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Hand-Book on STATISTICAL DISTRIBUTIONS for Experimentalists

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Hand-Book on STATISTICAL DISTRIBUTIONS for Experimentalists Internal Report SUF–PFY/96–01 Stockholm, 11 December 1996 1st revision, 31 October 1998 last modification 10 September 2007 Hand-book on STATISTICAL DISTRIBUTIONS for experimentalists by Christian Walck Particle Physics Group Fysikum University of Stockholm (e-mail: [email protected]) Contents 1 Introduction 1 1.1 Random Number Generation .............................. 1 2 Probability Density Functions 3 2.1 Introduction ........................................ 3 2.2 Moments ......................................... 3 2.2.1 Errors of Moments ................................ 4 2.3 Characteristic Function ................................. 4 2.4 Probability Generating Function ............................ 5 2.5 Cumulants ......................................... 6 2.6 Random Number Generation .............................. 7 2.6.1 Cumulative Technique .............................. 7 2.6.2 Accept-Reject technique ............................. 7 2.6.3 Composition Techniques ............................. 8 2.7 Multivariate Distributions ................................ 9 2.7.1 Multivariate Moments .............................. 9 2.7.2 Errors of Bivariate Moments .......................... 9 2.7.3 Joint Characteristic Function .......................... 10 2.7.4 Random Number Generation .......................... 11 3 Bernoulli Distribution 12 3.1 Introduction ........................................ 12 3.2 Relation to Other Distributions ............................. 12 4 Beta distribution 13 4.1 Introduction ........................................ 13 4.2 Derivation of the Beta Distribution ........................... 13 4.3 Characteristic Function ................................. 14 4.4 Moments ......................................... 14 4.5 Probability Content ................................... 14 4.6 Random Number Generation .............................. 15 5 Binomial Distribution 16 5.1 Introduction ........................................ 16 5.2 Moments ......................................... 16 5.3 Probability Generating Function ............................ 16 5.4 Cumulative Function ................................... 17 5.5 Random Number Generation .............................. 17 5.6 Estimation of Parameters ................................ 17 5.7 Probability Content ................................... 18 6 Binormal Distribution 20 6.1 Introduction ........................................ 20 6.2 Conditional Probability Density ............................. 21 6.3 Characteristic Function ................................. 21 6.4 Moments ......................................... 21 6.5 Box-Muller Transformation ............................... 22 6.6 Probability Content ................................... 23 i 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 ii 11 Doubly Non-Central F -Distribution 49 11.1 Introduction ........................................ 49 11.2 Moments ......................................... 49 11.3 Cumulative Distribution ................................. 50 11.4 Random Number Generation .............................. 50 12 Doubly Non-Central t-Distribution 51 12.1 Introduction ........................................ 51 12.2 Moments ......................................... 51 12.3 Cumulative Distribution ................................. 52 12.4 Random Number Generation .............................. 52 13 Error Function 53 13.1 Introduction ........................................ 53 13.2 Probability Density Function .............................. 53 14 Exponential Distribution 54 14.1 Introduction ........................................ 54 14.2 Cumulative Function ................................... 54 14.3 Moments ......................................... 54 14.4 Characteristic Function ................................. 54 14.5 Random Number Generation .............................. 55 14.5.1 Method by von Neumann ............................ 55 14.5.2 Method by Marsaglia .............................. 55 14.5.3 Method by Ahrens ................................ 56 15 Extreme Value Distribution 57 15.1 Introduction ........................................ 57 15.2 Cumulative Distribution ................................. 58 15.3 Characteristic Function ................................. 58 15.4 Moments ......................................... 58 15.5 Random Number Generation .............................. 60 16 F-distribution 61 16.1 Introduction ........................................ 61 16.2 Relations to Other Distributions ............................ 62 16.3 1/F ............................................ 62 16.4 Characteristic Function ................................. 62 16.5 Moments ......................................... 62 16.6 F-ratio ........................................... 63 16.7 Variance Ratio ...................................... 64 16.8 Analysis of Variance ................................... 65 16.9 Calculation of Probability Content ........................... 65 16.9.1 The Incomplete Beta function .......................... 66 16.9.2 Final Formulæ .................................. 67 16.10 Random Number Generation .............................. 68 iii 17 Gamma Distribution 69 17.1 Introduction ........................................ 69 17.2 Derivation of the Gamma Distribution ......................... 69 17.3 Moments ......................................... 70 17.4 Characteristic Function ................................. 70 17.5 Probability Content ................................... 71 17.6 Random Number Generation .............................. 71 17.6.1 Erlangian distribution .............................. 71 17.6.2 General case ................................... 71 17.6.3 Asymptotic Approximation ........................... 72 18 Generalized Gamma Distribution 73 18.1 Introduction ........................................ 73 18.2 Cumulative Function ................................... 73 18.3 Moments ......................................... 74 18.4 Relation to Other Distributions ............................. 74 19 Geometric Distribution 75 19.1 Introduction ........................................ 75 19.2 Moments ......................................... 75 19.3 Probability Generating Function ............................ 75 19.4 Random Number Generation .............................. 75 20 Hyperexponential Distribution 77 20.1 Introduction ........................................ 77 20.2 Moments ......................................... 77 20.3 Characteristic Function ................................. 77 20.4 Random Number Generation .............................. 78 21 Hypergeometric Distribution 79 21.1 Introduction ........................................ 79 21.2 Probability Generating Function ............................ 79 21.3 Moments ......................................... 79 21.4 Random Number Generation .............................. 80 22 Logarithmic Distribution 81 22.1 Introduction .......................................
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