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Random process for Engineers, Bruce Hajek

Random

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process

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Random process for Engineers is written by Bruce Hajek at UIUC, which is useful for the learning of stochastic process.

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Random Processes for

Engineers

Bruce Hajek

Illinois

This is a preproduction copy of the text of the same title published by Cambridge

University Press, March 2015. See http://www.cambridge.org/us/academic/subjects/

engineering/communications-and-signal-processing/random-processes-engineers

The book supercedes “Notes for ECE 534: An Exploration of Random Processes for

Engineers.” Cambridge University Press has kindly allowed the author to make this

version of the book freely available on his webpage. It does not incorporate ﬁnal sets of

edits and corrections. Permission is hereby given to freely print and circulate copies of

these notes so long as the notes are left intact and not reproduced for commercial

purposes. Email to b-hajek@illinois.edu, pointing out errors or hard to understand

passages or providing comments, is welcome.

To Beth, for her loving support.

Contents

Preface page vii

1 A Selective Review of Basic Probability 1

1.1 The axioms of probability theory 1

1.2 Independence and conditional probability 5

1.3 Random variables and their distribution 8

1.4 Functions of a random variable 11

1.5 Expectation of a random variable 17

1.6 Frequently used distributions 22

1.7 Failure rate functions 25

1.8 Jointly distributed random variables 26

1.9 Conditional densities 28

1.10 Correlation and covariance 28

1.11 Transformation of random vectors 30

2 Convergence of a Sequence of Random Variables 42

2.1 Four deﬁnitions of convergence of random variables 42

2.2 Cauchy criteria for convergence of random variables 54

2.3 Limit theorems for sums of independent random variables 58

2.4 Convex functions and Jensen’s inequality 61

2.5 Chernoﬀ bound and large deviations theory 62

3 Random Vectors and Minimum Mean Squared Error Estimation 77

3.1 Basic deﬁnitions and properties 77

3.2 The orthogonality principle for minimum mean square error

estimation 79

3.3 Conditional expectation and linear estimators 83

3.3.1 Conditional expectation as a projection 83

3.3.2 Linear estimators 85

3.3.3 Comparison of the estimators 86

3.4 Joint Gaussian distribution and Gaussian random vectors 89

3.5 Linear innovations sequences 95

3.6 Discrete-time Kalman ﬁltering 96

Contents v

4 Random Processes 109

4.1 Deﬁnition of a random process 109

4.2 Random walks and gambler’s ruin 112

4.3 Processes with independent increments and martingales 115

4.4 Brownian motion 116

4.5 Counting processes and the Poisson process 118

4.6 Stationarity 121

4.7 Joint properties of random processes 124

4.8 Conditional independence and Markov processes 125

4.9 Discrete-state Markov processes 129

4.10 Space-time structure of discrete-state Markov processes 136

5 Inference for Markov Models 151

5.1 A bit of estimation theory 151

5.2 The expectation-maximization (EM) algorithm 156

5.3 Hidden Markov models 161

5.3.1 Posterior state probabilities and the forward-backward

algorithm 162

5.3.2 Most likely state sequence – Viterbi algorithm 166

5.3.3 The Baum-Welch algorithm, or EM algorithm for HMM 167

5.4 Notes 169

6 Dynamics of Countable-State Markov Models 177

6.1 Examples with ﬁnite state space 177

6.2 Classiﬁcation and convergence of discrete-time Markov processes 179

6.3 Classiﬁcation and convergence of continuous-time Markov processes 182

6.4 Classiﬁcation of birth-death processes 185

6.5 Time averages vs. statistical averages 187

6.6 Queueing systems, M/M/1 queue and Little’s law 189

6.7 Mean arrival rate, distributions seen by arrivals, and PASTA 192

6.8 More examples of queueing systems modeled as Markov birth-

death processes 194

6.9 Foster-Lyapunov stability criterion and moment bounds 196

6.9.1 Stability criteria for discrete-time processes 197

6.9.2 Stability criteria for continuous time processes 205

7 Basic Calculus of Random Processes 218

7.1 Continuity of random processes 218

7.2 Mean square diﬀerentiation of random processes 224

7.3 Integration of random processes 229

7.4 Ergodicity 236

7.5 Complexiﬁcation, Part I 242

7.6 The Karhunen-Lo`eve expansion 244

7.7 Periodic WSS random processes 252

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the solution of Probability, Statistics, and Random Processes for Engineers

Bruce Hajek's notes on Random processes

工程师随机过程的探索An Exploration of Random Processes for Engineers

Probability+,+Statistics+,+and+Random+Processes+for+Engineers

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