Book description
Probability, Random Variables, and Random Processes is a comprehensive textbook on probability theory for engineers that provides a more rigorous mathematical framework than is usually encountered in undergraduate courses. It is intended for first-year graduate students who have some familiarity with probability and random variables, though not necessarily of random processes and systems that operate on random signals. It is also appropriate for advanced undergraduate students who have a strong mathematical background.
The book has the following features:
Several appendices include related material on integration, important inequalities and identities, frequency-domain transforms, and linear algebra. These topics have been included so that the book is relatively self-contained. One appendix contains an extensive summary of 33 random variables and their properties such as moments, characteristic functions, and entropy.
Unlike most books on probability, numerous figures have been included to clarify and expand upon important points. Over 600 illustrations and MATLAB plots have been designed to reinforce the material and illustrate the various characterizations and properties of random quantities.
Sufficient statistics are covered in detail, as is their connection to parameter estimation techniques. These include classical Bayesian estimation and several optimality criteria: mean-square error, mean-absolute error, maximum likelihood, method of moments, and least squares.
The last four chapters provide an introduction to several topics usually studied in subsequent engineering courses: communication systems and information theory; optimal filtering (Wiener and Kalman); adaptive filtering (FIR and IIR); and antenna beamforming, channel equalization, and direction finding. This material is available electronically at the companion website.
Probability, Random Variables, and Random Processes is the only textbook on probability for engineers that includes relevant background material, provides extensive summaries of key results, and extends various statistical techniques to a range of applications in signal processing.
Note: The ebook version does not provide access to the companion files.
Table of contents
- Cover
- Title Page
- Copyright
- Dedication
- Preface
- Notation
- Chapter 1: Overview and Background
-
Part I: Probability, Random Variables, and Expectation
-
Chapter 2: Probability Theory
- 2.1 INTRODUCTION
- 2.2 SETS AND SAMPLE SPACES
- 2.3 SET OPERATIONS
- 2.4 EVENTS AND FIELDS
- 2.5 SUMMARY OF A RANDOM EXPERIMENT
- 2.6 MEASURE THEORY
- 2.7 AXIOMS OF PROBABILITY
- 2.8 BASIC PROBABILITY RESULTS
- 2.9 CONDITIONAL PROBABILITY
- 2.10 INDEPENDENCE
- 2.11 BAYES' FORMULA
- 2.12 TOTAL PROBABILITY
- 2.13 DISCRETE SAMPLE SPACES
- 2.14 CONTINUOUS SAMPLE SPACES
- 2.15 NONMEASURABLE SUBSETS OF R
- PROBLEMS
- FURTHER READING
- Chapter 3: Random Variables
-
Chapter 4: Multiple Random Variables
- 4.1 INTRODUCTION
- 4.2 RANDOM VARIABLE APPROXIMATIONS
- 4.3 JOINT AND MARGINAL DISTRIBUTIONS
- 4.4 INDEPENDENT RANDOM VARIABLES
- 4.5 CONDITIONAL DISTRIBUTION
- 4.6 RANDOM VECTORS
- 4.7 GENERATING DEPENDENT RANDOM VARIABLES
- 4.8 RANDOM VARIABLE TRANSFORMATIONS
- 4.9 IMPORTANT FUNCTIONS OF TWO RANDOM VARIABLES
- 4.10 TRANSFORMATIONS OF RANDOM VARIABLE FAMILIES
- 4.11 TRANSFORMATIONS OF RANDOM VECTORS
- 4.12 SAMPLE MEAN X AND SAMPLE VARIANCE S_2
- 4.13 MINIMUM, MAXIMUM, AND ORDER STATISTICS
- 4.14 MIXTURES
- PROBLEMS
- FURTHER READING
-
Chapter 5: Expectation and Moments
- 5.1 INTRODUCTION
- 5.2 EXPECTATION AND INTEGRATION
- 5.3 INDICATOR RANDOM VARIABLE
- 5.4 SIMPLE RANDOM VARIABLE
- 5.5 EXPECTATION FOR DISCRETE SAMPLE SPACES
- 5.6 EXPECTATION FOR CONTINUOUS SAMPLE SPACES
- 5.7 SUMMARY OF EXPECTATION
- 5.8 FUNCTIONAL VIEW OF THE MEAN
- 5.9 PROPERTIES OF EXPECTATION
- 5.10 EXPECTATION OF A FUNCTION
- 5.11 CHARACTERISTIC FUNCTION
- 5.12 CONDITIONAL EXPECTATION
- 5.13 PROPERTIES OF CONDITIONAL EXPECTATION
- 5.14 LOCATION PARAMETERS: MEAN, MEDIAN, AND MODE
- 5.15 VARIANCE, COVARIANCE, AND CORRELATION
- 5.16 FUNCTIONAL VIEW OF THE VARIANCE
- 5.17 EXPECTATION AND THE INDICATOR FUNCTION
- 5.18 CORRELATION COEFFICIENTS
- 5.19 ORTHOGONALITY
- 5.20 CORRELATION AND COVARIANCE MATRICES
- 5.21 HIGHER ORDER MOMENTS AND CUMULANTS
- 5.22 FUNCTIONAL VIEW OF SKEWNESS
- 5.23 FUNCTIONAL VIEW OF KURTOSIS
- 5.24 GENERATING FUNCTIONS
- 5.25 FOURTH-ORDER GAUSSIAN MOMENT
- 5.26 EXPECTATIONS OF NONLINEAR TRANSFORMATIONS
- PROBLEMS
- FURTHER READING
-
Chapter 2: Probability Theory
-
Part II: Random Processes, Systems, and Parameter Estimation
-
Chapter 6: Random Processes
- 6.1 INTRODUCTION
- 6.2 CHARACTERIZATIONS OF A RANDOM PROCESS
- 6.3 CONSISTENCY AND EXTENSION
- 6.4 TYPES OF RANDOM PROCESSES
- 6.5 STATIONARITY
- 6.6 INDEPENDENT AND IDENTICALLY DISTRIBUTED
- 6.7 INDEPENDENT INCREMENTS
- 6.8 MARTINGALES
- 6.9 MARKOV SEQUENCE
- 6.10 MARKOV PROCESS
- 6.11 RANDOM SEQUENCES
- 6.12 RANDOM PROCESSES
- PROBLEMS
- FURTHER READING
-
Chapter 7: Stochastic Convergence, Calculus, and Decompositions
- 7.1 INTRODUCTION
- 7.2 STOCHASTIC CONVERGENCE
- 7.3 LAWS OF LARGE NUMBERS
- 7.4 CENTRAL LIMIT THEOREM
- 7.5 STOCHASTIC CONTINUITY
- 7.6 DERIVATIVES AND INTEGRALS
- 7.7 DIFFERENTIAL EQUATIONS
- 7.8 DIFFERENCE EQUATIONS
- 7.9 INNOVATIONS AND MEAN-SQUARE PREDICTABILITY
- 7.10 DOOB–MEYER DECOMPOSITION
- 7.11 KARHUNEN–LOÈVE EXPANSION
- PROBLEMS
- FURTHER READING
-
Chapter 8: Systems, Noise, and Spectrum Estimation
- 8.1 INTRODUCTION
- 8.2 CORRELATION REVISITED
- 8.3 ERGODICITY
- 8.4 EIGENFUNCTIONS OF R_XX(τ)
- 8.5 POWER SPECTRAL DENSITY
- 8.6 POWER SPECTRAL DISTRIBUTION
- 8.7 CROSS-POWER SPECTRAL DENSITY
- 8.8 SYSTEMS WITH RANDOM INPUTS
- 8.9 PASSBAND SIGNALS
- 8.10 WHITE NOISE
- 8.11 BANDWIDTH
- 8.12 SPECTRUM ESTIMATION
- 8.13 PARAMETRIC MODELS
- 8.14 SYSTEM IDENTIFICATION
- PROBLEMS
- FURTHER READING
-
Chapter 9: Sufficient Statistics and Parameter Estimation
- 9.1 INTRODUCTION
- 9.2 STATISTICS
- 9.3 SUFFICIENT STATISTICS
- 9.4 MINIMAL SUFFICIENT STATISTIC
- 9.5 EXPONENTIAL FAMILIES
- 9.6 LOCATION-SCALE FAMILIES
- 9.7 COMPLETE STATISTIC
- 9.8 RAO–BLACKWELL THEOREM
- 9.9 LEHMANN–SCHEFFÉ THEOREM
- 9.10 BAYES ESTIMATION
- 9.11 MEAN-SQUARE-ERROR ESTIMATION
- 9.12 MEAN-ABSOLUTE-ERROR ESTIMATION
- 9.13 ORTHOGONALITY CONDITION
- 9.14 PROPERTIES OF ESTIMATORS
- 9.15 MAXIMUM A POSTERIORI ESTIMATION
- 9.16 MAXIMUM LIKELIHOOD ESTIMATION
- 9.17 LIKELIHOOD RATIO TEST
- 9.18 EXPECTATION–MAXIMIZATION ALGORITHM
- 9.19 METHOD OF MOMENTS
- 9.20 LEAST-SQUARES ESTIMATION
- 9.21 PROPERTIES OF LS ESTIMATORS
- 9.22 BEST LINEAR UNBIASED ESTIMATION
- 9.23 PROPERTIES OF BLU ESTIMATORS
- PROBLEMS
- FURTHER READING
- A NOTE ON PART III OF THE BOOK
-
Chapter 6: Random Processes
-
Appendices
- Appendix A: Summaries of Univariate Parametric Distributions
-
Appendix B: Functions and Properties
- B.1 CONTINUITY AND BOUNDED VARIATION
- B.2 SUPREMUM AND INFIMUM
- B.3 ORDER NOTATION
- B.4 FLOOR AND CEILING FUNCTIONS
- B.5 CONVEX AND CONCAVE FUNCTIONS
- B.6 EVEN AND ODD FUNCTIONS
- B.7 SIGNUM FUNCTION
- B.8 DIRAC DELTA FUNCTION
- B.9 KRONECKER DELTA FUNCTION
- B.10 UNIT-STEP FUNCTIONS
- B.11 RECTANGLE FUNCTIONS
- B.12 TRIANGLE AND RAMP FUNCTIONS
- B.13 INDICATOR FUNCTIONS
- B.14 SINC FUNCTION
- B.15 LOGARITHM FUNCTIONS
- B.16 GAMMA FUNCTIONS
- B.17 BETA FUNCTIONS
- B.18 BESSEL FUNCTIONS
- B.19 Q-FUNCTION AND ERROR FUNCTIONS
- B.20 MARCUM Q-FUNCTION
- B.21 ZETA FUNCTION
- B.22 RISING AND FALLING FACTORIALS
- B.23 LAGUERRE POLYNOMIALS
- B.24 HYPERGEOMETRIC FUNCTIONS
- B.25 BERNOULLI NUMBERS
- B.26 HARMONIC NUMBERS
- B.27 EULER–MASCHERONI CONSTANT
- B.28 DIRICHLET FUNCTION
- FURTHER READING
- Appendix C: Frequency-Domain Transforms and Properties
- Appendix D: Integration and Integrals
- Appendix E: Identities and Infinite Series
- Appendix F: Inequalities and Bounds for Expectations
-
Appendix G: Matrix and Vector Properties
- G.1 BASIC PROPERTIES
- G.2 FOUR FUNDAMENTAL SUBSPACES
- G.3 EIGENDECOMPOSITION
- G.4 LU, LDU, AND CHOLESKY DECOMPOSITIONS
- G.5 JACOBIAN MATRIX AND THE JACOBIAN
- G.6 KRONECKER AND SCHUR PRODUCTS
- G.7 PROPERTIES OF TRACE AND DETERMINANT
- G.8 MATRIX INVERSION LEMMA
- G.9 CAUCHY–SCHWARZ INEQUALITY
- G.10 DIFFERENTIATION
- G.11 COMPLEX DIFFERENTIATION
- FURTHER READING
- Glossary
- References
- Index
Product information
- Title: Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
- Author(s):
- Release date: November 2012
- Publisher(s): Wiley-Interscience
- ISBN: 9780470242094
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