Probability distributions involving Gaussian random variables: a handbook for engineers and scientists

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