718
12.4 Multi-Server Systems: M/M/c, M/M/c/c, And 727
12.5 Finite-Source Queueing Systems 734
12.6 M/G/1 Queueing Systems 738
12.7 M/G/1 Analysis Using Embedded Markov Chains 745
12.8 Burke’s Theorem: Departures From M/M/c Systems 754
12.9 Networks of Queues: Jackson’s Theorem 758
12.10 Simulation and Data Analysis of Queueing Systems 771
Summary 782
Problems 784
Appendices
A. Mathematical Tables 797
B. Tables of Fourier Transforms 800
C. Matrices and Linear Algebra 802
Index 805

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