Probability Distributions Involving Gaussian Random Variables: A Handbook for Engineers and Scientists (The Springer International Series in Engineering and Computer Science (683))

"Probability Distributions Involving Gaussian Random Variables: A Handbook for Engineers and Scientists brings together a vast and comprehensive collection of mathematical material in one location, as well as offering a variety of new results interpreted in a form that is particularly useful to engineers and scientists."--BOOK JACKET. Contents......Page 11 A Brief Biography of Carl Friedrich Gauss......Page 17 Preface......Page 24 Acknowledgment......Page 25 INTRODUCTION......Page 26 1. BASIC DEFINITIONS AND NOTATION......Page 29 B. Rayleigh......Page 33 C. Rician......Page 35 D. Central Chi-square......Page 37 E. Noncentral Chi-square......Page 38 F. Log-Normal......Page 39 A. Gaussian......Page 41 B. Rayleigh......Page 42 C. Rician......Page 44 D. Central Chi-Square......Page 45 E. Noncentral Chi-Square......Page 46 F. Log-Normal......Page 47 A. Independent Central Chi-Square (-) Central Chi-Square......Page 48 B. Dependent Central Chi-Square (-) Central Chi-Square......Page 51 C. Independent Noncentral Chi-Square (-) Central Chi-Square......Page 53 D. Independent Central Chi-Square (-) Noncentral Chi-Square......Page 56 E. Independent Noncentral Chi-Square (-) Noncentral Chi-Square......Page 57 5. SUM OF CHI-SQUARE RANDOM VARIABLES......Page 58 A. Independent Central Chi-Square (+) Central Chi-Square......Page 59 B. Dependent Central Chi-Square (+) Central Chi-Square......Page 63 C. Independent Noncentral Chi-Square (+) Central Chi-Square......Page 66 D. Independent Noncentral Chi-Square (+) Noncentral Chi-Square......Page 69 A. Independent Gaussian (×) Gaussian (Both Have Zero Mean)......Page 71 B. Dependent Gaussian (×) Gaussian (Both Have Zero Mean)......Page 73 C. Independent Gaussian (×) Gaussian (One Has Zero Mean, Both Have Identical Variance)......Page 75 E. Independent Rayleigh (×) Rayleigh......Page 77 F. Dependent Rayleigh (×) Rayleigh......Page 79 G. Independent Rice (×) Rayleigh......Page 80 H. Independent Rice (×) Rice......Page 81 I. Dependent Rayleigh Products......Page 82 B. Independent Gaussian (÷) Gaussian (One Has Zero Mean)......Page 83 C. Independent Gaussian (÷) Gaussian (Both Have Nonzero Mean)......Page 84 E. Dependent Gaussian (÷) Gaussian (One Has Zero Mean)......Page 85 G. Independent Gaussian (Zero Mean) (÷) Rayleigh......Page 86 H. Independent Gaussian (Zero Mean) (÷) Rice......Page 88 I. Independent Rayleigh (÷) Rayleigh......Page 89 J. Dependent Rayleigh (÷) Rayleigh......Page 92 K. Independent Rice (÷) Rayleigh......Page 94 L. Independent Rice (÷) Rice......Page 96 M. Dependent Rayleigh Ratios......Page 99 8. MAXIMUM AND MINIMUM OF PAIRS OF RANDOM VARIABLES......Page 100 B. Dependent Gaussian......Page 101 C. Independent Rayleigh......Page 102 D. Dependent Rayleigh......Page 104 E. Independent Log-Normal......Page 106 F. Dependent Log-Normal......Page 107 9. QUADRATIC FORMS......Page 109 A. Both Vectors Have Zero Mean......Page 111 B. One or Both Vectors Have Nonzero Mean......Page 112 C. A Reduced Quadratic Form Where the Vectors Have Different Numbers of Dimensions......Page 114 D. General Hermetian Quadratic Forms......Page 116 C. Independent Gaussian (×) Rayleigh (+) Gaussian......Page 119 E. General Products of Ratios of Independent Gaussians......Page 120 A. The Gaussian Q-Function......Page 122 B. The Marcum Q-Function......Page 123 C. The Nuttall Q-Function......Page 130 D. The Complementary Incomplete Gamma Function......Page 132 A. The Gaussian Q-Function......Page 133 1. THE GAUSSIAN Q-FUNCTION......Page 136 B. Q-Function with Exponentials and x......Page 137 B. Q-Function with One Linear Argument and Exponentials......Page 138 C. Q-Function with One Linear Argument and x......Page 139 D. Q-Function with One Linear Argument, Exponentials and Powers of x......Page 140 E. Q-Function with One Linear Argument, Bessel Functions, Exponentials and Powers of x......Page 142 F. Product of Two Q-Functions with One Linear Argument......Page 144 H. Q-Function with Two Linear Arguments, Exponentials and x......Page 145 B. Q-Function with One Linear Argument, Exponentials and Powers of x......Page 146 C. Q-Function with One Linear Argument, Bessel Functions, Exponentials and Powers of x......Page 147 D. Q-Function with Two Linear Arguments, Exponentials and Powers of x......Page 148 1. THE GAUSSIAN Q-FUNCTION......Page 150 2. THE MARCUM Q-FUNCTION......Page 154 REFERENCES......Page 158 ILLUSTRATIONS......Page 161 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. 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. Brings together a comprehensive collection of mathematical material in one location. This book also offers a variety of results interpreted in a form that is particularly useful to engineers, scientists, and applied mathematicians