Probability Distributions Involving Gaussian Random Variables

1. Front Matter

   Pages i-xxiii

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2. Introduction

   Pages 1-3

3. Basic Definitions and Notation

   Pages 5-8

4. Fundamental One-Dimensional Variables

   Pages 9-16

5. Fundamental Multidimensional Variables

   Pages 17-23

6. Difference of Chi-Square Random Variables

   Pages 25-34

7. Sum of Chi-Square Random Variables

   Pages 35-47

8. Products of Random Variables

   Pages 49-60

9. Ratios of Random Variables

   Pages 61-77

10. Maximum and Minimum of Pairs of Random Variables

    Pages 79-87

11. Quadratic Forms

    Pages 89-98

12. Other Miscellaneous Forms

    Pages 99-101

13. Back Matter

    Pages 103-200

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This book is intended for use by students, academicians and practicing engineers who in the course of their daily study or research have need for the probability distributions and associated statistics of random variables that are themselves Gaussian or in various forms derived from them. The format of the book is primarily that of a handbook in that, for the most part, the results are merely presented in their final form without derivation or discussion. As such the reader must rely on the typographical accuracy of the documented expressions, which the author has taken great pains to assure. Also included at the end of the book are numerous curves illustrating the behavior of a variety of the probability measures presented in mathematical form. The author wishes to acknowledge his many colleagues in industry and academia for the encouragement and support they provided for this project without which it might never have gotten started. INTRODUCTION There are certain reference works that engineers and scientists alike find invaluable in their day-to-day work activities. Many of these reference volumes are of a generic nature such as tables of integrals, tables of series, handbooks of mathematical formulas and transforms, etc. (see Refs. 1, 2, 3, and 4 for example), whereas others are collections of technical papers and textbooks that directly relate to the individual's specific field of specialty.

• MATLAB
• Mathematica
• Probability distribution
• RSI
• Random variable
• Signal
• communication
• electrical engineering
• image processing
• modeling
• optimization

This handbook, now available in paperback, brings together a comprehensive collection of mathematical material in one location. It also offers a variety of new results interpreted in a form that is particularly useful to engineers, scientists, and applied mathematicians. The handbook is not specific to fixed research areas, but rather it has a generic flavor that can be applied by anyone working with probabilistic and stochastic analysis and modeling.

• Classic results are presented in their final form without derivation or discussion, allowing for much material to be condensed into one volume.
• Concise compilation of disparate formulae saves time in searching different sources
• Focused application has broad interest for many disciplines: engineers, computer scientists, statisticians, physicists; as well as any researcher working in probabilistic and stochastic analysis and modeling in the natural or social sciences.
• The material is timeless and has intrinsic value to today’s and tomorrow’s practicing engineers and scientists
• Endorsed by top experts in field worldwide, ensuring quality and value

Excerpts from reviews of the hardbound edition:

The reference book "Probability Distributions Involving Gaussian Random Variables," authored by Dr. Marvin Simon, has become, in a very short time frame, one of the most useful aids to research in the field of digital communications that has come out in many years. It has replaced various other well known resources because of its concentration of relevant, timely, and easily accessible results. —Dr. Larry Milstein, UCSD

This is a unique book and an invaluable reference for engineers and scientists in the fields of electrical engineering, mathematics and statistics. There is no other single reference book that covers Gaussian andGaussian-related distributions of random variables that are encountered in research work and practice. This is a reference book that I recommend to all my graduate students working in the area of telecommunication system analysis and design. —Dr. John Proakis, Northeastern University and UCSD

… this book collected all these results (after double-checking them) with unified notations and in an organized fashion made it very useful for many of the graduate students and researchers doing analytical type of investigations. Not only did I use it, I also recommend it to my graduate students and many of our papers benefited from the formulas available in the handbook…. The handbook saves valuable time for all the researchers that rely on it to simplify their results or write them in terms of well-known functions that are available in popular mathematical packages such as Matlab and Mathematica. —Dr. Mohamed-Slim Alouini, Texas A&M University

The distributions and associated statistics described in the book apply to systems like mobile and cellular communications, vehicular technology, radar systems, spread spectrum systems, and optical communication systems. The book, with its comprehensive information in analytical, tabular, and graphical form, is an invaluable tool for scientists and engineers. —Dr. Sergio Benedetto, Politecnico di Torino

The book "Probability Distributions Involving Gaussian Random Variables" is a handy research reference in areas such as communication systems.  I have found the book useful for my own work, since it presents probability distributions that are difficult to find elsewhere and that have non-obvious derivations. —Dr. Gordon Stüber, Georgia Institute of Technology

A unique and invaluable reference for researchers in communications, signal processing, math, statistics, and many other scientific fields. I have used this book in my own research and alsorecommended it to colleagues in universities and in industry, as well as to graduate students. —Dr. Andrea Goldsmith, Stanford University

In my research in wireless communications, I have often found that I can get all the mathematical forms and material that I need from one source, where before I had to consult five or six references to get the complete information. There are a small number of reference works that have proved so invaluable that I have purchased a home copy in addition to my office copy. This handbook is one of them. —Dr. Norman C. Beaulieu, University of Alberta

Our research Group at the University of Canterbury is engaged in both theoretical and experimental work on advanced wireless communications systems. We have found that the book by Simon, "Probability Distributions Involving Gaussian Random Variables" is an indispensable and often used reference for this work. We do significant amounts of analysis involving Gaussian statistics and quadratic forms. The book is the one source that brings almost all of the necessary background material together in a readily usable form and is thus an extremely valuable reference. —Dr. Desmond P. Taylor, University of Canterbury

Dr. Simon's book is by far the most comprehensive compilation of Gaussian probability distributions and their associated statistics - in fact to my knowledge this is the only book of its type in publication now. This handbook is a much needed and most welcome entry to the toolbox of any engineer or scientist. For those working in digital and wireless communications, signal and image processing, pattern recognition, and many other technical fields, the text is absolutely essential. —Payman Arabshahi, Applied Physics Laboratory

As a researcher and teacher in the digital communication theory area for the last forty years, I can state that Simon’s book contains materials in his ten chapters and appendices mostrelevant to researchers and developers of modern communication systems. These sophisticated Gaussian related expressions are needed in the analysis, design, and optimization of modern microwave and cellular communications. Many of these results are unavailable elsewhere, since these results were originated by Simon himself. —Dr. Kung Yao, UCLA

After I spent a few minutes trying to come up with my own derivation, I decided to consult Probability Distributions Involving Gaussian Random Variables, where I found the needed result within a few seconds. On this and many other occasions, I have found Marvin Simon's book to be a unique and extremely valuable resource for my own research. The book is even more crucial for the graduate students who, unlike those in my generation, did not grow up with the Gaussian Q function, Marcum's Q-function, Rayleigh and Rician distributions, chi-squared random variables, and all the other important functions, distributions, and concepts that have their roots in Gaussian random variables. —Dr. Michael B. Pursley, Clemson University

Because the source of randomness and noise is usually modeled as Gaussian, engineers and scientists often need the probability distributions of functions of Gaussian random variables. Marvin Simon has written an extremely useful handbook for everyone working in these fields. It is by far the most complete and easy-to-use collection of these characterizations of functions of Gaussian random variables. …an important and very popular book that is useful to a very large community of engineers and scientists. This work is also very carefully done with many corrections to earlier works that were not easily available to this community. —Jim Omura, Technology Strategist, Gordon and Betty Moore Foundation

The handbook "Probability Distributions Involving Gaussian Random Variables" by Marvin K. Simon is a wonderful resource for both theoreticians andpractitioners in that it contains a wealth of information summarized in one book. —Dr. Costas N. Georghiades, Texas A&M University

• Jet Propulsion Laboratory, Pasadena, USA

  Marvin K. Simon

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