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Probability and Random Processes for Electrical and Computer Engineers

Book Description

The theory of probability is a powerful tool that helps electrical and computer engineers to explain, model, analyze, and design the technology they develop. The text begins at the advanced undergraduate level, assuming only a modest knowledge of probability, and progresses through more complex topics mastered at graduate level. The first five chapters cover the basics of probability and both discrete and continuous random variables. The later chapters have a more specialized coverage, including random vectors, Gaussian random vectors, random processes, Markov Chains, and convergence. Describing tools and results that are used extensively in the field, this is more than a textbook; it is also a reference for researchers working in communications, signal processing, and computer network traffic analysis. With over 300 worked examples, some 800 homework problems, and sections for exam preparation, this is an essential companion for advanced undergraduate and graduate students. Further resources for this title, including solutions (for Instructors only), are available online at www.cambridge.org/9780521864701.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Chapter dependencies
  8. Preface
  9. 1. Introduction to probability
    1. 1.1 Sample spaces, outcomes, and events
    2. 1.2 Review of set notation
    3. 1.3 Probability models
    4. 1.4 Axioms and properties of probability
    5. 1.5 Conditional probability
    6. 1.6 Independence
    7. 1.7 Combinatorics and probability
    8. Notes
    9. Problems
    10. Exam preparation
  10. 2. Introduction to discrete random variables
    1. 2.1 Probabilities involving random variables
    2. 2.2 Discrete random variables
    3. 2.3 Multiple random variables
    4. 2.4 Expectation
    5. Notes
    6. Problems
    7. Exam preparation
  11. 3. More about discrete random variables
    1. 3.1 Probability generating functions
    2. 3.2 The binomial random variable
    3. 3.3 The weak law of large numbers
    4. 3.4 Conditional probability
    5. 3.5 Conditional expectation
    6. Notes
    7. Problems
    8. Exam preparation
  12. 4. Continuous random variables
    1. 4.1 Densities and probabilities
    2. 4.2 Expectation of a single random variable
    3. 4.3 Transform methods
    4. 4.4 Expectation of multiple random variables
    5. 4.5 Probability bounds
    6. Notes
    7. Problems
    8. Exam preparation
  13. 5. Cumulative distribution functions and their applications
    1. 5.1 Continuous random variables
    2. 5.2 Discrete random variables
    3. 5.3 Mixed random variables
    4. 5.4 Functions of random variables and their cdfs
    5. 5.5 Properties of cdfs
    6. 5.6 The central limit theorem
    7. 5.7 Reliability
    8. Notes
    9. Problems
    10. Exam preparation
  14. 6. Statistics
    1. 6.1 Parameter estimators and their properties
    2. 6.2 Histograms
    3. 6.3 Confidence intervals for the mean – known variance
    4. 6.4 Confidence intervals for the mean – unknown variance
    5. 6.5 Confidence intervals for Gaussian data
    6. 6.6 Hypothesis tests for the mean
    7. 6.7 Regression and curve fitting
    8. 6.8 Monte Carlo estimation
    9. Notes
    10. Problems
    11. Exam preparation
  15. 7. Bivariate random variables
    1. 7.1 Joint and marginal probabilities
    2. 7.2 Jointly continuous random variables
    3. 7.3 Conditional probability and expectation
    4. 7.4 The bivariate normal
    5. 7.5 Extension to three or more random variables
    6. Notes
    7. Problems
    8. Exam preparation
  16. 8. Introduction to random vectors
    1. 8.1 Review of matrix operations
    2. 8.2 Random vectors and random matrices
    3. 8.3 Transformations of random vectors
    4. 8.4 Linear estimation of random vectors (Wiener filters)
    5. 8.5 Estimation of covariance matrices
    6. 8.6 Nonlinear estimation of random vectors
    7. Notes
    8. Problems
    9. Exam preparation
  17. 9. Gaussian random vectors
    1. 9.1 Introduction
    2. 9.2 Definition of the multivariate Gaussian
    3. 9.3 Characteristic function
    4. 9.4 Density function
    5. 9.5 Conditional expectation and conditional probability
    6. 9.6 Complex random variables and vectors
    7. Notes
    8. Problems
    9. Exam preparation
  18. 10. Introduction to random processes
    1. 10.1 Definition and examples
    2. 10.2 Characterization of random processes
    3. 10.3 Strict-sense and wide-sense stationary processes
    4. 10.4 WSS processes through LTI systems
    5. 10.5 Power spectral densities for WSS processes
    6. 10.6 Characterization of correlation functions
    7. 10.7 The matched filter
    8. 10.8 The Wiener filter
    9. 10.9 The Wiener–Khinchin theorem
    10. 10.10 Mean-square ergodic theorem for WSS processes
    11. 10.11 Power spectral densities for non-WSS processes
    12. Notes
    13. Problems
    14. Exam preparation
  19. 11. Advanced concepts in random processes
    1. 11.1 The Poisson process
    2. 11.2 Renewal processes
    3. 11.3 The Wiener process
    4. 11.4 Specification of random processes
    5. Notes
    6. Problems
    7. Exam preparation
  20. 12. Introduction to Markov chains
    1. 12.1 Preliminary results
    2. 12.2 Discrete-time Markov chains
    3. 12.3 Recurrent and transient states
    4. 12.4 Limiting n-step transition probabilities
    5. 12.5 Continuous-time Markov chains
    6. Notes
    7. Problems
    8. Exam preparation
  21. 13. Mean convergence and applications
    1. 13.1 Convergence in mean of order p
    2. 13.2 Normed vector spaces of random variables
    3. 13.3 The Karhunen–Loève expansion
    4. 13.4 The Wiener integral (again)
    5. 13.5 Projections, orthogonality principle, projection theorem
    6. 13.6 Conditional expectation and probability
    7. 13.7 The spectral representation
    8. Notes
    9. Problems
    10. Exam preparation
  22. 14. Other modes of convergence
    1. 14.1 Convergence in probability
    2. 14.2 Convergence in distribution
    3. 14.3 Almost-sure convergence
    4. Notes
    5. Problems
    6. Exam preparation
  23. 15. Self similarity and long-range dependence
    1. 15.1 Self similarity in continuous time
    2. 15.2 Self similarity in discrete time
    3. 15.3 Asymptotic second-order self similarity
    4. 15.4 Long-range dependence
    5. 15.5 ARMA processes
    6. 15.6 ARIMA processes
    7. Problems
    8. Exam preparation
  24. Bibliography
  25. Index