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Library of Congress Cataloging-in-Publication Data Andrews, Larry C. Field guide to probability, random processes, and random data analysis / Larry C. Andrews, Ronald L. Phillips. p. cm. – (Field guide series) Includes bibliographical references and index. ISBN 978-0-8194-8701-8 1. Mathematical analysis. 2. Probabilities. 3. Random data (Statistics) I. Phillips, Ronald L. II. Title. QA300.A5583 2012 519.2—dc23 2011051386 Published by SPIE P.O. Box 10 Bellingham, Washington 98227-0010 USA Phone: +1.360.676.3290 Fax: +1.360.647.1445 Email: books@spie.org Web: http://spie.org The content of this book reflects the work and thoughts of the author. Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon. Printed in the United States of America. First printing Introduction to the SeriesWelcome to the SPIE Field Guides—a series of publications written directly for the practicing engineer or scientist. Many textbooks and professional reference books cover optical principles and techniques in depth. The aim of the SPIE Field Guides is to distill this information, providing readers with a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. A significant effort will be made to provide a consistent notation and style between volumes in the series. Each SPIE Field Guide addresses a major field of optical science and technology. The concept of these Field Guides is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page, and provides full coverage of that topic on that page. Highlights, insights, and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each Field Guide provide additional information such as related material outside the main scope of the volume, key mathematical relationships, and alternative methods. While complete in their coverage, the concise presentation may not be appropriate for those new to the field. The SPIE Field Guides are intended to be living documents. The modular page-based presentation format allows them to be easily updated and expanded. We are interested in your suggestions for new Field Guide topics as well as what material should be added to an individual volume to make these Field Guides more useful to you. Please contact us at fieldguides@SPIE.org. John E. Greivenkamp, Series Editor College of Optical Sciences The University of Arizona The Field Guide SeriesKeep information at your fingertips with all of the titles in the Field Guide Series: Field Guide to Adaptive Optics, Robert Tyson & Benjamin Frazier Atmospheric Optics, Larry Andrews Binoculars and Scopes, Paul Yoder, Jr. & Daniel Vukobratovich Diffractive Optics, Yakov G. Soskind Geometrical Optics, John Greivenkamp Illumination, Angelo Arecchi, Tahar Messadi, & John Koshel Infrared Systems, Detectors, and FPAs, Second Edition, Arnold Daniels Interferometric Optical Testing, Eric Goodwin & Jim Wyant Laser Pulse Generation, Rüdiger Paschotta Lasers, Rüdiger Paschotta Microscopy, Tomasz Tkaczyk Optical Fabrication, Ray Williamson Optical Fiber Technology, Rüdiger Paschotta Optical Lithography, Chris Mack Optical Thin Films, Ronald Willey Polarization, Edward Collett Radiometry, Barbara Grant Special Functions for Engineers, Larry Andrews Spectroscopy, David Ball Visual and Ophthalmic Optics, Jim Schwiegerling Field Guide to Probability, Random Processes, and Random Data AnalysisDeveloped in basic courses in engineering and science, mathematical theory usually involves deterministic phenomena. Such is the case for solving a differential equation that describes a linear system where both input and output are deterministic quantities. In practice, however, the input to a linear system, such as imaging or radar systems, can contain a “random” quantity that yields uncertainty about the output. Such systems must be treated by probabilistic rather than deterministic methods. For this reason, probability theory and random-process theory have become indispensable tools in the mathematical analysis of these kinds of engineering systems. Topics included in this Field Guide are basic probability theory, random processes, random fields, and random data analysis. The analysis of random data is less well known than the other topics, particularly some of the tests for stationarity, periodicity, and normality. Much of the material is condensed from the authors’ earlier text Mathematical Techniques for Engineers and Scientists (SPIE Press, 2003). As is the case for other volumes in this series, it is assumed that the reader has some basic knowledge of the subject. Larry C. Andrews Professor Emeritus Townes Laser Institute CREOL College of Optics University of Central Florida Ronald L. Phillips Professor Emeritus Townes Laser Institute CREOL College of Optics University of Central Florida Table of ContentsGlossary of Symbols and Notation x Probability: One Random Variable 1 Terms and Axioms 2 Random Variables and Cumulative Distribution 3 Probability Density Function 4 Expected Value: Moments 5 Example: Expected Value 6 Expected Value: Characteristic Function 7 Gaussian or Normal Distribution 8 Other Examples of PDFs: Continuous RV 9 Other Examples of PDFs: Discrete RV 12 Chebyshev Inequality 13 Law of Large Numbers 14 Functions of One RV 15 Example: Square-Law Device 16 Example: Half-Wave Rectifier 17 Conditional Probabilities 18 Conditional Probability: Independent Events 19 Conditional CDF and PDF 20 Conditional Expected Values 21 Example: Conditional Expected Value 22 Probability: Two Random Variables 23 Joint and Marginal Cumulative Distributions 24 Joint and Marginal Density Functions 25 Conditional Distributions and Density Functions 26 Example: Conditional PDF 27 Principle of Maximum Likelihood 28 Independent RVs 29 Expected Value: Moments 30 Example: Expected Value 31 Bivariate Gaussian Distribution 32 Example: Rician Distribution 33 Functions of Two RVs 34 Sum of Two RVs 35 Product and Quotient of Two RVs 36 Conditional Expectations and Mean-Square Estimation 37 Sums of N Complex Random Variables 38 Central Limit Theorem 39 Example: Central Limit Theorem 40 Phases Uniformly Distributed on (−π, π) 41 Phases Not Uniformly Distributed on (−π, π) 42 Example: Phases Uniformly Distributed on (− α, α) 43 Central Limit Theorem Does Not Apply 45 Example: Non-Gaussian Limit 46 Random Processes 48 Random Processes Terminology 49 First- and Second-Order Statistics 50 Stationary Random Processes 51 Autocorrelation and Autocovariance Functions 52 Wide-Sense Stationary Process 53 Example: Correlation and PDF 54 Time Averages and Ergodicity 55 Structure Functions 56 Cross-Correlation and Cross-Covariance Functions 57 Power Spectral Density 58 Example: PSD 59 PSD Estimation 60 Bivariate Gaussian Processes 61 Multivariate Gaussian Processes 62 Examples of Covariance Function and PSD 63 Interpretations of Statistical Averages 64 Random Fields 65 Random Fields Terminology 66 Mean and Spatial Covariance Functions 67 1D and 3D Spatial Power Spectrums 68 2D Spatial Power Spectrum 69 Structure Functions 70 Example: PSD 71 Transformations of Random Processes 72 Memoryless Nonlinear Transformations 73 Linear Systems 74 Expected Values of a Linear System 75 Example: White Noise 76 Detection Devices 77 Zero-Crossing Problem 78 Random Data Analysis 79 Tests for Stationarity, Periodicity, and Normality 80 Nonstationary Data Analysis for Mean 81 Analysis for Single Time Record 82 Runs Test for Stationarity 83 Equation Summary 85 Biography 90 Index 91 Glossary of Symbols and Notationa, x, u, etc. Random variable, process, or field Bu(R) Autocovariance or covariance function of random field Cx(τ) Autocovariance or covariance function of random process C xy(τ) Cross-covariance function CDF Cumulative distribution function Cov Covariance Dx(τ) Structure function E[.] Expectation operator E[g(x)|A] Conditional expectation operator fx(x), fx(x, t) Probability density function fx(x|A) Conditional probability density Fx(x), Fx(x, t) Cumulative distribution function Fx(x|A) Conditional cumulative distribution function Generalized hypergeometric function h(t) Impulse response function H(ω) Transfer function Ip(x) Modified Bessel function of the first kind Jp(x) Bessel function of the first kind Kp(x) Modified Bessel function of the second kind m, m(t) Mean value mk k’th standard statistical moment n! Factorial function Probability density function Pr Probability Pr(B|A) Conditional probability PSD Power spectral density RV Random variable Rx(τ) Autocorrelation or correlation function Rxy(τ) Cross-correlation function Long-time-average correlation function Sx(ω), Su(κ) Power spectral density function U(x−a) Unit step function Var Variance Var[x|A] Conditional variance Time average Complex conjugate of z γ(c, x) Incomplete gamma function Γ(x) Gamma function δ(x−a) Dirac delta function (impulse function) μk k’th central statistical moment Estimator of mean value Variance τ Time difference t2−t1 Φx(s) Characteristic function || Absolute value ∈ Belonging to Binomial coefficient 〈 〉 Ensemble average { } Event Intersection |
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Data analysis
Spectral data processing
Data processing
Adaptive optics
Atmospheric optics
Geometrical optics
Probability theory