6.7 Random Number Generation .............................. 24
7 Cauchy Distribution 26
7.1 Introduction ........................................ 26
7.2 Moments ......................................... 26
7.3 Normalization ....................................... 27
7.4 Characteristic Function ................................. 27
7.5 Location and Scale Parameters ............................. 27
7.6 Breit-Wigner Distribution ................................ 28
7.7 Comparison to Other Distributions ........................... 28
7.8 Truncation ........................................ 29
7.9 Sum and Average of Cauchy Variables ......................... 30
7.10 Estimation of the Median ................................ 31
7.11 Estimation of the HWHM ................................ 31
7.12 Random Number Generation .............................. 33
7.13 Physical Picture ..................................... 34
7.14 Ratio Between Two Standard Normal Variables .................... 35
8 Chi-square Distribution 36
8.1 Introduction ........................................ 36
8.2 Moments ......................................... 36
8.3 Characteristic Function ................................. 37
8.4 Cumulative Function ................................... 38
8.5 Origin of the Chi-square Distribution .......................... 38
8.6 Approximations ...................................... 39
8.7 Random Number Generation .............................. 39
8.8 Confidence Intervals for the Variance .......................... 40
8.9 Hypothesis Testing .................................... 40
8.10 Probability Content ................................... 40
8.11 Even Number of Degrees of Freedom .......................... 42
8.12 Odd Number of Degrees of Freedom .......................... 42
8.13 Final Algorithm ...................................... 43
8.14 Chi Distribution ..................................... 43
9 Compound Poisson Distribution 45
9.1 Introduction ........................................ 45
9.2 Branching Process .................................... 45
9.3 Moments ......................................... 45
9.4 Probability Generating Function ............................ 45
9.5 Random Number Generation .............................. 46
10 Double-Exponential Distribution 47
10.1 Introduction ........................................ 47
10.2 Moments ......................................... 47
10.3 Characteristic Function ................................. 47
10.4 Cumulative Function ................................... 48
10.5 Random Number Generation .............................. 48
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