Field Guide To Continuous Probability Distributions
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Field Guide to Continuous Probability Distributions Gavin E. Crooks v 0.12 2019 G. E. Crooks – Field Guide to Probability Distributions v 0.12 Copyright © 2010-2019 Gavin E. Crooks http://threeplusone.com/fieldguide https://github.com/gecrooks/fieldguide typeset on 2019-02-23 with XeTeX version 0.99999 fonts: Trump Mediaeval (text), Euler (math) 27182818284590 2 G. E. Crooks – Field Guide to Probability Distributions Preface: The search for GUD A common problem is that of describing the probability distribution of a single, continuous variable. A few distributions, such as the normal and exponential, were discovered in the 1800’s or earlier. But about a century ago the great statistician, Karl Pearson, realized that the known probability distributions were not sufficient to handle all of the phenomena then under investigation, and set out to create new distributions with useful properties. During the 20th century this process continued with abandon and a vast menagerie of distinct mathematical forms were discovered and invented, investigated, analyzed, rediscovered and renamed, all for the purpose of describing the probability of some interesting variable. There are hundreds of named distributions and synonyms in current usage. The apparent diversity is unending and disorienting. Fortunately, the situation is less confused than it might at first appear. Most common, continuous, univariate, unimodal distributions can be organized into a small number of distinct families, which are all special cases of a single Grand Unified Distribution. This compendium details these hundred or so simple distributions, their properties and their interrelations. Gavin E. Crooks 3 G. E. Crooks – Field Guide to Probability Distributions Acknowledgments In curating this collection of distributions, I have benefited greatly from Johnson, Kotz, and Balakrishnan’s monumental compendiums [2, 3], Eric Weisstein’s MathWorld, the Leemis chart of Univariate Distribution Relationships [8, 9], and myriad pseudo-anonymous contributors to Wikipedia. Additional contributions are noted in the version history below. Version History 0.12 (2019-02-23) Added Porter-Thomas (7.5), Epanechnikov (12.9), biweight (12.10), triweight (12.11), Libby-Novick (20.9), Gauss hypergeometric (20.10), confluent hypergeometric (20.11) Johnson SU (21.10), and log-Cauchy (21.12) distributions. 0.11 (2017-06-19) Added hyperbola (20.5), Halphen (20.4), Halphen B (20.6), inverse Halphen B (20.7), generalized Halphen (20.12), Sichel (20.8) and Appell Beta (20.13) distributions. Thanks to Saralees Nadarajah. 0.10 (2017-02-08) Added K (21.8) and generalized K (21.5) distributions. Clarified notation and nomenclature. Thanks to Harish Vangala. 0.9 (2016-10-18) Added pseudo Voigt (21.17), and Student’s t3 (9.4) distributions. Reparameterized hyperbolic sine (14.3) distribution. Renamed inverse Burr to Dagum (18.4). Derived limit of unit gamma to log-normal (p68). Corrected spelling of “arrises” (sharp edges formed by the meeting of surfaces) to “arises” (emerge; become apparent). 0.8 (2016-08-30) The Unprincipled edition: Added Moyal distribution (8.8), a special case of the gamma-exponential distribution. Corrected spelling of “principle” to “principal”. Thanks to Matthew Hankins and Mara Averick. 0.7 (2016-04-05) Added Hohlfeld distribution. Added appendix on limits. Reformatted and rationalized distribution hierarchy diagrams. Thanks to Phill Geissler. 0.6 (2014-12-22) Total of 147 named simple, unimodal, univariate, continuous probability distributions, and at least as many synonyms. Added appendix on the algebra of random variables. Added Box-Muller transformation. For the gammaexponential distribution, switched the sign on the parameter α. Fixed the relation between beta distributions and ratios of gamma distributions (α and γ were switched in most cases). Thanks to Fabian Krüger and Lawrence Leemis. 0.5 (2013-07-01) Added uniform product, half generalized Pearson VII, half exponential power distributions, GUD and q-Type distributions. Moved Pearson IV to 4 own section. Fixed errors in Inverse Gaussian. Added random variate generation to appendix. Thanks to David Sivak, Dieter Grientschnig, Srividya Iyer-Biswas, and Shervin Fatehi. 0.4 (2012-03-01) Added erratics. Moved gamma distribution to own section. Renamed log-gamma to gamma-exponential. Added permalink. Added new tree of distributions. Thanks to David Sivak and Frederik Beaujean. 0.3 (2011-06-40) Added tree of distributions. 0.2 (2011-03-01) Expanded families. Thanks to David Sivak. 0.1 (2011-01-16) Initial release. Organize over 100 simple, continuous, univariate probability distributions into 14 families. Greatly expands on previous paper that discussed the Amoroso and gamma-exponential families [10]. Thanks to David Sivak, Edward E. Ayoub, and Francis J. O’Brien. G. E. Crooks – Field Guide to Probability Distributions 5 6 G. E. Crooks – Field Guide to Probability Distributions Endorsements “Ridiculously useful” – Mara Averick1 “I can’t stress how useful I’ve found this. I wish I’d had a printout of it by my desk every day for the last 6 years”– Guillermo Roditi Dominguez2 “Abramowitz and Stegun for probability distributions”– Kranthi K. Mandadapu3 “I had no idea how much I needed this guide.”– Daniel J. Harris4 “Who are you? How did you get in my house?” – Donald Knuth5 1 https://twitter.com/dataandme/status/770732084872810496 https://twitter.com/groditi/status/772266190190194688 3 Thursday Lunch with Scientists 4 https://twitter.com/DHarrisPsyc/status/870614354529370112 5 https://xkcd.com/163/ 2 G. E. Crooks – Field Guide to Probability Distributions 7 G. E. Crooks – Field Guide to Probability Distributions Contents Preface: The search for GUD 3 Acknowledgments & Version History 4 Contents 8 List of figures 17 List of tables 18 Distribution hierarchies Hierarchy of principal distributions Pearson distributions . . . . . . . . Extreme order statistics . . . . . . . Symmetric simple distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 20 21 22 23 Zero shape parameters 1 Uniform Distribution Uniform . . . . . . Special cases . . . . . . . Half uniform . . . . Unbounded uniform Degenerate . . . . . Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 24 24 24 24 24 24 2 Exponential Distribution Exponential . . . . . . Special cases . . . . . . . . Anchored exponential Standard exponential Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 28 28 28 28 28 3 Laplace Distribution Laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Laplace . . . . . . . . . . . . . . . . . . . . . . . . 31 31 31 31 8 Contents Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Normal Distribution Normal . . . . . Special cases . . . . . Error function . Standard normal Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 . . . . . 34 34 34 34 34 34 One shape parameter 5 Power Function Distribution Power function . . . . . . Alternative parameterizations Generalized Pareto . . . . q-exponential . . . . . . . Special cases: Positive β . . . Pearson IX . . . . . . . . Pearson VIII . . . . . . . . Wedge . . . . . . . . . . . Ascending wedge . . . . . Descending wedge . . . . Special cases: Negative β . . . Pareto . . . . . . . . . . . Lomax . . . . . . . . . . . Exponential ratio . . . . . Uniform-prime . . . . . . Limits and subfamilies . . . . Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 37 37 37 37 38 38 38 38 38 38 38 38 40 41 41 41 42 6 Log-Normal Distribution Log-normal . . . . . . Special cases . . . . . . . . Anchored log-normal Gibrat . . . . . . . . . Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 45 45 45 45 45 . . . . . . . . . . G. E. Crooks – Field Guide to Probability Distributions 9 Contents 7 Gamma Distribution Gamma . . . . . . Special cases . . . . . . Wein . . . . . . . Erlang . . . . . . . Standard gamma . Chi-square . . . . Scaled chi-square . Porter-Thomas . . Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 48 48 48 48 48 49 50 51 52 8 Gamma-Exponential Distribution Gamma-exponential . . . . . Special cases . . . . . . . . . . . . Standard gamma-exponential Chi-square-exponential . . . Generalized Gumbel . . . . . Gumbel . . . . . . . . . . . . Standard Gumbel . . . . . . BHP . . . . . . . . . . . . . . Moyal . . . . . . . . . . . . . Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 54 54 54 55 55 55 57 58 58 59 9 Pearson VII Distribution Pearson VII . . . . . . . . Special cases . . . . . . . . . . Student’s t . . . . . . . . Student’s t2 . . . . . . . . Student’s t3 . . . . . . . . Student’s z . . . . . . . . Cauchy . . . . . . . . . . Standard Cauchy . . . . . Relativistic Breit-Wigner Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 60 60 60 61 62 62 63 63 64 64 . . . . . . . . . . . . . . . . . . . . Two shape parameters 10 G. E. Crooks – Field Guide to Probability Distributions Contents 10 Unit Gamma Distribution Unit gamma . . . . Special cases . . . . . . . Uniform product . . Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 67 67 67 67 11 Amoroso Distribution Amoroso . . . . . . . . . . . . . . Special cases: Miscellaneous . . . . . . Stacy . . . . . . . . . . . . . . . . Pseudo-Weibull . . . . . . . . . . Half exponential power . . . . . . Hohlfeld . . . . . . . . . . . . . . Special cases: Positive integer β . . . . Nakagami . . . . . . . . . . . . . . Half normal . . . . . . . . . . . . Chi . . . . . . . . . . . . . . . . . Scaled chi . . . . . . . . . . . . . . Rayleigh . . . . . . . . . . . . . . Maxwell . . . . . . . . . . . . . . Wilson-Hilferty . . . . . . . . . . Special cases: Negative integer β . . . Inverse gamma . . . . . . . . . . . Inverse exponential . . . . . . . . Lévy . . . . . . . . . . . . . . . . . Scaled inverse chi-square . . . . . Inverse chi-square . . . . . . . . . Scaled inverse chi . . . . . . . . . Inverse chi . . . . . . . . . . . . . Inverse Rayleigh . . . . . . . . . . Special cases: Extreme order statistics Generalized Fisher-Tippett . . . . Fisher-Tippett . . . . . . . . . . . Generalized Weibull . . . . . . . . Weibull . . . . . . . . . . . . . . . Reversed Weibull . . . . . . . . . Generalized Fréchet . . . . . . . . Fréchet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 72 72 72 74 74 75 75 76 76 76 77 78 78 78 79 79 79 79 80 81 81 81 82 83 83 83 84 84 85 85 85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. E. Crooks – Field Guide to Probability Distributions 11 Contents Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Beta Distribution Beta . . . . . . Special cases . . . . U-shaped beta J-shaped beta . Standard beta . Pert . . . . . . Pearson XII . . Pearson II . . . Arcsine . . . . Central arcsine Semicircle . . Epanechnikov Biweight . . . Triweight . . . Interrelations . . . 87 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 88 88 88 88 88 89 90 92 92 93 93 93 94 94 94 13 Beta Prime Distribution Beta prime . . . . . Special cases . . . . . . . Standard beta prime F . . . . . . . . . . . Inverse Lomax . . . Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 96 96 96 97 98 98 14 Beta-Exponential Distribution Beta-exponential . . . . . . Standard beta-exponential Special cases . . . . . . . . . . . Exponentiated exponential Hyperbolic sine . . . . . . Nadarajah-Kotz . . . . . . . Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 101 101 101 101 103 103 104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Beta-Logistic Distribution 107 Beta-Logistic . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Standard Beta-Logistic . . . . . . . . . . . . . . . . . . . . . 107 12 G. E. Crooks – Field Guide to Probability Distributions Contents Special cases . . . . . . . . . . Burr type II . . . . . . . . Reversed Burr type II . . Symmetric Beta-Logistic Logistic . . . . . . . . . . Hyperbolic secant . . . . Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 107 108 108 110 110 111 16 Pearson IV Distribution 113 Pearson IV . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Three (or more) shape parameters 17 Generalized Beta Distribution Generalized beta . . . . Special Cases . . . . . . . . . Kumaraswamy . . . . . Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 116 116 116 119 18 Gen. Beta Prime Distribution Generalized beta prime . . . Special cases . . . . . . . . . . . . Transformed beta . . . . . . Burr . . . . . . . . . . . . . . Dagum . . . . . . . . . . . . Paralogistic . . . . . . . . . . Inverse paralogistic . . . . . Log-logistic . . . . . . . . . . Half-Pearson VII . . . . . . . Half-Cauchy . . . . . . . . . Half generalized Pearson VII Half Laha . . . . . . . . . . . Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 121 121 121 121 122 124 124 124 125 125 126 126 126 . . . . . . . . 19 Pearson Distribution 128 Pearson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 q-Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 G. E. Crooks – Field Guide to Probability Distributions 13 Contents 20 Grand Unified Distribution Special cases . . . . . . . . . . . . . . Extended Pearson . . . . . . . . Inverse Gaussian . . . . . . . . . Halphen . . . . . . . . . . . . . . Hyperbola . . . . . . . . . . . . . Halphen B . . . . . . . . . . . . Inverse Halphen B . . . . . . . . Sichel . . . . . . . . . . . . . . . Libby-Novick . . . . . . . . . . . Gauss hypergeometric . . . . . . Confluent hypergeometric . . . Generalized Halphen . . . . . . Greater Grand Unified Distributions Appell Beta . . . . . . . . . . . . Laha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 132 132 132 135 135 135 136 136 136 137 137 137 138 138 138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 139 139 140 140 141 141 142 142 143 143 144 144 144 145 145 145 146 146 Miscellanea 21 Miscellaneous Distributions Bates . . . . . . . . . . Beta-Fisher-Tippett . . Birnbaum-Saunders . . Exponential power . . . Generalized K . . . . . Generalized Pearson VII Holtsmark . . . . . . . K. . . . . . . . . . . . . Irwin-Hall . . . . . . . Johnson SU . . . . . . . Landau . . . . . . . . . Log-Cauchy . . . . . . . Meridian . . . . . . . . Noncentral chi-square . Non-central F . . . . . Pseudo Voigt . . . . . . Rice . . . . . . . . . . . Slash . . . . . . . . . . 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. E. Crooks – Field Guide to Probability Distributions Contents Stable . . . . . . . . Suzuki . . . . . . . Triangular . . . . . Uniform difference Voigt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 148 148 148 148 Appendix A Notation and Nomenclature 149 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 B Properties of Distributions 152 C Order statistics 157 Order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Extreme order statistics . . . . . . . . . . . . . . . . . . . . . . . 158 Median statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 158 D Limits Exponential function limit Logarithmic function limit Gaussian function limit . Miscellaneous limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 160 161 161 162 E Algebra of Random Variables Transformations . . . . . . Combinations . . . . . . . Transmutations . . . . . . Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 164 166 169 170 F Miscellaneous mathematics 171 Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Bibliography 177 Index of distributions 191 Subject Index 201 G. E. Crooks – Field Guide to Probability Distributions 15 List of Figures List of Figures 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 16 Hierarchy of principal distributions . . . . . . . . . . Hierarchy of principal Pearson distributions . . . . . Extreme order statistics . . . . . . . . . . . . . . . . Hierarchies of symmetric simple distributions . . . . Uniform distribution . . . . . . . . . . . . . . . . . . Standard exponential distribution . . . . . . . . . . . Standard Laplace distribution . . . . . . . . . . . . . Normal distributions . . . . . . . . . . . . . . . . . . Pearson IX distributions . . . . . . . . . . . . . . . . Pearson VIII distributions . . . . . . . . . . . . . . . Pareto distributions . . . . . . . . . . . . . . . . . . . Log normal distributions . . . . . . . . . . . . . . . . Gamma distributions, unit variance . . . . . . . . . . Chi-square distributions . . . . . . . . . . . . . . . . Gamma exponential distributions . . . . . . . . . . . Gumbel distribution . . . . . . . . . . . . . . . . . . Student’s t distributions . . . . . . . . . . . . . . . . Standard Cauchy distribution . . . . . . . . . . . . . Unit gamma, finite support. . . . . . . . . . . . . . . Unit gamma, semi-infinite support. . . . . . . . . . . Gamma, scaled chi and Wilson-Hilferty distributions Half normal, Rayleigh and Maxwell distributions . . Inverse gamma and scaled inverse-chi distributions . Extreme value distributions of maxima. . . . . . . . Beta distribution . . . . . . . . . . . . . . . . . . . . Pearson XII distribution . . . . . . . . . . . . . . . . Pearson II distributions . . . . . . . . . . . . . . . . . Beta prime distribution . . . . . . . . . . . . . . . . . Inverse Lomax distribution . . . . . . . . . . . . . . . Beta-exponential distributions . . . . . . . . . . . . . Exponentiated exponential distribution . . . . . . . . Hyperbolic sine and Nadarajah-Kotz distributions. . Burr II distributions . . . . . . . . . . . . . . . . . . . Symmetric beta-logistic distributions . . . . . . . . . Kumaraswamy distribution . . . . . . . . . . . . . . Log-logistic distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. E. Crooks – Field Guide to Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 21 22 23 25 30 32 35 39 39 40 46 49 50 57 58 61 64 69 70 75 77 80 82 89 90 92 97 98 102 102 103 108 111 119 125 List of Figures 37 38 39 Grand Unified Distributions . . . . . . . . . . . . . . . . . . 133 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 159 Limits and special cases of principal distributions . . . . . . 163 G. E. Crooks – Field Guide to Probability Distributions 17 List of Tables List of Tables 1.1 2.1 3.1 4.1 5.1 5.2 6.1 7.1 8.1 8.2 9.1 9.2 10.1 11.1 11.2 12.1 13.1 14.1 14.2 15.1 15.2 16.1 17.1 17.2 18.1 18.2 19.1 19.2 20.1 21.1 18 Uniform distribution – Properties . . . . . . . . . . Exponential distribution – Properties . . . . . . . . Laplace distribution – Properties . . . . . . . . . . Normal distribution – Properties . . . . . . . . . . Power function distribution – Special cases . . . . Power function distribution – Properties . . . . . . Log-normal distribution – Properties . . . . . . . . Gamma distribution – Properties . . . . . . . . . . Gamma-exponential distribution – Special cases . Gamma-exponential distribution – Properties . . . Pearson VII distribution – Special cases . . . . . . . Pearson VII distribution – Properties . . . . . . . . Unit gamma distribution – Properties . . . . . . . . Amoroso and gamma distributions – Special cases Amoroso distribution – Properties . . . . . . . . . . Beta distribution – Properties . . . . . . . . . . . . Beta prime distribution – Properties . . . . . . . . . Beta-exponential distribution – Special cases . . . . Beta-exponential distribution – Properties . . . . . Beta-logistic distribution – Special cases . . . . . . Beta-logistic distribution – Properties . . . . . . . . Pearson IV distribution – Properties . . . . . . . . . Generalized beta distributions – Special cases . . . Generalized beta distribution– Properties . . . . . . Generalized beta prime distribution – Special cases Generalized beta prime distribution – Properties . Pearson’s categorization . . . . . . . . . . . . . . . Pearson distribution – Special cases . . . . . . . . . Grand Unified Distribution – Special cases . . . . . Stable distribution – Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. E. Crooks – Field Guide to Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 29 33 36 41 44 47 51 55 56 62 66 71 73 86 91 99 104 105 109 109 115 117 118 122 123 131 131 133 147 List of Tables G. E. Crooks – Field Guide to Probability Distributions 19 G. E. Crooks – Field Guide to Probability Distributions 4 3 shape parameters Figure 1: Hierarchy of principal distributions GUD Beta Gen. Beta Pearson Gen. Beta Prime Beta Exp. Beta Prime Beta-Logistic Pearson IV 2 Unit Gamma Pearson II Amoroso Gamma Gamma-Exp. Inv. gamma Pearson VII 1 Power Func. 0 20 Uniform Exponential Log Normal Normal Inv. Exponential Cauchy G. E. Crooks – Field Guide to Probability Distributions 4 3 shape parameters Figure 2: Hierarchy of principal Pearson distributions Pearson Beta Pearson IV Beta Prime 2 Pearson II Inv. gamma Gamma Pearson VII 1 0 Uniform Exponential Normal Inv. Exponential Cauchy 21 G. E. Crooks – Field Guide to Probability Distributions 4 3 shape parameters Figure 3: Extreme order statistics 2 Gen. Fisher-Tippett Gen. Weibull Gen. Frechet Fisher-Tippett 1 Weibull 0 22 Exponential Gen. Gumbel Gumbel Frechet Inv. Exponential G. E. Crooks – Field Guide to Probability Distributions 4 3 shape parameters Figure 4: Hierarchies of symmetric simple distributions 2 q-Gaussian Sym. Beta-Logistic Pearson II Pearson VII 1 Uniform Normal Laplace Logistic Cauchy 0 23 G. E. Crooks – Field Guide to Probability Distributions 1 Uniform Distribution The simplest continuous distribution is a uniform density over an interval. Uniform (flat, rectangular) distribution: Uniform(x ; a, s) = 1 |s| (1.1) for a, s in R, support x ∈ [a, a + s], s>0 x ∈ [a + s, a], s<0 The uniform distribution is also commonly parameterized with the boundary points, a and b = a + s, rather than location a and scale s as here. Note that the discrete analog of the continuous uniform distribution is also often referred to as the uniform distribution. Special cases The standard uniform distribution covers the unit interval, x ∈ [0, 1]. StdUniform(x) = Uniform(x ; 0, 1) (1.2) The standardized uniform distribution, with zero mean and unit variance, √ √ is Uniform(x ; − 3, 2 3). Three limits of the uniform distribution are important. If one of the boundary points is infinite (infinite scale), then we obtain an improper (unnormalizable) half-uniform distribution. In the limit that both boundary points reach infinity (with opposite signs) we obtain an unbounded uniform distribution. In the alternative limit that the boundary points converge, we obtain a degenerate (delta, Dirac) distribution, wherein the entire probability density is concentrated on a single point. Interrelations Uniform distributions, with finite, semi-infinite, or infinite support, are limits of many distribution families. The finite uniform distribution is a 24 1 Uniform Distribution 1/s 0 a a+s Figure 5: Uniform distribution, Uniform(x ; a, s) (1.1) special case of the beta distribution (12.1). Uniform(x ; a, s) = Beta(x ; a, s, 1, 1) = PearsonII(x ; a + s2 , s) Similarly, the semi-infinite uniform distribution is a limit of the Pareto (5.5), beta prime (13.1), Amoroso (11.1), gamma (7.1), and exponential (2.1) distributions, and the infinite support uniform distribution is a limit of the normal (4.1), Cauchy (9.6), logistic (15.5) and gamma-exponential (8.1) distributions, among others. The order statistics (§C) of the uniform distribution is the beta distribution (12.1). OrderStatisticUniform(a,s) (x ; α, γ) = Beta(x ; a, s, α, γ) The standard uniform distribution is related to every other continuous distribution via the inverse probability integral transform (Smirnov transform). If X is a random variable and F−1 X (z) the inverse of the corresponding G. E. Crooks – Field Guide to Probability Distributions 25 1 Uniform Distribution cumulative distribution function then ( ) X ∼ F−1 X StdUniform() . If the inverse cumulative distribution function has a tractable closed form, then inverse transform sampling can provide an efficient method of sampling random numbers from the distribution of interest. See appendix (§E). The power function distribution (5.1) is related to the uniform distribution via a Weibull transform. 1 PowerFn(a, s, β) ∼ a + s StdUniform() β The sum of n independent standard uniform variates is the Irwin-Hall (21.9) distribution, n ∑ Uniformi (0, 1) ∼ IrwinHall(n) i=1 and the product is the uniform-product distribution (10.2). n ∏ Uniformi (0, 1) ∼ UniformProduct(n) i=1 26 G. E. Crooks – Field Guide to Probability Distributions 1 Uniform Distribution Table 1.1: Properties of the uniform distribution Properties notation PDF / CDF CCDF parameters Uniform(x ; a, s) 1 |s| x−a s s>0 / s<0 a, s in R a⩽x⩽a+s support a+s⩽x⩽a s>0 s<0 median a + 12 s mode any supported value mean a + 21 s variance skew ex. kurtosis entropy MGF CF 1 2 12 s 0 − 6 5 ln |s| eat (est − 1) |s|t iat ist e (e ) − 1 i|s|t G. E. Crooks – Field Guide to Probability Distributions 27 G. E. Crooks – Field Guide to Probability Distributions 2 Exponential Distribution Exponential (Pearson type X, waiting time, negative exponential, inverse exponential) distribution [7, 11, 2]: Exp(x ; a, θ) = { } 1 x−a exp − |θ| θ (2.1) a, θ, in R support x > a, θ>0 x < a, θ<0 An important property of the exponential distribution is that it is memoryless: assuming positive scale and zero location (a = 0, θ > 0) the conditional probability given that x > c, where c is a positive content, is again an exponential distribution with the same scale parameter. The only other distribution with this property is the geometric distribution, the discrete analog of the exponential distribution. The exponential is the maximum entropy distribution given the mean and semi-infinite support. Special cases The exponential distribution is commonly defined with zero location and positive scale (anchored exponential). With a = 0 and θ = 1 we obtain the standard exponential distribution. Interrelations The exponential distribution is common limit of many distributions. Exp(x ; a, θ) = Amoroso(x ; a, θ, 1, 1) = Gamma(x ; a, θ, 1) Exp(x ; 0, θ) = Amoroso(x ; 0, θ, 1, 1) = Gamma(x ; 0, θ, 1) Exp(x ; a, θ) = lim PowerFn(x ; a − βθ, βθ, β) β→∞ The sum of independent exponentials is an Erlang distribution, a special 28 2 Exponential Distribution Table 2.1: Properties of the exponential distribution Properties notation Exp(x ; a, θ) { } x−a 1 exp − PDF |θ| θ { } / x−a CDF CCDF 1 − exp − θ parameters a, θ, in R θ>0 / θ<0 support [a, +∞] θ>0 [−∞, a] θ<0 median a + θ ln 2 mode a mean a + θ variance θ2 skew ex. kurtosis sgn(θ) 2 6 entropy 1 + ln |θ| MGF CF exp(at) (1 − θt) exp(iat) (1 − iθt) case of the gamma distribution (7.1). n ∑ Expi (0, θ) ∼ Gamma(0, θ, n) i=1 The minima of a collection of exponentials, with positive scales θi > 0, is also exponential, ( ) min Exp1 (0, θ1 ), Exp2 (0, θ2 ), . . . , Expn (0, θn ) ∼ Exp(0, θ ′ ) , where θ ′ = ( ∑n 1 −1 . i=1 θi ) G. E. Crooks – Field Guide to Probability Distributions 29 2 Exponential Distribution 1 0.5 0 0 1 2 3 4 Figure 6: Standard exponential distribution, Exp(x ; 0, 1) The order statistics (§C) of the exponential distribution are the betaexponential distribution (14.1). OrderStatisticExp(ζ,λ) (x ; α, γ) = BetaExp(x ; ζ, λ, α, γ) A Weibull transform of the standard exponential distribution yields the Weibull distribution (11.24). 1 Weibull(a, θ, β) ∼ a + θ StdExp() β The ratio of independent anchored exponential distributions is the exponential ratio distribution (5.7), a special case of the beta prime distribution (13.1). BetaPrime(0, θθ21 , 1, 1) ∼ ExpRatio(0, θθ12 ) ∼ 30 Exp1 (0, θ1 ) Exp2 (0, θ2 ) G. E. Crooks – Field Guide to Probability Distributions G. E. Crooks – Field Guide to Probability Distributions 3 Laplace Distribution Laplace (Laplacian, double exponential, Laplace’s first law of error, twotailed exponential, bilateral exponential, biexponential) distribution [12, 13, 14] is a two parameter, symmetric, continuous, univariate, unimodal probability density with infinite support, smooth expect for a single cusp. The functional form is Laplace(x ; ζ, θ) = 1 −| x−ζ e θ | 2|θ| (3.1) for x, ζ, θ in R The two real parameters consist of a location parameter ζ, and a scale parameter θ. Special cases The standard Laplace (Poisson’s first law of error) distribution occurs when ζ = 0 and θ = 1. Interrelations The Laplace distribution is a limit of the symmetric beta-logistic (15.4), exponential power (21.4) and generalized Pearson VII (21.6) distributions. As θ limits to infinity, the Laplace distribution limits to a degenerate distribution. In the alternative limit that θ limits to zero, we obtain an indefinite uniform distribution. The difference between two independent identically distributed exponential random variables is Laplace, and therefore so is the time difference between two independent Poisson events. Laplace(ζ, θ) ∼ Exp1 (ζ, θ) − Exp2 (ζ, θ) Conversely, the absolute value (about the centre of symmetry) is exponential. Exp(ζ, |θ|) ∼ Laplace(ζ, θ) − ζ + ζ 31 3 Laplace Distribution 0.5 0 -3 -2 -1 0 1 2 3 Figure 7: Standard Laplace distribution, Laplace(x ; 0, 1) The log ratio of standard uniform distributions is a standard Laplace. Laplace(0, 1) ∼ ln StdUniform1 () StdUniform2 () The Fourier transform of a standard Laplace distribution is the standard Cauchy distribution (9.6). ∫ +∞ −∞ 32 1 −|x| itx 1 e e dx = 2 1 + t2 G. E. Crooks – Field Guide to Probability Distributions 3 Laplace Distribution Table 3.1: Properties of the Laplace distribution Properties notation PDF CDF parameters support Laplace(x ; ζ, θ) 1 −| x−ζ e θ | 2|θ| { 1 −| x−ζ e θ | x⩽ζ 2 1 − 21 e−| x−ζ θ | x⩾ζ ζ, θ in R x ∈ [−∞, +∞] median ζ mode ζ mean ζ variance 2θ2 skew 0 ex. kurtosis 3 entropy MGF CF 1 + ln(2|θ|) exp(ζt) 1 − θ2 t2 exp(iζt) 1 + θ2 t2 G. E. Crooks – Field Guide to Probability Distributions 33 G. E. Crooks – Field Guide to Probability Distributions 4 Normal Distribution The Normal (Gauss, Gaussian, bell curve, Laplace-Gauss, de Moivre, error, Laplace’s second law of error, law of error) [15, 2] distribution is a ubiquitous two parameter, continuous, univariate unimodal probability distribution with infinite support, and an iconic bell shaped curve. { } (x − µ)2 1 exp − Normal(x ; µ, σ) = √ 2σ2 2πσ2 for x, µ, σ in R (4.1) The location parameter µ is the mean, and the scale parameter σ is the standard deviation. Note that the normal distribution is often parameterized with the variance σ2 rather than the standard deviation. Herein, we choose to consistently parameterize distributions with a scale parameter. The normal distribution most often arises as a consequence of the famous central limit theorem, which states (in its simplest form) that the mean of independent and identically distribution random variables, with finite mean and variance, limit to the normal distribution as the sample size becomes large. The normal distribution is also the maximum entropy distribution for fixed mean and variance. Special cases √ With µ = 0 and σ = 1/ 2h we obtain the error function distribution, and with µ = 0 and σ = 1 we obtain the standard normal (Φ, z, unit normal) distribution. Interrelations In the limit that σ → ∞ we obtain an unbounded uniform (flat) distribution, and in the limit σ → 0 we obtain a degenerate (delta) distribution. The normal distribution is a limiting form of many distributions, including the gamma-exponential (8.1), Amoroso (11.1) and Pearson IV (16.1) families and their superfamilies. 34 4 Normal Distribution 1 σ=2 0.5 σ=1 σ=0.5 0 -4 -2 0 2 4 6 Figure 8: Normal distributions, Normal(x ; 0, σ) Many distributions are transforms of normal distributions. ( ) exp Normal(µ, σ) ∼ LogNormal(0, eµ , σ) (6.1) Normal(0, σ) ∼ HalfNormal(σ) (11.7) ∑ 2 StdNormal() ∼ ChiSqr(1) (7.3) StdNormali ()2 ∼ ChiSqr(k) (7.3) i=1,k Normal(0, σ)−2 ∼ Lévy(0, σ12 ) Normal(0, σ) 2 β ∼ 1 Stacy((2σ2 ) β , 12 , β) StdNormal1 () ∼ StdCauchy() StdNormal2 () (11.15) (11.2) (9.7) The normal distribution is stable (21.20). That is a sum of independent normal random variables is also normally distributed. Normal1 (µ1 , σ1 ) + Normal2 (µ2 , σ2 ) ∼ Normal3 (µ1 + µ2 , σ1 + σ2 ) G. E. Crooks – Field Guide to Probability Distributions 35 4 Normal Distribution Table 4.1: Properties of the normal distribution Properties notation Normal(x ; µ, σ) { } 1 (x − µ)2 PDF √ exp − 2σ2 2πσ2 )] [ ( 1 x−µ CDF 1 + erf √ 2 2σ2 parameters µ, σ in R support x ∈ [−∞, +∞] median µ mode µ mean µ variance σ2 skew 0 ex. kurtosis 0 entropy 1 2 MGF CF ln(2πeσ2 ) ( ) exp µt + 12 σ2 t2 ) ( exp iµt − 12 σ2 t2 The Box-Muller transform [16] generates pairs of independent normal variates from pairs of uniform random variates. ( ) StdNormal1 () ∼ ChiSqr(1) cos 2π StdUniform2 () ( ) StdNormal2 () ∼ ChiSqr(1) sin 2π StdUniform2 () √ where ChiSqr(1) ∼ −2 ln StdUniform1 () Nowadays more efficient random normal generation methods are generally employed (§E). 36 G. E. Crooks – Field Guide to Probability Distributions G. E. Crooks – Field Guide to Probability Distributions 5 Power Function Distribution Power function (power) distribution [7, 17, 3] is a three parameter, continuous, univariate, unimodal probability density, with finite or semi-infinite support. The functional form in most straightforward parameterization consists of a single power function. PowerFn(x ; a, s, β) = β s ( x−a s )β−1 (5.1) for x, a, s, β in R support x ∈ [a, a + s], s > 0, β > 0 or x ∈ [a + s, a], s < 0, β > 0 or x ∈ [a + s, +∞], s > 0, β < 0 or x ∈ [−∞, a + s], s < 0, β < 0 With positive β we obtain a distribution with finite support. But by allowing β to extend to negative numbers, the power function distribution also encompasses the semi-infinite Pareto distribution (5.5), and in the limit β → ∞ the exponential distribution (2.1). Alternative parameterizations Generalized Pareto distribution: An alternative parameterization that emphasizes the limit to exponential. GenPareto(x ; a ′ , s ′ , ξ) ) 1 1( x−ζ − ξ −1 1 + ξ ξ ̸= 0 θ = |θ| 1 exp(− x−ζ ) ξ=0 |θ| θ (5.2) = PowerFn(x ; ζ − θξ , θξ , − ξ1 ) q-exponential (generalized Pareto) distribution is an alternative parameterization of the power function distribution, utilizing the Tsallis generalized 37 5 Power Function Distribution q-exponential function, expq (x) (§D). (5.3) QExp(x ; ζ, θ, q) ( ) (2 − q) x−ζ = expq − |θ| θ 1 ( (2−q) 1 − (1 − q) x−ζ ) 1−q |θ| θ = ( ) 1 exp − x−ζ |θ| θ = PowerFn(x ; ζ + q ̸= 1 q=1 θ θ 2−q ,− , ) 1−q 1−q 1−q for x, ζ, θ, q in R Special cases: Positive β Pearson [7, 2] noted two special cases, the monotonically decreasing Pearson type VIII 0 < β < 1, and the monotonically increasing Pearson type IX distribution [7, 2] with β > 1. Wedge distribution [2]: x−a s2 = PowerFn(x ; a, s, 2) Wedge(x ; a, s) = 2 sgn(s) (5.4) With a positive scale we obtain an ascending wedge (right triangular) distribution, and with negative scale a descending wedge (left triangular). Special cases: Negative β Pareto (Pearson XI, Pareto type I) distribution [18, 7, 2]: Pareto(x ; a, s, γ) = β̄ s ( x−a s )−β̄−1 β̄ > 0 x > a + s, s > 0 x < a + s, s < 0 = PowerFn(x ; a, s, −β̄) 38 G. E. Crooks – Field Guide to Probability Distributions (5.5) 5 Power Function Distribution 4 4.0 3.5 3.5 3 3.0 2.5 2.5 2 2.0 1.5 1.5 β 1 0.5 0 0 0.2 0.4 0.6 0.8 1 Figure 9: Pearson type IX, PowerFn(x ; 0, 1, β), β > 1 4 3.5 3 2.5 2 1.5 β 1 0.8 0.6 0.4 0.2 0.5 0 0 0.2 0.4 0.6 0.8 1 Figure 10: Pearson type VIII, PowerFn(x ; 0, 1, β), 0 < β < 1. G. E. Crooks – Field Guide to Probability Distributions 39 5 Power Function Distribution 4 3.5 3 2.5 2 1.5 1 0.5 0 1 1.5 2 2.5 3 Figure 11: Pareto distributions, Pareto(x ; 0, 1, β̄), β̄ left axis. The most important special case is the Pareto distribution, which has a semi-infinite support with a power-law tail. The Zipf distribution is the discrete analog of the Pareto distribution. Lomax (Pareto type II, ballasted Pareto) distribution [19]: Lomax(x ; a, s, β̄) = ( )−β̄−1 β̄ x−a 1+ |s| s (5.6) = Pareto(x ; a − s, s, β̄) = PowerFn(x ; a − s, s, −β̄) Originally explored as a model of business failure. The alternative name “ballasted Pareto” arises since this distribution is a shifted Pareto distribution (5.5) whose origin is fixed at zero, and no longer moves with changes in scale. 40 G. E. Crooks – Field Guide to Probability Distributions 5 Power Function Distribution Table 5.1: Special cases of the power function distribution (5.1) power function (5.5) (5.8) a s β Pareto . . <0 uniform prime . . -1 (5.1) Pearson type VIII 0 . (0, 1) (1.1) uniform . . 1 (5.1) Pearson type IX 0 . >1 (5.4) wedge . . 2 (2.1) exponential . . +∞ Exponential ratio distribution [1]: ExpRatio(x ; s) = 1 1 ( ) |s| 1 + x 2 (5.7) s = Lomax(x ; 0, s, 1) = PowerFn(x ; −s, s, 1) Arises as the ratio of independent exponential distributions (p 30). Uniform-prime distribution [20, 1]: UniPrime(x ; a, s) = 1 1 ( ) |s| 1 + x−a 2 (5.8) s = Lomax(x ; a, s, 1) = PowerFn(x ; a − s, s, −1) An exponential ratio (5.7) distribution with a shift parameter. So named since this distribution is related to the uniform distribution as beta is to beta prime. The ordering distribution (§C) of the beta-prime distribution. Limits and subfamilies With β = 1 we recover the uniform distribution. PowerFn(a, s, 1) ∼ Uniform(a, s) G. E. Crooks – Field Guide to Probability Distributions 41 5 Power Function Distribution As β limits to infinity, the power function distribution limits to the exponential distribution (2.1). Exp(x ; ν, λ) = lim PowerFn(x ; ν − βλ, βλ, β) β→∞ 1 = lim β→∞ λ ( Recall that limc→∞ 1 + ) x c c ( )β−1 x−ν 1+ βλ = ex . Interrelations With positive β, the power function distribution is a special case of the beta distribution (12.1), with negative beta, a special case of the beta prime distribution (13.1), and with either sign a special case of the generalized beta (17.1) and unit gamma (10.1) distributions. PowerFn(x ; a, s, β) = GenBeta(x ; a, s, 1, 1, β) = GenBeta(x ; a, s, β, 1, 1) = Beta(x ; a, s, β, 1) β>0 β>0 = GenBeta(x ; a + s, s, 1, −β, −1) β<0 = BetaPrime(x ; a + s, s, 1, −β) β<0 = UnitGamma(x ; a, s, 1, β) The order statistics (§C) of the power function distribution yields the generalized beta distribution (17.1). OrderStatisticPowerFn(a,s,β) (x ; α, γ) = GenBeta(x ; a, s, α, γ, β) Since the power function distribution is a special case of the generalized beta distribution (17.1), GenBeta(x ; a, s, α, 1, β) = PowerFn(x ; a, s, αβ) it follows that the power function family is closed under maximization for β β s > 0 and minimization for s < 0. The product of independent power function distributions (With zero lo- 42 G. E. Crooks – Field Guide to Probability Distributions 5 Power Function Distribution cation parameter, and the same β) is a unit-gamma distribution (10.1) [21]. α ∏ PowerFni (0, si , β) ∼ UnitGamma(0, i=1 α ∏ si , α, β) i=1 Consequently, the geometric mean of independent, anchored power function distributions (with common β) is also unit-gamma. v u α α ∏ u∏ α t PowerFni (0, si , β) ∼ UnitGamma(0, si , α, αβ) i=1 i=1 The power function distribution can be obtained from the Weibull transβ form x → ( x−a s ) of the uniform distribution (1.1). 1 PowerFn(a, s, β) ∼ a + s StdUniform() β The power function distribution limits to the exponential distribution (§2). Exp(x ; a, θ) = lim PowerFn(x ; a + βθ, −βθ, β) β→∞ G. E. Crooks – Field Guide to Probability Distributions 43 5 Power Function Distribution Table 5.2: Properties of the power function distribution Properties notation PowerFn(x ; a, s, β) )β−1 ( β x−a PDF s s ( )β / x−a CDF CCDF s parameters a, s, β in R support mode mean variance skew s β 44 / s β <0 x ∈ [a, a + s] s > 0, β > 0 x ∈ [a + s, a] s < 0, β > 0 x ∈ [a + s, +∞] s > 0, β < 0 x ∈ [−∞, a + s] s < 0, β < 0 a β>0 a+s β<0 sβ β+1 s2 β (β + 1)2 (β + 2) β∈ / [−1, 0] a+ sgn( βs ) 2(1 − β) (β + 3) β∈ / [−2, 0] √ β+2 β 6(β3 − β2 − 6β + 2) β(β + 3)(β + 4) MGF undefined ex. kurtosis >0 β∈ / [−3, 0] β∈ / [−4, 0] G. E. Crooks – Field Guide to Probability Distributions G. E. Crooks – Field Guide to Probability Distributions 6 Log-Normal Distribution Log-normal (Galton, Galton-McAlister, anti-log-normal, Cobb-Douglas, log-Gaussian, logarithmic-normal, logarithmico-normal, Λ, Gibrat) distribution [22, 23, 2] is a three parameter, continuous, univariate, unimodal probability density with semi-infinite support. The functional form in the standard parameterization is LogNormal(x ; a, ϑ, β) { ( )−1 )2 } ( |β| x−a x−a 1 =√ exp − β ln ϑ 2 ϑ 2πϑ2 (6.1) for x, a, ϑ, β in R, x−a ϑ >0 The log-normal is so called because the log transform of the log-normal variate is a normal random variable. The distribution should, perhaps, be more accurately called the anti-log-normal distribution, but the nomenclature is now standard. Special cases The anchored log-normal (two-parameter log-normal) distribution (a = 0) arises from the multiplicative version of the central limit theorem: When the sum of independent random variables limits to normal, the product of those random variables limits to log-normal. With a = 0, ϑ = 1, σ = 1 we obtain the standard log-normal (Gibrat) distribution [24]. Interrelations The log-normal forms a location-scale-power distribution family. LogNormal(a, ϑ, β) ∼ a + ϑ StdLogNormal()1/β The log-normal distribution is the anti-log transform of a normal random variable. ( ) LogNormal(a, ϑ, β) ∼ a + exp − Normal(− ln ϑ, 1/β) 45 6 Log-Normal Distribution 1.5 β=4 1 β=2 β=1 0.5 0 0.5 1 1.5 2 2.5 3 Figure 12: Log normal distributions, LogNormal(x ; 0, 1, β) Because of this close connection to the normal distribution, the log-normal is often parameterized with the mean and standard deviation of the corresponding normal distribution, µ = ln ϑ, σ = 1/β rather than standard scale and power parameters. The log-normal distribution is a limiting form of the Unit gamma (10.1) and Amoroso (11.1), distributions (And therefore also of the generalized beta and generalized beta prime distributions) and limits to the normal distribution (§D). Normal(x ; µ, σ) = lim LogNormal(x ; µ + βσ, −βσ, β) β→∞ A product of log-normal distributions (With zero location parameter) is again a log-normal distribution. This follows from the fact that the sum of normal distributions is normal. n ∏ i=1 46 LogNormali (0, ϑi , βi ) ∼ LogNormali (0, n ∏ i=1 ϑi , ( n ∑ 1 −2 β−2 ) i ) i=0 G. E. Crooks – Field Guide to Probability Distributions 6 Log-Normal Distribution Table 6.1: Properties of the log-normal distribution Properties notation LogNormal(x ; a, ϑ, β) { ( )−1 )2 } ( |β| x−a 1 x−a PDF √ exp − β ln ϑ 2 ϑ 2πϑ2 ( ) / / x−a CDF CCDF 12 + 12 erf √12 β ln ϑ>0 ϑ<0 ϑ parameters a, ϑ, β in R support x ∈ [a, +∞] ϑ>0 x ∈ [−∞, a] ϑ < 0 median a + ϑ mode a + ϑe−β 1 −2 mean a + ϑe 2 β variance ϑ2 (eβ skew ex. kurtosis entropy −2 −2 − 1)eβ sgn(ϑ) (eβ e4β 1 2 −2 + 1 2 −2 + 2e3β −2 + 2) √ eβ−2 − 1 −2 + 3e2β −2 ) + ln |ϑ| ln(2πβ −2 −6 MGF doesn’t exist in general CF no simple closed form expression G. E. Crooks – Field Guide to Probability Distributions 47 G. E. Crooks – Field Guide to Probability Distributions 7 Gamma Distribution Gamma (Γ , Pearson type III) distribution [4, 5, 2] : Gamma(x ; a, θ, α) = ( )α−1 { } 1 x−a x−a exp − Γ (α)|θ| θ θ for x, a, θ, α in R, (7.1) α>0 = Amoroso(x ; a, θ, α, 1) The name of this distribution derives from the normalization constant. Special cases Special cases of the beta prime distribution are listed in table 11, under β = 1. The gamma distribution often appear as a solution to problems in statistical physics. For example, the energy density of a classical ideal gas, or the Wien (Vienna) distribution Wien(x ; T ) = Gamma(x ; 0, T , 4), an approximation to the relative intensity of black body radiation as a function of the frequency. The Erlang (m-Erlang) distribution [25] is a gamma distribution with integer α, which models the waiting time to observe α events from a Poisson process with rate 1/θ (θ > 0). For α = 1 we obtain an exponential distribution (2.1). Standard gamma (standard Amoroso) distribution [2]: StdGamma(x ; α) = 1 α−1 −x x e Γ (α) = Gamma(x ; 0, 1, α) 48 (7.2) 7 Gamma Distribution 1.5 α=8 α=6 α=1 α=4 α=2 1 0.5 0 0 1 2 Figure 13: Gamma distributions, unit variance Gamma(x ; 3 1 α , α) Chi-square (χ2 ) distribution [26, 2]: ChiSqr(x ; k) = k { ( x )} 1 ( x ) 2 −1 exp − 2 2Γ ( k2 ) 2 (7.3) for positive integer k = Gamma(x ; 0, 2, k2 ) = Stacy(x ; 2, k2 , 1) = Amoroso(x ; 0, 2, k2 , 1) The distribution of a sum of squares of k independent standard normal random variables. The chi-square distribution is important for statistical hypothesis testing in the frequentist approach to statistical inference. G. E. Crooks – Field Guide to Probability Distributions 49 7 Gamma Distribution 0.5 k=1 k=2 k=3 k=4 k=5 0 0 1 2 3 4 5 6 7 8 Figure 14: Chi-square distributions, ChiSqr(x ; k) Scaled chi-square distribution [27]: ScaledChiSqr(x ; σ, k) = ( x ) k −1 { ( x )} 1 2 exp − 2σ2 2σ2 Γ ( k2 ) 2σ2 (7.4) for positive integer k = Stacy(x ; 2σ2 , k2 , 1) = Gamma(x ; 0, 2σ2 , k2 ) = Amoroso(x ; 0, 2σ2 , k2 , 1) The distribution of a sum of squares of k independent normal random variables with variance σ2 . 50 G. E. Crooks – Field Guide to Probability Distributions 7 Gamma Distribution Table 7.1: Properties of the gamma distribution Properties notation Gamma(x ; a, θ, α) { ( )α−1 } 1 x−a x−a PDF exp − Γ (α)|θ| θ θ ( x−a ) / CDF / CCDF 1 − Q α, θ θ>0 θ<0 parameters support mode a, θ, α, in R, α > 0 x⩾a θ>0 x⩽a θ<0 a + θ(α − 1) α⩾1 a α⩽1 mean a + θα variance skew ex. kurtosis entropy θ2 α 2 sgn(θ) √ α 6 α( ) ln |θ|Γ (α) + α + (1 − α)ψ(α) MGF eat (1 − θt)−α CF eiat (1 − iθt)−α Porter-Thomas distribution [28]: PorterThomas(x ; σ) = ( x )− 1 { ( x )} 1 2 exp − 1 2 2 2σ2 2σ Γ ( 2 ) 2σ (7.5) = Stacy(x ; 2σ2 , 12 , 1) = Gamma(x ; 0, 2σ2 , 12 ) = Amoroso(x ; 0, 2σ2 , 12 , 1) A chi-square distribution with a single degree of freedom. Used to model fluctuations in decay mode strengths of excited nuclei [28] G. E. Crooks – Field Guide to Probability Distributions 51 7 Gamma Distribution Interrelations Gamma distributions with common scale obey an addition property: Gamma1 (0, θ, α1 ) + Gamma2 (0, θ, α2 ) ∼ Gamma3 (0, θ, α1 + α2 ) The sum of two independent, gamma distributed random variables (with common θ’s, but possibly different α’s) is again a gamma random variable [2]. The Amoroso distribution can be obtained from the standard gamma ( )β . distribution by the Weibull change of variables, x 7→ x−a θ [ ]1/β Amoroso(a, θ, α, β) ∼ a + θ StdGamma(α) For large α the gamma distribution limits to normal (4.1). √ Normal(x ; µ, σ) = lim Gamma(x ; µ − σ α, √σα , α) α→∞ Conversely, the sum of squares of normal distributions is a gamma distribution. See chi-square (7.3). ∑ i=1,k k StdNormali ()2 ∼ ChiSqr(k) ∼ Gamma(0, 2, ) 2 A large variety of distributions can be obtained from transformations of 1 or 2 gamma distributions, which is convenient for generating pseudo- 52 G. E. Crooks – Field Guide to Probability Distributions 7 Gamma Distribution random numbers from those distributions (See appendix (§E)). √ Normal(µ, σ) ∼ µ + σ Sgn() 2 StdGamma( 12 ) (4.1) ( ) GammaExp(a, s, α) ∼ a − s ln StdGamma(α) (8.1) √ StdGamma1 ( 12 ) PearsonVII(a, s, m) ∼ a + s Sgn() (9.1) StdGamma2 (m − 21 ) √ StdGamma1 ( 21 ) Cauchy(a, s) ∼ a + s Sgn() (9.6) StdGamma2 ( 21 ) ( ) UnitGamma(a, s, α, β) ∼ a + s exp − β1 StdGamma(α) (10.1) ( )−1 StdGamma2 (γ) Beta(a, s, α, γ) ∼ a + s 1 + (12.1) StdGamma1 (α) StdGamma1 (α) BetaPrime(a, s, α, γ) ∼ a + s (13.1) StdGamma2 (γ) 1 Amoroso(a, θ, α, β) ∼ a + θ StdGamma(α) β ( )−1 StdGamma2 (γ) BetaExp(a, s, α, γ) ∼ a − s ln 1 + StdGamma1 (α) ) ( StdGamma1 (α) BetaLogistic(a, s, α, γ) ∼ a − s ln StdGamma2 (γ) )− 1 ( β StdGamma2 (γ) GenBeta(a, s, α, γ, β) ∼ a + s 1 + StdGamma1 (α) ( )1 StdGamma1 (α) β GenBetaPrime(a, s, α, γ, β) ∼ a + s StdGamma2 (γ) (11.1) (14.1) (15.1) (17.1) (18.1) Here, Sgn() is the sign (or Rademacher) discrete random variable: 50% chance −1, 50% chance +1. G. E. Crooks – Field Guide to Probability Distributions 53 G. E. Crooks – Field Guide to Probability Distributions 8 Gamma-Exponential Distribution The gamma-exponential (log-gamma, generalized Gompertz, generalized Gompertz-Verhulst type I, Coale-McNeil, exponential gamma) distribution [29, 30, 3] is a three parameter, continuous, univariate, unimodal probability density with infinite support. The functional form in the most straightforward parameterization is GammaExp(x ; ν, λ, α) { ( ) ( )} x−ν x−ν 1 exp −α − exp − = Γ (α)|λ| λ λ (8.1) for x, ν, λ, α, in R, α > 0, support − ∞ ⩽ x ⩽ ∞ The three real parameters consist of a location parameter ν, a scale parameter λ, and a shape parameter α. Note that this distribution is often called the “log-gamma” distribution. This naming convention is the opposite of that used for the log-normal distribution (6.1). The name “log-gamma” has also been used for the antilog transform of the generalized gamma distribution, which leads to the unit-gamma distribution (10.1). Also note that the gamma-exponential is often defined with the sign of the scale λ flipped. The parameterization used here is consistent with other log-transformed distributions. (See Log and anti-log transformation, p.165) Special cases Standard gamma-exponential distribution: StdGammaExp(x ; α) = 1 exp{−α x − exp(−x)} Γ (α) (8.2) = GammaExp(x ; 0, 1, α) The gamma-exponential distribution with zero location and unit scale. 54 8 Gamma-Exponential Distribution Table 8.1: Special cases of the gamma-exponential family (8.1) gamma-exponential (8.2) standard gamma-exponential (8.3) chi-square-exponential (8.4) ν λ α 0 1 α ln 2 1 k 2 generalized Gumbel . . n (8.5) Gumbel . . 1 (8.6) standard Gumbel 0 1 1 (8.7) BHP . . (8.8) Moyal . . π 2 1 2 Chi-square-exponential (log-chi-square) distribution [27]: ChiSqrExp(x ; k) = { } k 1 exp − x − exp(−x) k 2 2 2 2 Γ(k) 1 2 for positive integer k = GammaExp(x ; ln 2, 1, (8.3) k 2) The log transform of the chi-square distribution (7.3). Generalized Gumbel distribution [31, 3]: GenGumbel(x ; u, λ, n) { ( ) ( )} nn x−u x−u = exp −n − n exp − Γ (n)|λ| λ λ (8.4) for positive integer n = GammaExp(x ; u − λ ln n, λ, n) The limiting distribution of the nth largest value of a large number of unbounded identically distributed random variables whose probability distribution has an exponentially decaying tail. Gumbel (Fisher-Tippett type I, Fisher-Tippett-Gumbel, Gumbel-FisherTippett, FTG, log-Weibull, extreme value (type I), doubly exponential, dou- G. E. Crooks – Field Guide to Probability Distributions 55 8 Gamma-Exponential Distribution Table 8.2: Properties of the gamma-exponential distribution Properties notation GammaExp(x ; ν, λ, α) { ( ) ( )} 1 x−ν x−ν PDF exp −α − exp − Γ (α)|λ| λ λ ( ) / / − x−ν λ>0 λ<0 CDF CCDF Q α, e λ parameters support mode ν, λ, α, in R, α > 0, x ∈ [−∞, +∞] ν − λ ln α mean ν − λψ(α) variance skew ex. kurtosis λ2 ψ1 (α) 2 (α) − sgn(λ) ψψ 3/2 1 (α) ψ3 (α) ψ1 (α)2 Γ (α − λt) Γ (α) Γ (α − iλt) CF eiνt Γ (α) MGF eνt 56 G. E. Crooks – Field Guide to Probability Distributions [3] 8 Gamma-Exponential Distribution 1 α=5 α=4 α=3 0.5 α=2 α=1 0 -3 -2 -1 0 1 2 3 Figure 15: Gamma exponential distributions, GammaExp(x ; 0, 1, α) ble exponential) distribution [32, 31, 3]: Gumbel(x ; u, λ) = { ( ) ( )} 1 x−u x−u exp − − exp − |λ| λ λ (8.5) = GammaExp(x ; u, λ, 1) This is the asymptotic extreme value distribution for variables of “exponential type”, unbounded with finite moments [31]. With positive scale λ > 0, this is an extreme value distribution of the maximum, with negative scale λ < 0 an extreme value distribution of the minimum. Note that the Gumbel is sometimes defined with the negative of the scale used here. The term “double exponential distribution” can refer to either Laplace or Gumbel distributions [3]. Standard Gumbel (Gumbel) distribution [31]: { } StdGumbel(x) = exp −x − e−x (8.6) = GammaExp(x ; 0, 1, 1) The Gumbel distribution with zero location and a unit scale. G. E. Crooks – Field Guide to Probability Distributions 57 8 Gamma-Exponential Distribution 0.5 0 -3 -2 -1 0 1 2 3 4 5 6 7 8 Figure 16: Standard Gumbel distribution, StdGumbel(x) BHP (Bramwell-Holdsworth-Pinton) distribution [33, 34]: { ( ) ( )} 1 π x−ν x−ν exp − − exp − Γ ( π2 )|λ| 2 λ λ π = GammaExp(x ; ν, λ, ) 2 BHP(x ; ν, λ) = (8.7) Proposed as a model of rare fluctuations in turbulence and other correlated systems. Moyal distribution [35]: ) { ( 1 x−µ Moyal(x ; µ, λ) = √ − exp − 12 λ 2π|λ| 1 2 ( )} x−µ exp − λ (8.8) = GammaExp(x ; µ + λ ln 2, λ, 21 ) Introduced as analytic approximation to the Landau distribution (21.11) [35]. 58 G. E. Crooks – Field Guide to Probability Distributions 8 Gamma-Exponential Distribution Interrelations The name “log-gamma” arises because the standard log-gamma distribution is the logarithmic transform of the standard gamma distribution ( ) StdGammaExp(α) ∼ − ln StdGamma(α) ) ( GammaExp(ν, λ, α) ∼ − ln Amoroso(0, e−ν , α, λ1 ) The difference of two gamma-exponential distribution (with common scale) is a beta-logistic distribution (15.1) [3]. BetaLogistic(x ; ζ1 − ζ2 , λ, α, γ) ∼ GammaExp1 (x ; ζ1 , λ, α) − GammaExp2 (x ; ζ2 , λ, γ) It follows that the difference of two Gumbel distributions (8.5) is a logistic distribution (15.5). The gamma-exponential distribution is a limit of the Amoroso distribution (11.1), and itself contains the normal (4.1) distribution as a limiting case. √ √ lim GammaExp(x ; µ + σ α ln α, σ α, α) = Normal(x ; µ, σ) α→∞ G. E. Crooks – Field Guide to Probability Distributions 59 G. E. Crooks – Field Guide to Probability Distributions 9 Pearson VII Distribution The Pearson type VII distribution [7] is a three parameter, continuous, univariate, unimodal, symmetric probability distribution, with infinite support. The functional form in the most straight forward parameterization is ( ( )2 )−m 1 x−a 1+ PearsonVII(x ; a, s, m) = s |s|B(m − 12 , 12 ) m> (9.1) 1 2 = PearsonIV(x ; a, s, m, 0) This distribution family is notable for having long power-law tails in both directions. Special cases Student’s t (Student, t, Student-Fisher, Fisher) distribution [36, 37, 38, 39] : )− 12 (k+1) ( 1 x2 + StudentsT(x ; k) = √ 1 k kB( 21 , 12 k) √ = PearsonVII(x ; 0, k, 12 (k + 1)) (9.2) integer k ⩾ 0 The distribution of the statistic t, which arises when considering the error of samples means drawn from normal random variables. √ x̄ − µ t= n s̄ n ∑ x̄ = n1 Normali (µ, σ) i=1 s̄2 = 1 n−1 n ∑ )2 ( Normali (µ, σ) − x̄ i=1 Here, x̄ is the sample mean of n independent normal (4.1) random variables with mean µ and variance σ2 , s̄ is the sample variance, and k = n − 1 is the 60 9 Pearson VII Distribution 0.5 0 -3 -2 -1 0 1 2 3 4 Figure 17: Student’s t distributions (9.2): Cauchy (k = 1), t2 (k = 2), t3 (k = 3), normal (k → ∞) (low to high peak). ‘degrees of freedom’. Student’s t2 (t2 ) distribution [40] : StudentsT2 (x) = 1 (9.3) 3 (2 + x2 ) 2 = StudentsT(x ; 2) = PearsonVII(x ; 0, √ 3 2, 2 ) Student’s t distribution with 2 degrees of freedom has a particularly simple form. G. E. Crooks – Field Guide to Probability Distributions 61 9 Pearson VII Distribution Table 9.1: Special cases of the Pearson type VII distribution (9.1) Pearson type VII a (9.2) Student’s t 0 (9.3) Student’s t2 0 (9.4) Student’s t3 0 s √ k √ 2 √ 3 m (9.5) Student’s z 0 1 n/2 (9.6) Cauchy . . 1 (9.7) standard Cauchy 0 1 1 (9.8) relativistic Breit-Wigner . . 2 k+1 2 3 2 2 Student’s t3 (t3 ) distribution [41] : StudentsT3 (x) = 2 )2 ( 2 π 1 + x3 (9.4) = StudentsT(x ; 3) √ = RelBreitWigner(x ; 0, 3) √ = PearsonVII(x ; 0, 3, 2) Student’s t distribution with 3 degrees of freedom. Notable since the cumulative distribution function has a relatively simple form [41, p37]. StudentsT3 CDF(x) = 1 2 + √1 3π ( arctan( √x3 ) + x √ 3 2 1+ x3 ) Student’s z distribution [36, 38]: StudentsZ(z ; n) = 1 1 B( n−1 2 , 2) ( )− n 1 + z2 2 (9.5) = PearsonVII(z ; 0, 1, n2 ) The distribution of the statistic z, which was the original distribution investigated by Gosset (aka Student)6 in his famous 1908 paper on the statis6 Gosset’s employer, the Guinness Brewing Company, insisted that he publish under a pseudonym. 62 G. E. Crooks – Field Guide to Probability Distributions 9 Pearson VII Distribution tical error of sample means [36]. z= x̄ = s2 = x̄ − µ s n ∑ 1 n 1 n Normali (µ, σ) , i=1 n ∑ ( )2 Normali (µ, σ) − x̄ i=1 Here, x̄ is the sample mean of n independent normal (4.1) random variables with mean µ and variance σ2 , and s2 is the sample variance, except normalized by n rather than the now conventional n − 1. Latter work by Student √ and Fisher [37] resulted in a switch to the statistic t = z/ n − 1. Cauchy (Lorentz, Lorentzian, Cauchy-Lorentz, Breit-Wigner, normal ratio, Witch of Agnesi) distribution [42, 43, 3]: ( ( )2 )−1 1 x−a Cauchy(x ; a, s) = 1+ sπ s (9.6) = PearsonVII(x ; a, s, 1) The Cauchy distribution is stable (21.20): a sum of independent Cauchy random variables is also Cauchy distributed. Cauchy1 (a1 , s1 ) + Cauchy2 (a2 , s2 ) ∼ Cauchy3 (a1 + a2 , s1 + s2 ) Standard Cauchy distribution [3]: 1 1 π 1 + x2 1 = (x + i)−1 (x − i)−1 π = Cauchy(x ; 0, 1) StdCauchy(x) = (9.7) = PearsonVII(x ; 0, 1, 1) G. E. Crooks – Field Guide to Probability Distributions 63 9 Pearson VII Distribution 0.5 0 -3 -2 -1 0 1 2 3 4 Figure 18: Standard Cauchy distribution, StdCauchy(x). Relativistic Breit-Wigner (modified Lorentzian) distribution [44]: ( ( )2 )−2 2 x−a RelBreitWigner(x ; a, s) = 1+ |s|π s (9.8) = PearsonVII(x ; a, s, 2) Used to model the energy distribution of unstable particles in high-energy physics. Interrelations The Pearson VII distribution is a special case of the Pearson IV distribution (16.1). At high shape parameter m the Pearson VII limits to the normal distribution. √ Normal(x ; µ, σ) = lim PearsonVII(x ; µ, σ 2m, m) m→∞ The Pearson type VII distribution is given by a ratio of normal and 64 G. E. Crooks – Field Guide to Probability Distributions 9 Pearson VII Distribution gamma random variables [41, p445]. √ StdNormal() PearsonVII(a, s, m) ∼ a + s 2m − 1 √ StdGamma(m − 12 ) The Cauchy distribution can be generated as a ratio of normal distributions Cauchy(0, 1) ∼ Normal1 (0, 1) Normal2 (0, 1) and as a ratio of gamma distributions [41, p427]. )2 StdGamma ( 1 ) ( 1 2 Cauchy(0, 1) ∼ StdGamma2 ( 12 ) G. E. Crooks – Field Guide to Probability Distributions 65 9 Pearson VII Distribution Table 9.2: Properties of the Pearson VII distribution Properties notation PearsonVII(x ; a, s, m) PDF CDF / CCDF parameters ( ( )2 )−m x−a 1 1+ s |s|B(m − 12 , 12 ) ( ( ) ( )2 ) 1 3 1 x−a 1 x−a + , m; ; − 2 F1 2 s 2 2 s B(m − 12 , 12 ) a, s, m ∈ R m> 1 2 support − ∞ < x < +∞ median a mode a mean a m>1 2 s 2m − 3 skew 0 variance MGF CF 66 m> m>2 undefined e 3 2 2Km− 12 (s|t|) · iat (1 )m− 21 s|t| 2 Γ (m − 21 ) G. E. Crooks – Field Guide to Probability Distributions m> 1 2 G. E. Crooks – Field Guide to Probability Distributions 10 Unit Gamma Distribution Unit gamma (log-gamma) distribution [45, 21, 46, 47]: UnitGamma(x ; a, s, α, β) ( )β−1 ( )α−1 1 β x−a x−a = −β ln Γ (α) s s s (10.1) for x, a, s, α, β in R, α > 0 support x ∈ [a, a + s], s > 0, β > 0 or x ∈ [a + s, a], s < 0, β > 0 or x ∈ [a + s, +∞], s > 0, β < 0 or x ∈ [−∞, a + s], s < 0, β < 0 A curious distribution that occurs as a limit of the generalized beta (17.1), and as the anti-log transform of the gamma distribution (7.1). For this reason, it is also sometimes called the log-gamma distribution. Special cases Uniform product distribution [48]: UniformProduct(x ; n) = 1 (− ln x)n−1 Γ (n) (10.2) = UnitGamma(x ; 0, 1, n, 1) 0 > x > 1, n = 1, 2, 3, . . . The product of n standard uniform distributions (1.2). Interrelations With α = 1 we obtain the power function distribution (5.1) as a special case. UnitGamma(x ; a, s, 1, β) = PowerFn(x ; a, s, β) The unit gamma is the anti-log transform of the standard gamma dis- 67 10 Unit Gamma Distribution tribution (7.2). ( ) UnitGamma(0, 1, α, β) ∼ exp − Gamma(0, β1 , α) ( ) UnitGamma(0, 1, α, 1) ∼ exp − StdGamma(α) The unit gamma distribution is a limit of the generalized beta distribution (17.1), and limits to the gamma (7.1) and log-normal (6.1) [1] distributions. Gamma(x ; a, s, α) = lim UnitGamma(x ; a + βs, −βs, α, β) β→∞ lim UnitGamma(x ; a, ϑeσ α→∞ ( √ α , α, √ √ α σ ) ) α −1 ( √ )α−1 σ α x−a x−a √ ∝ lim − ln σ√α α→∞ ϑeσ α σ ϑe ( )−1 { }( )α−1 √ 1 1 x−a x−a 1 x−a 1− √ ln ∝ lim exp α ln α→∞ ϑ σ ϑ ϑ ασ ( )−1 √ ( z )α x−a , z = − σ1 ln x−a ∝ lim e−z α 1 + √ ϑ α→∞ ϑ α { ( )−1 ( )2 } x−a 1 x−a ∝ exp − 2 ln ϑ 2σ ϑ = LogNormal(x ; a, ϑ, σ) Here we utilize the Gaussian function limit limc→∞ e−z √ ( c 1+ )c √z c = − 21 z2 (§D). The product of two unit-gamma distributions with common β is again a unit-gamma distribution [21, 1]. e UnitGamma1 (0, s1 , α1 , β) UnitGamma2 (0, s2 , α2 , β) ∼ UnitGamma3 (0, s1 s2 , α1 + α2 , β) The property is related to the analogous additive relation of the gamma 68 G. E. Crooks – Field Guide to Probability Distributions 10 Unit Gamma Distribution 3 2.5 2 α=1.5, β=1 α=5, β=8 α=2, β=2 1.5 1 0.5 0 0.5 1 Figure 19: Unit gamma, finite support, UnitGamma(x ; 0, 1, α, β), β > 0. distribution. UnitGamma1 (0, s1 , α1 , β) UnitGamma2 (0, s2 , α2 , β) 1 ∼ s1 s2 (UnitGamma1 (0, 1, α1 , 1) UnitGamma2 (0, 1, α2 , 1)) β )1 ( β ∼ s1 s2 e− StdGamma1 (α1 )−StdGamma2 (α2 ) ( )1 β ∼ s1 s2 e− StdGamma3 (α1 +α2 ) ∼ UnitGamma3 (0, s1 s2 , α1 + α2 , β) G. E. Crooks – Field Guide to Probability Distributions 69 10 Unit Gamma Distribution 1 α=5, β=-8 0.5 α=2, β=-1 α=1.5, β=-1 0 1 1.5 2 2.5 3 3.5 4 Figure 20: Unit gamma, semi-infinite support. UnitGamma(x ; 0, 1, α, β), β<0 70 G. E. Crooks – Field Guide to Probability Distributions 10 Unit Gamma Distribution Table 10.1: Properties of the unit gamma distribution Properties notation UnitGamma(x ; a, s, α, β) )β−1 ( )α−1 ( 1 β x−a x−a −β ln Γ (α) s s s ) / ( x−a CDF CCDF 1 − Q α, −β ln s PDF β s >0 / β s <0 parameters a, s, α, β in R, α, β > 0 support [a, a + s], s > 0, β > 0 [a + s, a], s < 0, β > 0 [a + s, +∞]s > 0, β < 0 [−∞, a + s], s < 0, β < 0 ( )α β mean a + s β+1 ( )α ( )2α β β variance s2 β+2 − s2 β+1 skew not simple ex. kurtosis E(Xh ) not simple ( β β+h )α a = 0 [46] G. E. Crooks – Field Guide to Probability Distributions 71 G. E. Crooks – Field Guide to Probability Distributions 11 Amoroso Distribution The Amoroso (generalized gamma, Stacy-Mihram) distribution [49, 2, 50] is a four parameter, continuous, univariate, unimodal probability density, with semi-infinite support. The functional form in the most straightforward parameterization is Amoroso(x ; a, θ, α, β) = 1 β Γ (α) θ ( x−a θ )αβ−1 { ( )β } x−a exp − θ (11.1) for x, a, θ, α, β in R, α > 0, support x ⩾ a if θ > 0, x ⩽ a if θ < 0. The Amoroso distribution was originally developed to model lifetimes [49]. It occurs as the Weibullization of the standard gamma distribution (7.1) and, with integer α, in extreme value statistics (11.21). The Amoroso distribution is itself a limiting form of various more general distributions, most notable the generalized beta (17.1) and generalized beta prime (18.1) distributions [51]. Many common and interesting probability distributions are special cases or limiting forms of the Amoroso (See Table 11). The four real parameters of the Amoroso distribution consist of a location parameter a, a scale parameter θ, and two shape parameters, α and β. Whenever these symbols appears in special cases or limiting forms, they refer directly to the parameters of the Amoroso distribution. The shape parameter α is positive, and in many special cases an integer, α = n, or half-integer, α = k2 . The negation of a standard parameter is indicated by a bar, e.g. β̄ = −β. The chi, chi-squared and related distributions are traditionally parameterized with the scale parameter σ, where θ = (2σ2 )1/β , and σ is the standard deviation of a related normal distribution. Additional alternative parameters are introduced as necessary. Special cases: Miscellaneous The gamma distribution (β = 1) and it’s special cases are detailed in (§7). Stacy (hyper gamma, generalized Weibull, Nukiyama-Tanasawa, generalized gamma, generalized semi-normal, hydrograph, Leonard hydrograph, 72 11 Amoroso Distribution Table 11.1: Special cases of the Amoroso and gamma families (11.1) (11.2) (11.4) (11.21) (11.22) (11.26) (11.25) (11.18) (11.19) Amoroso Stacy half exponential power gen. Fisher-Tippett Fisher-Tippett Fréchet generalized Fréchet scaled inverse chi inverse chi (11.20) (11.13) (11.16) (11.17) (11.15) (11.14) (7.1) (7.1) (7.2) (7.5) (7.4) (7.3) (2.1) (7.1) (11.5) (11.6) (11.9) (11.8) (11.7) (11.10) (11.11) (11.12) (11.23) (11.24) (11.3) inverse Rayleigh inverse gamma scaled inverse chi-square inverse chi-square Lévy inverse exponential gamma Erlang standard gamma Porter-Thomas scaled chi-square chi-square exponential Wien Hohlfeld Nakagami scaled chi chi half normal Rayleigh Maxwell Wilson-Hilferty generalized Weibull Weibull pseudo-Weibull (k, n positive integers) a 0 . . . . . 0 0 0 . 0 0 . 0 . 0 0 0 0 0 . 0 0 . 0 0 0 0 0 0 . . . θ . . . . . . . α . 4 β . . . . <0 <0 -2 -2 -2 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 2 3 3 2 1 β 1 2 n 1 1 n 1 2k 1 2k 1 . 1 2k 1 2k . . . 1 . √1 2 . . . >0 1 2 . 2 . . . . . √ 2 . . . . . . . 1 2 n . 1 2 1 2k 1 2k 1 . 2 1 2k 1 2k 1 2 2 2 2 2 2 3 1 3 2 . n 1 1+ β1 >0 >0 >0 G. E. Crooks – Field Guide to Probability Distributions 73 11 Amoroso Distribution transformed gamma) distribution [52, 53]: Stacy(x ; θ, α, β) = { ( ) } 1 β ( x )αβ−1 x β exp − Γ (α) θ θ θ (11.2) = Amoroso(x ; 0, θ, α, β) If we drop the location parameter from Amoroso, then we obtain the Stacy, or generalized gamma distribution, the parent of the gamma family of distributions. If β is negative then the distribution is generalized inverse gamma, the parent of various inverse distributions, including the inverse gamma (11.13) and inverse chi (11.19). The Stacy distribution is obtained as the positive even powers, modulus, and powers of the modulus of a centered, normal random variable (4.1), ) ( 1 Stacy (2σ2 ) β , 12 , β ∼ Normal(0, σ) 2 β and as powers of the sum of squares of k centered, normal random variables. ) ( ) (∑ k ( )2 β 1 2 β 1 Stacy (2σ ) , 2 k, β ∼ Normal(0, σ) 1 i=1 Pseudo-Weibull distribution [54]: 1 PseudoWeibull(x ; a, θ, β) = Γ (1 + { ( ( )β )β } x−a β x−a exp − 1 |θ| θ θ β) (11.3) for β > 0 = Amoroso(x ; a, θ, 1 + 1 β , β) Proposed as another model of failure times. Half exponential power (half Subbotin) distribution [55]: { ( )β } 1 β x−a HalfExpPower(x ; a, θ, β) = 1 exp − θ Γ(β) θ = Amoroso(x ; a, θ, β1 , β) 74 G. E. Crooks – Field Guide to Probability Distributions (11.4) 11 Amoroso Distribution 2 1.5 β=4 β=3, Wilson-Hilferty 1 β=2, scaled chi 0.5 0 β=1, gamma 0 1 2 3 Figure 21: Gamma, scaled chi and Wilson-Hilferty distributions, Amoroso(x ; 0, 1, 2, β) As the name implies, half an exponential power (21.4) distribution. Special cases include β = −1 inverse exponential (11.14), β = 1 exponential (2.1), β = 23 Hohlfeld (11.5) and β = 2 half normal (11.7) distributions. Hohlfeld distribution [56]: { ( )3/2 } x−a 1 3 exp − Hohlfeld(x ; a, θ) = 2 θ Γ ( 3 ) 2θ (11.5) = HalfExpPower(x ; a, θ, 32 ) = Amoroso(x ; a, θ, 23 , 32 ) Occurs in the extreme statistics of Brownian ratchets [56, Suppl. p.5]. Special cases: Positive integer β With β = 1 we obtain the gamma family of distributions: gamma (7.1), standard gamma (7.2) and chi square (7.3) distributions. See (§7). G. E. Crooks – Field Guide to Probability Distributions 75 11 Amoroso Distribution Nakagami (generalized normal, Nakagami-m, m) distribution [57]: Nakagami(x ; a, θ, α) { ( ( )2α−1 )2 } 2 x−a x−a = exp − Γ (α)|θ| θ θ (11.6) = Amoroso(x ; a, θ, α, 2) Used to model attenuation of radio signals that reach a receiver by multiple paths [57]. Half normal (semi-normal, positive definite normal, one-sided normal) distribution [2]: )} { ( 2 (x − a)2 HalfNormal(x ; a, σ) = √ exp − 2σ2 2πσ2 (x − a)/σ > 0 √ = Amoroso(x ; a, 2σ2 , 12 , 2) (11.7) The modulus of a normal distribution about the mean. Chi (χ) distribution [2]: √ ( )k−1 { ( 2 )} 2 x x Chi(x ; k) = k √ exp − 2 Γ( 2 ) 2 (11.8) for positive integer k = ScaledChi(x ; 1, k) √ = Stacy(x ; 2, k2 , 2) √ = Amoroso(x ; 0, 2, k2 , 2) The root-mean-square of k independent standard normal variables, or the square root of a chi-square random variable. Chi(k) ∼ 76 √ ChiSqr(k) G. E. Crooks – Field Guide to Probability Distributions 11 Amoroso Distribution 1.5 α=1/2, half-normal 1 α=1, Rayleigh α=3/2, Maxwell 0.5 0 0 1 2 3 Figure 22: Half normal, Rayleigh and Maxwell distributions, Amoroso(x ; 0, 1, α, 2) Scaled chi (generalized Rayleigh) distribution [58, 2]: 2 ScaledChi(x ; σ, k) = k √ Γ ( 2 ) 2σ2 ( x )k−1 √ 2σ2 { ( 2 )} x exp − 2σ2 for positive integer k √ 2σ2 , k2 , 2) √ = Amoroso(x ; 0, 2σ2 , k2 , 2) = Stacy(x ; (11.9) The root-mean-square of k independent and identically distributed normal variables with zero mean and variance σ2 . G. E. Crooks – Field Guide to Probability Distributions 77 11 Amoroso Distribution Rayleigh (circular normal) distribution [59, 2]: Rayleigh(x ; σ) = { ( 2 )} 1 x x exp − σ2 2σ2 (11.10) = ScaledChi(x ; σ, 2) √ = Stacy(x ; 2σ2 , 1, 2) √ = Amoroso(x ; 0, 2σ2 , 1, 2) The root-mean-square of two independent and identically distributed normal variables with zero mean and variance σ2 . For instance, wind speeds are approximately Rayleigh distributed, since the horizontal components of the velocity are approximately normal, and the vertical component is typically small [60]. Maxwell (Maxwell-Boltzmann, Maxwell speed, spherical normal) distribution [61, 62]: √ { ( 2 )} 2 2 x √ Maxwell(x ; σ) = x exp − 2σ2 πσ3 (11.11) = ScaledChi(x ; σ, 3) √ = Stacy(x ; 2σ2 , 32 , 2) √ = Amoroso(x ; 0, 2σ2 , 32 , 2) The speed distribution of molecules in thermal equilibrium. The rootmean-square of three independent and identically distributed normal variables with zero mean and variance σ2 . Wilson-Hilferty distribution [63, 2]: { ( ) } x 3 3 ( x )3α−1 exp − WilsonHilferty(x ; θ, α) = Γ (α)|θ| θ θ (11.12) = Stacy(x ; θ, α, 3) = Amoroso(x ; 0, θ, α, 3) The cube root of a gamma variable follows the Wilson-Hilferty distribution [63], which has been used to approximate a normal distribution if α is 78 G. E. Crooks – Field Guide to Probability Distributions 11 Amoroso Distribution not too small. WilsonHilferty(x ; θ, α) ≈ Normal(x ; 1 − 2 2 9α , 9α ) A related approximation using quartic roots of gamma variables [64] leads to Amoroso(x ; 0, θ, α, 4). Special cases: Negative integer β With negative β we obtain various “inverse” distributions related to distriθ butions with positive β by the reciprocal transformation ( x−a θ ) 7→ ( x−a ). Inverse gamma (Pearson type V, March, Vinci) distribution [6, 2]: InvGamma(x ; θ, α) = ( )α+1 )} { ( θ 1 θ exp − Γ (α)|θ| x − a x−a (11.13) = Amoroso(x ; a, θ, α, −1) Occurs as the conjugate prior for an exponential distribution’s scale parameter [2], or the prior for variance of a normal distribution with known mean [65]. Frequently defined with zero scale parameter. Inverse exponential distribution [66]: { ( ( )2 )} θ 1 θ exp − InvExp(x ; a, θ) = |θ| x − a x−a (11.14) = InvGamma(x ; a, θ, 1) = Amoroso(x ; a, θ, 1, −1) Note that the name “inverse exponential” is occasionally used for the ordinary exponential distribution (2.1). Lévy distribution (van der Waals profile) [67]: √ Lévy(x ; a, c) = { } |c| 1 c exp − 2π (x − a)3/2 2(x − a) (11.15) = Amoroso(x ; a, c2 , 12 , −1) The Lévy distribution is notable for being stable: a linear combination of identically distributed Lévy distributions is again a Lévy distribution. The G. E. Crooks – Field Guide to Probability Distributions 79 11 Amoroso Distribution 2.5 2 β=-3 β=-2 scaled β=-1 inverse-chi inverse gamma 1.5 1 0.5 0 0 1 2 Figure 23: Inverse gamma and scaled inverse-chi distributions, Amoroso(x ; 0, 1, 2, β), negative β. other stable distributions with analytic forms are the normal distribution (4.1), which is also a limit of the Amoroso distribution, and the Cauchy distribution (9.6), which is not. Lévy distributions describe first passage times in one dimension [67]. See also the inverse Gaussian distribution (20.3), the first passage time distribution for Brownian diffusion with drift. Scaled inverse chi-square distribution [65]: ScaledInvChiSqr(x ; σ, k) ( ) k2 +1 { ( )} 2σ2 1 1 = k exp − 2σ2 x Γ ( 2 ) 2σ2 x for positive integer k = InvGamma(x ; 0, 2σ1 2 , k2 ) 1 k 2σ2 , 2 , −1) Amoroso(x ; 0, 2σ1 2 , k2 , −1) = Stacy(x ; = 80 G. E. Crooks – Field Guide to Probability Distributions (11.16) 11 Amoroso Distribution A special case of the inverse gamma distribution with half-integer α. Used as a prior for variance parameters in normal models [65]. Inverse chi-square distribution [65]: ( ) k2 +1 { ( )} 2 1 1 InvChiSqr(x ; k) = k exp − 2x Γ ( 2 ) 2x (11.17) for positive integer k = ScaledInvChiSqr(x ; 1, k) = InvGamma(x ; 0, 21 , k2 ) = Stacy(x ; 21 , k2 , −1) = Amoroso(x ; 0, 21 , k2 , −1) A standard scaled inverse chi-square distribution. Scaled inverse chi distribution [27]: ScaledInvChi(x ; σ, k) √ ( )k+1 { ( )} 2 2σ2 1 1 √ = exp − 2σ2 x2 Γ ( k2 ) 2σ2 x (11.18) √ 1 , k , −2) 2σ2 2 1 Amoroso(x ; 0, √2σ , k2 , −2) 2 = Stacy(x ; = Used as a prior for the standard deviation of a normal distribution. Inverse chi distribution [27]: InvChi(x ; k) = √ ( { ( )k+1 )} 1 2 2 1 √ exp − 2x2 Γ ( k2 ) 2x (11.19) √1 , k , −2) 2 2 Amoroso(x ; 0, √12 , k2 , −2) = Stacy(x ; = G. E. Crooks – Field Guide to Probability Distributions 81 11 Amoroso Distribution 1 standard Gumbel reversed Weibull, β=2 Frechet, β=-2 0.5 0 -3 -2 -1 0 1 2 3 Figure 24: Extreme value distributions of maxima. Inverse Rayleigh distribution [68]: ( )} )3 { ( √ 1 1 2 InvRayleigh(x ; σ) = 2 2σ √ exp − 2σ2 x2 2σ2 x 1 = Stacy(x ; √2σ2 , 1, −2) (11.20) 1 = Fréchet(x ; 0, √2σ , 2) 2 1 , 1, −2) = Amoroso(x ; 0, √2σ 2 The inverse Rayleigh distribution has been used to model failure time [69]. 82 G. E. Crooks – Field Guide to Probability Distributions 11 Amoroso Distribution Special cases: Extreme order statistics Generalized Fisher-Tippett distribution [70, 71]: GenFisherTippett(x ; a, ω, n, β) { ( ( )nβ−1 )β } nn β x − a x−a = exp −n Γ (n) ω ω ω for positive integer n (11.21) 1 β = Amoroso(x ; a, ω/n , n, β) If we take N samples from a probability distribution, then asymptotically for large N and n ≪ N, the distribution of the nth largest (or smallest) sample follows a generalized Fisher-Tippett distribution. The parameter β depends on the tail behavior of the sampled distribution. Roughly speaking, if the tail is unbounded and decays exponentially then β limits to ∞, if the tail scales as a power law then β < 0, and if the tail is finite β > 0 [31]. In these three limits we obtain the Gumbel (8.5, 8.4), Fréchet (11.26, 11.25) and Weibull (11.24,11.23) families of extreme value distribution (Extreme value distributions types I, II and III) respectively. If β/ω is negative we obtain distributions for the nth maxima, if positive then the nth minima. Fisher-Tippett (Generalized extreme value, GEV, von Mises-Jenkinson, von Mises extreme value, log-Gumbel, Brody) distribution [32, 72, 31, 3, 73]: FisherTippett(x ; a, ω, β) { ( ( )β−1 )β } x−a β x−a exp − = ω ω ω (11.22) = GenFisherTippett(x ; a, ω, 1, β) = Amoroso(x ; a, ω, 1, β) The asymptotic distribution of the extreme value from a large sample. The superclass of type I, II and III (Gumbel, Fréchet, Weibull) extreme value distributions [72]. This is the max stable distribution (distribution of maxima) with β/ω < 0 and the min stable distribution (distribution of minima) for β/ω > 0. The maximum of two Fisher-Tippett random variables (minimum if G. E. Crooks – Field Guide to Probability Distributions 83 11 Amoroso Distribution β/ω > 0) is again a Fisher-Tippett random variable. [ ] max FisherTippett(a, ω1 , β), FisherTippett(a, ω2 , β) ∼ FisherTippett(a, ω1 ω2 (ωβ 1 1/β + ωβ 2) , β) This follows since taking the maximum of two random variables is equivalent to multiplying their cumulative distribution{functions,}and the Fisher- ( x−a )β Tippett cumulative distribution function is exp − ω . Generalized Weibull distribution [70, 71]: GenWeibull(x ; a, ω, n, β) (11.23) { ( ( )nβ−1 )β } n n β x−a x−a = exp −n Γ (n) |ω| ω ω for β > 0 = GenFisherTippett(x ; a, ω, n, β) 1 = Amoroso(x ; a, ω/n β , n, β) The limiting distribution of the nth smallest value of a large number of identically distributed random variables that are at least a. If ω is negative we obtain the distribution of the nth largest value. Weibull (Fisher-Tippett type III, Gumbel type III, Rosin-Rammler, RosinRammler-Weibull, extreme value type III, Weibull-Gnedenko, stretched exponential) distribution [74, 3]: { ( ( )β−1 )β } β x−a x−a Weibull(x ; a, ω, β) = exp − |ω| ω ω (11.24) for β > 0 = FisherTippett(x ; a, ω, β) = Amoroso(x ; a, ω, 1, β) Weibull7 is the limiting distribution of the minimum of a large number of identically distributed random variables that are at least a. If ω is nega7 84 Pronounced variously as vay-bull or wye-bull. G. E. Crooks – Field Guide to Probability Distributions 11 Amoroso Distribution tive we obtain a reversed Weibull (extreme value type III) distribution for maxima. Special cases of the Weibull distribution include the exponential (β = 1) and Rayleigh (β = 2) distributions. Generalized Fréchet distribution [70, 71]: GenFréchet(x ; a, ω, n, β̄) (11.25) { ( } ( ) ) −nβ̄−1 −β̄ x−a nn β̄ x − a exp −n = Γ (n) |ω| ω ω for β̄ > 0 = GenFisherTippett(x ; a, ω, n, −β̄) 1 = Amoroso(x ; a, ω/n β , n, −β̄), The limiting distribution of the nth largest value of a large number identically distributed random variables whose moments are not all finite (i.e. heavy tailed distributions). (If the shape parameter ω is negative then minimum rather than maxima.) Fréchet (extreme value type II, Fisher-Tippett type II, Gumbel type II, inverse Weibull) distribution [75, 31]: { ( ( )−β̄−1 )−β̄ } x−a β̄ x − a exp − Fréchet(x ; a, ω, β̄) = |ω| ω ω (11.26) for β̄ > 0 = FisherTippett(x ; a, ω, −β̄) = Amoroso(x ; a, ω, 1, −β̄) The limiting distribution of the maximum of a large number identically distributed random variables whose moments are not all finite (i.e. heavy tailed distributions). (If the shape parameter ω is negative then minimum rather than maxima.) Special cases of the Fréchet distribution include the inverse exponential (β̄ = 1) and inverse Rayleigh (β̄ = 2) distributions. w G. E. Crooks – Field Guide to Probability Distributions 85 11 Amoroso Distribution Table 11.2: Properties of the Amoroso distribution Properties notation Amoroso(x ; a, θ, α, β) { ( )αβ−1 )β } ( 1 β x−a x−a PDF exp − Γ (α) θ θ θ ( ( ) ) β CDF / CCDF 1 − Q α, x−a θ θ β >0 / θ β <0 parameters a, θ, α, β in R, α > 0 support x ⩾ a θ>0 x⩽a θ<0 mode a + θ(α − 1 1 β β) αβ ⩾ 1 αβ ⩽ 1 a Γ (α + 1 β) α+ 1 β ⩾0 − α+ Γ (α) [ Γ (α+ 3 ) 2 1 1 3] Γ (α+ β )Γ (α+ β ) Γ (α+ β ) β skew sgn( β ) − 3 + 2 2 θ Γ (α) Γ (α) Γ (α)3 /[ Γ (α+ 2 ) Γ (α+ 1 )2 ]3/2 β β Γ (α) − Γ (α)2 [ 3 2 4 1 1 2 Γ (α+ β Γ (α+ β Γ (α+ β ) )Γ (α+ β ) )Γ (α+ β ) +6 ex. kurtosis Γ (α) − 4 Γ (α)2 Γ (α)3 ]/[ 2 1 2 ]2 Γ (α+ 1 )4 Γ (α+ β ) Γ (α+ β ) − 3 Γ (α)β4 −3 Γ (α) − Γ (α)2 2 β ⩾0 mean a + θ [ variance entropy 86 2 θ ln Γ (α) Γ (α + 2 β) 1 β) Γ (α)2 Γ (α + ] 2 ( ) |θ|Γ (α) + α + β1 − α ψ(α) |β| G. E. Crooks – Field Guide to Probability Distributions [53] 11 Amoroso Distribution Interrelations The Amoroso distribution is a limiting form of the generalized beta (17.1) and generalized beta prime (18.1) distributions [51]. Limits of the Amoroso distribution include gamma-exponential (8.1), log-normal (6.1), and normal (4.1) [2] and power function (5.1) distributions. GammaExp(x ; ν, λ, α) = lim Amoroso(x ; ν + βλ, −βλ, α, β) β→∞ √ 1 LogNormal(x ; a, ϑ, σ) = lim Amoroso(x ; a, ϑα−σ α , α, σ√ ) α α→∞ √ Normal(x ; µ, σ) = lim Amoroso(x ; 0, µ − σ α, √σα , α, 1) α→∞ The log-normal limit is particularly subtle [76], (§D). lim Amoroso(x ; a, ϑα−σ √ α α→∞ 1 , α, σ√ ) α Ignore normalization constants and rearrange, { x−a } ( x−a )−1 ln( θ )β β ∝ θ exp α ln( x−a θ ) −e make the requisite substitutions, { } ( x−a )−1 1 √ ln( x−a 1 x−a ϑ ) σ α ∝ ϑ exp α σ√ ln( ) − αe ϑ α expand second exponential to second order, (once more ignoring normalization terms) { ( )2 } ( x−a )−1 ∝ ϑ exp − 2σ1 2 ln x−a ϑ and reconstitute the normalization constant. = LogNormal(x ; a, ϑ, σ) G. E. Crooks – Field Guide to Probability Distributions 87 G. E. Crooks – Field Guide to Probability Distributions 12 Beta Distribution Beta (β, Beta type I, Pearson type I) distribution [5]: Beta(x ;a, s, α, γ) ( )α−1 ( ( ))γ−1 1 1 x−a x−a = 1− B(α, γ) |s| s s (12.1) = GenBeta(x ; a, s, α, γ, 1) The beta distribution is one member of Person’s distribution family, notable for having two roots located at the minimum and maximum of the distribution. The name arises from the beta function in the normalization constant. Special cases Special cases of the beta distribution are listed in table 17.1, under β = 1. With α < 1 and γ < 1 the distribution is U-shaped with a single anti-mode (U-shaped beta distribution). If (α − 1)(γ − 1) ⩽ 0 then the distribution is a monotonic J-shaped beta distribution. Standard beta (Beta) distribution: StdBeta(x ; α, γ) = 1 xα−1 (1 − x)γ−1 B(α, γ) (12.2) = Beta(x ; 0, 1, α, γ) = GenBeta(x ; 0, 1, α, γ, 1) The standard beta distribution has two shape parameters, α > 0 and γ > 0, and support x ∈ [0, 1]. 88 12 Beta Distribution 3 2.5 2 1.5 1 0.5 0 0 1 Figure 25: A beta distribution, Beta(0, 1, 2, 4) Pert (beta-pert) distribution [77, 78] is a subset of the beta distribution, parameterized by minimum (a), maximum (b) and mode (xmode ). Pert(x ; a, b, xmode ) ( )α−1 ( x−a b−x 1 B(α, γ)(b − a) b − a b−a a + 4xmode + b xmean = 6 (xmean − a)(2xmode − a − b) α= (xmode − xmean )(b − a) (b − xmean ) γ=α xmean − a = Beta(x ; a, b − a, α, γ) )γ−1 (12.3) = = GenBeta(x ; a, b − a, α, γ, 1) The PERT (Program Evaluation and Review Technique) distribution is used in project management to estimate task completion times. The modified mode +b pert distribution replaces the estimate of the mean with xmean = a+λx , 2+λ where λ is an additional parameter that controls the spread of the distribu- G. E. Crooks – Field Guide to Probability Distributions 89 12 Beta Distribution 3 2.5 2 1.5 1 0.5 0 0 1 Figure 26: A J-shaped Pearson XII distribution, Beta(0, 1, 41 , 1 34 ) tion [78]. Pearson XII distribution [7]: ( )α−1 1 1 x−a PearsonXII(x ; a, b, α) = B(α, −α + 2) |b − a| b − x (12.4) = Beta(x ; a, b − a, α, 2 − α) = GenBeta(x ; a, b − a, α, 2 − α, 1) 0<α<2 A monotonic, J-shaped special case of the beta distribution noted by Pearson [7]. 90 G. E. Crooks – Field Guide to Probability Distributions 12 Beta Distribution Table 12.1: Properties of the beta distribution Properties name Beta(x ; a, s, α, γ) )α−1 ( ( ))γ−1 ( 1 x−a x−a 1 1− PDF B(α, γ) |s| s s ( ) / B α, γ; x−a s ) s>0 s<0 CDF / CCDF = I(α, γ; x−a s B(α, γ) parameters a, s, α, γ, in R, α, γ ⩾ 0 support mode mean variance skew ex. kurtosis entropy a ⩾ x ⩾ a + s, s > 0 a + s ⩾ x ⩾ a, s < 0 α−1 a+s α+γ−2 α a+s α+γ αγ s2 (α + γ)2 (α + γ + 1) √ 2(γ − α) α + γ + 1 sgn(s) √ (α + γ + 2) αγ α, γ > 1 (α − γ)2 (α + γ + 1) − αγ(α + γ + 2) αγ(α + γ + 2)(α + γ + 3) ( ) ln(|s|) + ln B(α, γ) − (α − 1)ψ(α) 6 − (γ − 1)ψ(γ) + (α + γ − 2)ψ(α + γ) MGF not simple CF 1 F1 (α; α + γ; it) G. E. Crooks – Field Guide to Probability Distributions 91 12 Beta Distribution Triweight Biweight 1 Epanechnikov Semicircle Uniform 0.5 Arcsine 0 -1 0 1 Figure 27: Special cases of the Pearson II distribution, α = 21 , 1, 32 , 2, 3, 4. Pearson II (Symmetric beta) distribution [5]: ( ( )2 )α−1 Γ (2α) x−µ PearsonII(x ; µ, b, α) = 2α−1 1− 2 |b| Γ (α)2 b 1 (12.5) = Beta(x ; µ − b, 2b, α, α) = GenBeta(x ; µ − b, 2b, α, α, 1) A symmetric centered distribution with support [µ − b, µ + b]. Arcsine distribution [79]: Arcsine(x ; a, s) = 1 √ x−a π|s| ( s )(1 − (12.6) x−a s ) = Beta(x ; a, s, 12 , 12 ) = GenBeta(x ; a, s, 21 , 12 , 1) Describes the percentage of time spent ahead of the game in a fair coin tossing contest [3, 79]. The name comes from the inverse sine function in the 92 G. E. Crooks – Field Guide to Probability Distributions 12 Beta Distribution cumulative distribution function, ArcsineCDF(x ; 0, 1) = 2 π √ arcsin( x). Central arcsine distribution [79]: 1 √ 2π b2 − x2 = Beta(x ; b, −2b, 12 , 12 ) CentralArcsine(x ; b) = (12.7) = GenBeta(x ; b, −2b, 12 , 12 , 1) A common variant of the arcsin, with support x ∈ [−b, b] symmetric about the origin. Describes the position at a random time of a particle engaged in simple harmonic motion with amplitude b [79]. With b = 1, the limiting distribution of the proportion of time spent on the positive side of the starting position by a simple one dimensional random walk [80]. Semicircle (Wigner semicircle, Sato-Tate) distribution [81] 2 √ 2 b − x2 πb2 = Beta(x ; −b, 2b, 1 12 , 1 21 ) Semicircle(x ; b) = (12.8) = GenBeta(x ; −b, 2b, 1 12 , 1 21 , 1) As the name suggests, the probability density describes a semicircle, or more properly a half-ellipse. This distribution arises as the distribution of eigenvectors of various large random symmetric matrices. Epanechnikov (parabolic) distribution [82]: ( ( )2 ) x−µ 3 1 1− Epanechnikov(x ; µ, b) = 4 |b| b (12.9) = PearsonII(x ; µ, b, 2) = Beta(x ; µ − b, 2b, 2, 2) = GenBeta(x ; µ − b, 2b, 2, 2, 1) Used in non-parametric kernel density estimation. G. E. Crooks – Field Guide to Probability Distributions 93 12 Beta Distribution Biweight (Quartic) distribution: ( ( )2 )2 15 1 x−µ Biweight(x ; µ, b) = 1− 16 |b| b (12.10) = PearsonII(x ; µ, b, 3) = Beta(x ; µ − b, 2b, 3, 3) = GenBeta(x ; µ − b, 2b, 3, 3, 1) Used in non-parametric kernel density estimation. Triweight distribution: ( ( )2 )3 35 1 x−µ 1− Triweight(x ; µ, b) = 32 |b| b (12.11) = PearsonII(x ; µ, b, 4) = Beta(x ; µ − b, 2b, 4, 4) = GenBeta(x ; µ − b, 2b, 4, 4, 1) Used in non-parametric kernel density estimation. Interrelations The beta distribution describes the order statistics of a rectangular (1.1) distribution. OrderStatisticUniform(a,s) (x ; α, γ) = Beta(x ; a, s, α, γ) Conversely, the uniform (1.1) distribution is a special case of the beta distribution. Beta(x ; a, s, 1, 1) = Uniform(x ; a, s) The beta and gamma distributions are related by StdBeta(α, γ) ∼ StdGamma1 (α) StdGamma1 (α) + StdGamma2 (γ) which provides a convenient method of generating beta random variables, 94 G. E. Crooks – Field Guide to Probability Distributions 12 Beta Distribution given a source of gamma random variables. The Dirichlet distribution [83, 65] is a multivariate generalization of the beta distribution. The beta distribution is a special case of the generalized beta distribution (17.1), and limits to the gamma distribution (7.1). Gamma(x ; a, θ, α) = lim Beta(x ; a, θγ, α, γ) γ→∞ G. E. Crooks – Field Guide to Probability Distributions 95 G. E. Crooks – Field Guide to Probability Distributions 13 Beta Prime Distribution Beta prime (beta type II, Pearson type VI, inverse beta, variance ratio, gamma ratio, compound gamma,β ′ ) distribution [6, 3]: BetaPrime(x ; a, s, α, γ) ( )α−1 ( )−α−γ 1 x−a x−a 1 = 1+ B(α, γ) |s| s s (13.1) = GenBetaPrime(x ; a, s, α, γ, 1) for a, s, α, γ in R, α > 0, γ > 0 support x ⩾ a if s > 0, x ⩽ a if s < 0 A Pearson distribution (§19) with semi-infinite support, and both roots on the real line. Arises notable as the ratio of gamma distributions, and as the order statistics of the uniform-prime distribution (5.8). Special cases Special cases of the beta prime distribution are listed in table 18.1, under β = 1. Standard beta prime (beta prime) distribution [6]: StdBetaPrime(x ; α, γ) = 1 xα−1 (1 + x)−α−γ B(α, γ) = BetaPrime(x ; 0, 1, α, γ) = GenBetaPrime(x ; 0, 1, α, γ, 1) 96 (13.2) 13 Beta Prime Distribution 1.5 1 0.5 0 0 1 2 Figure 28: A beta prime distribution, BetaPrime(0, 1, 2, 4) F (Snedecor’s F, Fisher-Snedecor, Fisher, Fisher-F, variance-ratio, F-ratio) distribution [84, 85, 3]: k1 k2 k1 x 2 −1 k2 k2 F(x ; k1 , k2 ) = 1k1 2k2 B( 2 , 2 ) (k + k x) 12 (k1 +k2 ) 2 1 (13.3) = BetaPrime(x ; 0, kk21 , k21 , k22 ) = GenBetaPrime(x ; 0, kk12 , k21 , k22 , 1) for positive integers k1 , k2 An alternative parameterization of the beta prime distribution that derives from the ratio of two chi-squared distributions (7.3) with k1 and k2 degrees of freedom. F(k1 , k2 ) ∼ ChiSqr(k1 )/k1 ChiSqr(k2 )/k2 G. E. Crooks – Field Guide to Probability Distributions 97 13 Beta Prime Distribution 0 0 1 2 3 4 5 6 Figure 29: An inverse lomax distribution, InvLomax(0, 1, 2) Inverse Lomax (inverse Pareto) distribution [66]: ( )α−1 ( )−α−1 α x−a x−a InvLomax(x ; a, s, α) = 1+ |s| s s (13.4) = BetaPrime(x ; a, s, α, 1) = GenBetaPrime(x ; a, s, α, 1, 1) Interrelations The standard beta prime distribution is closed under inversion. StdBetaPrime(α, γ) ∼ 1 StdBetaPrime(γ, α) The beta and beta prime distributions are related by the transformation (§E) ( StdBetaPrime(α, γ) ∼ 98 1 −1 StdBeta(α, γ) )−1 G. E. Crooks – Field Guide to Probability Distributions 13 Beta Prime Distribution Table 13.1: Properties of the beta prime distribution Properties notation PDF CDF / CCDF BetaPrime(x ; a, s, α, γ) ( )α−1 ( )−α−γ 1 1 x−a x−a 1+ B(α, γ) |s| s s ) ( −1 −1 / ) ) B α, γ; (1 + ( x−a s s>0 s<0 B(α, γ) ( ) −1 −1 = I α, γ; (1 + ( x−a ) s ) parameters a, s, α, γ, in R α > 0, γ > 0 support x ⩾ a mode mean variance skew ex. kurtosis MGF x⩽a α−1 a+s γ+1 a α a+s γ−1 α(α + γ − 1) s2 (γ − 2)(γ − 1)2 not simple s>0 s<0 α⩾1 α<1 γ>1 γ>2 not simple none G. E. Crooks – Field Guide to Probability Distributions 99 13 Beta Prime Distribution and, therefore, the generalized beta prime can be realized as a transformation of the standard beta (12.2) distribution. ( )− 1 GenBetaPrime(a, s, α, γ, β) ∼ a + s StdBeta(α, γ)−1 − 1 β If the scale parameter of a gamma distribution (7.1) is also gamma distributed, the resulting compound distribution is beta prime [86]. ( ) BetaPrime(0, s, α, γ) ∼ Gamma2 0, Gamma1 (0, s, γ), α The name compound gamma distribution is occasionally used for the anchored beta prime distribution (scale parameter, but no location parameter) The beta prime distribution is a special case of both the generalized beta (17.1) and generalized beta prime (18.1) distributions, and itself limits to the gamma (7.1) and inverse gamma (11.13) distributions. Gamma(x ; 0, θ, α) = lim BetaPrime(x ; 0, θγ, α, γ) γ→∞ InvGamma(x ; θ, α) = lim BetaPrime(x ; 0, θ/γ, α, γ) γ→∞ 100 G. E. Crooks – Field Guide to Probability Distributions G. E. Crooks – Field Guide to Probability Distributions 14 Beta-Exponential Distribution The beta-exponential (Gompertz-Verhulst, generalized Gompertz-Verhulst type III, log-beta, exponential generalized beta type I) distribution [87, 88, 89] is a four parameter, continuous, univariate, unimodal probability density, with semi-infinite support. The functional form in the most straightforward parameterization is BetaExp(x ; ζ, λ, α, γ) = )γ−1 x−ζ 1 −α x−ζ ( 1 λ e 1 − e− λ B(α, γ) |λ| (14.1) for x, ζ, λ, α, γ in R, α, γ > 0, x−ζ λ >0 The four real parameters of the beta-exponential distribution consist of a location parameter ζ, a scale parameter λ, and two positive shape parameters α and γ. The standard beta-exponential distribution has zero location ζ = 0 and unit scale λ = 1. This distribution has a similar shape to the gamma (7.1) distribution. Near the boundary the density scales like xγ−1 , but decays exponentially in the wing. Special cases Exponentiated exponential tion [90, 87, 91]: (generalized exponential, Verhulst) distribu- ExpExp(x ; ζ, λ, γ) = )γ−1 x−ζ γ − x−ζ ( e λ 1 − e− λ |λ| (14.2) = BetaExp(x ; ζ, λ, 1, γ) A special case similar in shape to the gamma or Weibull (11.24) distribution. So named because the cumulative distribution function is equal to the exponential distribution function raise to a power. [ ]γ ExpExpCDF(x ; ζ, λ, γ) = ExpCDF(x ; ζ, λ) 101 14 Beta-Exponential Distribution 1 0.5 0 0 1 2 3 4 Figure 30: Beta-exponential distributions, (a) BetaExp(x ; 0, 1, 2, 2), (b) BetaExp(x ; 0, 1, 2, 4), (c) BetaExp(x ; 0, 1, 2, 8). 1 0.5 0 0 1 2 3 4 Figure 31: Exponentiated exponential distribution, ExpExp(x ; 0, 1, 2). 102 G. E. Crooks – Field Guide to Probability Distributions 14 Beta-Exponential Distribution 1 0.5 0 0 1 2 3 Figure 32: Hyperbolic sine HyperbolicSine(x ; NadarajahKotz(x) distributions. 1 2) 4 and Nadarajah-Kotz Hyperbolic sine distribution [1]: HyperbolicSine(x ; ζ, λ, γ) = = 1 B( 1−γ 2 , γ) γ−1 x−ζ )γ−1 1 ( + x−ζ e 2λ − e− 2λ |λ| 2 B( 1−γ 2 , γ)|λ| (14.3) ( )γ−1 sinh( x−ζ 2λ ) = BetaExp(x ; ζ, λ, 1−γ 2 , γ), 0<γ<1 Compare to the hyperbolic secant distribution (15.6). Nadarajah-Kotz distribution [88, 1] : 1 1 √ x−ζ π|λ| e λ − 1 = BetaExp(x ; ζ, λ, 12 , 12 ) NadarajahKotz(x ; ζ, λ) = A notable special case when α = γ = 1 2. (14.4) The cumulative distribution G. E. Crooks – Field Guide to Probability Distributions 103 14 Beta-Exponential Distribution Table 14.1: Special cases of the beta-exponential family (14.1) beta-exponential ζ λ std. beta-exponential 0 (14.2) exponentiated exponential . α γ 1 . . . 1 . . γ 1 (14.4) Nadarajah-Kotz . . 1 2 (1-γ) 1 2 (2.1) exponential . . . (14.3) hyperbolic sine . 0<γ<1 1 2 function has the simple form NadarajahKotzCDF(x ; 0, 1) = √ 2 arctan exp(x) − 1 . π Interrelations The beta-exponential distribution is a limit of the generalized beta distribution (§12). The analogous limit of the generalized beta prime distribution (§13) results in the beta-logistic family of distributions (§15). The beta-exponential distribution is the log transform of the beta distribution (12.1). ( ) StdBetaExp(α, γ) ∼ − ln StdBeta(α, γ) It follows that beta-exponential variates are related to ratios of gamma variates. StdBetaExp(α, γ) ∼ − ln StdGamma1 (α) StdGamma1 (α) + StdGamma2 (γ) The beta-exponential distribution describes the order statistics (§C) of the exponential distribution (2.1). OrderStatisticExp(ζ,λ) (x ; γ, α) = BetaExp(x ; ζ, λ, α, γ) With γ = 1 we recover the exponential distribution. λ BetaExp(x ; ζ, λ, α, 1) = Exp(x ; ζ, α ) 104 G. E. Crooks – Field Guide to Probability Distributions 14 Beta-Exponential Distribution Table 14.2: Properties of the beta-exponential distribution Properties notation BetaExp(x ; ζ, λ, α, γ) )γ−1 x−ζ 1 1 −α x−ζ ( λ PDF e 1 − e− λ B(α, γ) |λ| ) ( / x−ζ CDF CCDF I α, γ; e− λ λ>0 / λ<0 parameters ζ, λ, α, γ in R support α, γ > 0 x⩾ζ λ>0 x⩽ζ λ<0 mean ζ + λ[ψ(α + γ) − ψ(α)] variance [88] 2 λ [ψ1 (α) − ψ1 (α + γ)] [88] [ ] skew − sgn(λ) ψ2 (α) − ψ2 (α + γ) /[ ]3 ψ1 (α) − ψ1 (α + γ) 2 [88] [ 2 2 ex. kurtosis 3ψ1 (α) − 6ψ1 (α)ψ1 (α + γ) + 3ψ1 (α + γ) + ψ3 (α) ]/[ ]2 − ψ3 (α + γ) ψ1 (α) − ψ1 (α + γ) [88] entropy ln |λ| + ln B(α, γ) + (α + γ − 1)ψ(α + γ) − (γ − 1)ψ(γ) − αψ(α) B(α − λt, γ) B(α, γ) B(α − iλt, γ) CF eiζt B(α, γ) MGF eζt G. E. Crooks – Field Guide to Probability Distributions [88] [88] [88] 105 14 Beta-Exponential Distribution The beta-exponential distribution is a limit of the generalized beta distribution (17.1), and itself limits to the gamma-exponential distriution (8.1). GammaExp(x ; ν, λ, α) = lim BetaExp(x ; ν + λ/ ln γ, λ, α, γ) γ→∞ 106 G. E. Crooks – Field Guide to Probability Distributions G. E. Crooks – Field Guide to Probability Distributions 15 Beta-Logistic Distribution The beta-logistic (Prentice, beta prime exponential, generalized logistic type IV, exponential generalized beta prime, exponential generalized beta type II, log-F, generalized F, Fisher-Z, generalized Gompertz-Verhulst type II) distribution [92, 93, 3, 94] is a four parameter, continuous, univariate, unimodal probability density, with infinite support. The functional form in the most straightforward parameterization is x−ζ BetaLogistic(x ; ζ, λ, α, γ) = e−α λ 1 ( )α+γ B(α, γ)|λ| 1 + e− x−ζ λ x, ζ, λ, α, γ in R (15.1) α, γ > 0 The four real parameters consist of a location parameter ζ, a scale parameter λ, and two positive shape parameters α and γ. The standard beta-logistic distribution has zero location ζ = 0 and unit scale λ = 1. The beta-logistic distribution is perhaps most commonly referred to as ‘generalized logistic’, but this terminology is ambiguous, since many types of generalized logistic distribution have been investigated, and this distribution is not ‘generalized’ in the same sense used elsewhere in this survey (See ‘generalized’ §A). Therefore, we select the name ‘beta-logistic’ as a less ambiguous terminology that mirrors the names beta, beta-prime, and betaexponential. Special cases Burr type II (generalized logistic type I, exponential-Burr, skew-logistic) distribution [95, 2]: x−ζ γ e− λ BurrII(x ; ζ, λ, γ) = ( )γ+1 x−ζ |λ| 1 + e− λ (15.2) = BetaLogistic(x ; ζ, λ, 1, γ) 107 15 Beta-Logistic Distribution 0.5 γ=8 γ=2 0 -3 -2 -1 0 1 2 3 4 5 6 Figure 33: Burr type II distributions, BurrII(x ; 0, 1, γ) Reversed Burr type II (generalized logistic type II) distribution [2]: x−ζ e+ λ γ RevBurrII(x ; α) = ( )γ+1 x−ζ |λ| 1 + e+ λ (15.3) = BurrII(x ; ζ, −λ, γ) = BetaLogistic(x ; ζ, −λ, 1, γ) = BetaLogistic(x ; ζ, +λ, γ, 1) By setting the λ parameter to 1 (instead of α) we get a reversed Burr type II. Symmetric Beta-Logistic (generalized logistic type III, inverse cosh) distri- 108 G. E. Crooks – Field Guide to Probability Distributions 15 Beta-Logistic Distribution Table 15.1: Special cases of the beta-logistic distribution (15.1) Beta-Logistic ζ λ α γ (15.2) Burr type II . . 1 . (15.3) Reversed Burr type II . . . 1 (15.4) Symmetric Beta-Logistic . . α α (15.5) Logistic . . 1 1 (15.6) Hyperbolic secant . . 1 2 1 2 Table 15.2: Properties of the beta-logistic distribution Properties notation BetaLogistic(x ; ζ, λ, α, γ) x−ζ PDF CDF / CCDF 1 e−α λ ( )α+γ B(α, γ)|λ| 1 + e− x−ζ λ ( ) x−ζ B γ, α; (1 + e− λ )−1 B(α, γ) ) ( x−ζ = I γ, α; (1 + e− λ )−1 / λ > 0 λ < 0 [1] parameters ζ, λ, α, γ in R α, γ > 0 support x ∈ [−∞, +∞] mean ζ + λ[ψ(γ) − ψ(α)] variance λ2 [ψ1 (α) + ψ1 (γ)] ψ2 (γ) − ψ2 (α) [ψ1 (α) + ψ1 (γ)]3/2 ψ3 (α) + ψ3 (γ) ex. kurtosis [ψ1 (α) + ψ1 (γ)]2 Γ (α − λt)Γ (γ + λt) MGF eζt Γ (α)Γ (γ) Γ (α + iλt)Γ (γ − iλt) CF eiζt Γ (α)Γ (γ) skew sgn(λ) G. E. Crooks – Field Guide to Probability Distributions [3] 109 15 Beta-Logistic Distribution bution [3]: x−ζ e−α λ 1 SymBetaLogistic(x ; ζ, λ, α) = ( )2α x−ζ B(α, α)|λ| 1 + e− λ = (15.4) [1 ( )]2α 1 sech x−ζ 2λ B(α, α)|λ| 2 = BetaLogistic(x ; ζ, λ, α, α) With equal shape parameters the beta-logistic is symmetric. This distribution limits to the Laplace distribution (3.1). Logistic (sech-square, hyperbolic secant square, logit) distribution [96, 97, 3]: x−ζ 1 e− λ ( )2 x−ζ |λ| 1 + e− λ ( ) 1 x−ζ sech2 = 4|λ| λ Logistic(x ; ζ, λ) = (15.5) = BetaLogistic(x ; ζ, λ, 1, 1) Hyperbolic secant (Perks, inverse hyperbolic cosine, inverse cosh) distribution [98, 99, 3]: 1 1 x−ζ π|λ| e+ x−ζ 2λ + e− 2λ 1 sech( x−ζ = 2λ ) 2π|λ| HyperbolicSecant(x ; ζ, λ) = (15.6) = BetaLogistic(x ; ζ, λ, 12 , 12 ) The hyperbolic secant cumulative distribution function features the Gudermannian sigmoidal function, gd(z) . x−ζ 1 gd( ) π 2λ x−ζ 2 1 = arctan(e 2λ ) − π 2 HyperbolicSecantCDF(x ; ζ, λ) = The standardized hyperbolic secant distribution (zero mean, unit variance) 110 G. E. Crooks – Field Guide to Probability Distributions 15 Beta-Logistic Distribution 0.5 0 -3 -2 -1 0 1 2 3 Figure 34: Special cases of the symmetric beta-logistic distribution (15.4): Standardized (zero mean, unit variance) normal (α → ∞), logistic (α = 1), hyperbolic secant (α = 21 ), and Laplace (α → 0) (low to high peaks). is HyperbolicSecant(x ; 0, 1/π). Interrelations The beta-logistic distribution arises as a limit of the generalized beta prime distribution (§13). The analogous limit of the generalized beta distribution leads to the beta-exponential family (§14). The beta-logistic distribution is the log transform of the beta prime distribution. BetaLogistic(0, 1, α, γ) ∼ − ln BetaPrime(0, 1, α, γ) It follows that beta-logistic variates are related to ratios of gamma variates. BetaLogistic(ζ, λ, α, γ) ∼ ζ − λ ln StdGamma1 (γ) StdGamma2 (α) Negating the scale parameter is equivalent to interchanging the two G. E. Crooks – Field Guide to Probability Distributions 111 15 Beta-Logistic Distribution shape parameters. BetaLogistic(x ; ζ, +λ, α, γ) = BetaLogistic(x ; ζ, −λ, γ, α) The beta-logistic distribution, with integer α and γ is the logistic order statistics distribution [100, 20] (§C). OrderStatisticLogistic(ζ,λ) (x ; γ, α) = BetaLogistic(x ; ζ, λ, α, γ) The beta-logistic limits to the gamma exponential (8.1) and Laplace (3.1) distributions. GammaExp(x ; ν, λ, α) = lim BetaLogistic(x ; ν + λ/ ln γ, λ, α, γ) γ→∞ Laplace(x ; η, θ) = lim BetaLogistic(x ; η, θα α, α) α→0 112 G. E. Crooks – Field Guide to Probability Distributions G. E. Crooks – Field Guide to Probability Distributions 16 Pearson IV Distribution Pearson IV (skew-t) distribution [5, 101] is a four parameter, continuous, univariate, unimodal probability density, with infinite support. The functional form is PearsonIV(x ; a, s, m, v) (16.1) ( ( )2 )−m { ( )} x−a x−a 2 F1 (−iv, iv; m; 1) = 1+ exp −2v arctan 1 1 s s |s|B(m − 2 , 2 ) ( )−m+iv ( )−m−iv x−a x−a 2 F1 (−iv, iv; m; 1) 1+i = 1−i 1 1 s s |s|B(m − 2 , 2 ) x, a, s, m, v ∈ R m> 1 2 Note that the two forms are equivalent, since arctan(z) = 12 i ln 1−iz 1+iz . The first form is more conventional, but the second form displays the essential simplicity of this distribution. The density is an analytic function with two singularities, located at conjugate points in the complex plain, with conjugate, complex order. This is the one member of the Pearson distribution family that has not found significant utility. Interrelations The distribution parameters obey the symmetry PearsonIV(x ; a, s, m, v) = PearsonIV(x ; a, −s, m, −v) . Setting the complex part of the exponents to zero, v = 0, gives the Pearson VII family (9.1), which includes the Cauchy and Student’s t distributions. PearsonIV(x ; a, s, m, 0) = PearsonVII(x ; a, s, m) Suitable rescaled, the exponentiated arctan limits to an exponential of 113 16 Pearson IV Distribution the reciprocal argument. 1 lim exp(−2v arctan(−2vx) − πv) = e− x v→∞ Consequently, the high v limit of the Pearson IV distribution is an inverse gamma (Pearson V) distribution (11.13), which acts an intermediate distribution between the beta prime (Pearson VI) and Pearson IV distributions. θ α+1 lim PearsonIV(x ; 0, − 2v , 2 , v) = InvGamma(x ; θ, α) v→∞ The inverse exponential distribution (11.14) is therefore also a special case when α = 1 (m = 1). 114 G. E. Crooks – Field Guide to Probability Distributions 16 Pearson IV Distribution Table 16.1: Properties of the Pearson IV distribution Properties notation PearsonIV(x ; a, s, m, v) )2 )−m x−a PDF 1+ s { ( )} x−a × exp −2v arctan s CDF PearsonIV(x ; a, s, m, v) ( ) ( ) |s| x−a 2 × i− 2 F1 1, m + iv; 2m; x−a i−i s 2m − 1 s 2 F1 (−iv, iv; m; 1) |s|B(m − 21 , 12 ) ( ( parameters a, s, m, v in R m> support mode mean variance skew ex. kurtosis 1 2 x ∈ [−∞, +∞] sv a− m sv (m > 1) a− (m − 1) s2 v2 3 (1 + ) (m > ) 2m − 3 (m − 1)2 2 not simple not simple G. E. Crooks – Field Guide to Probability Distributions 115 G. E. Crooks – Field Guide to Probability Distributions 17 Generalized Beta Distribution The Generalized beta (beta-power) distribution [51] is a five parameter, continuous, univariate, unimodal probability density, with finite or semi infinite support. The functional form in the most straightforward parameterizaton is GenBeta(x ; a, s, α, γ, β) ( )αβ−1 ( )β )γ−1 ( 1 β x−a x−a = 1− B(α, γ) s s s (17.1) for x, a, θ, α, γ, β in R, α > 0, γ > 0 support x ∈ [a, a + s], s > 0, β > 0 x ∈ [a + s, a], s < 0, β > 0 x ∈ [a + s, +∞], s > 0, β < 0 x ∈ [−∞, a + s], s < 0, β < 0 The generalized beta distribution arises as the Weibullization of the stanβ dard beta distribution, x → ( x−a s ) , and as the order statistics of the power function distribution (5.1). The parameters consist of a location parameter a, shape parameter s and Weibull power parameter β, and two shape parameters α and γ. Special Cases The beta distribution (β=1) and specializations are described in (§12). Kumaraswamy (minimax) distribution [102, 8, 103]: β Kumaraswamy(x ; a, s, γ, β) = γ s ( x−a s )β−1 ( ( 1− x−a s )β )γ−1 (17.2) = GenBeta(x ; a, s, 1, γ, β) Proposed as an alternative to the beta distribution for modeling bounded variables, since the cumulative distribution function has a simple closed 116 17 Generalized Beta Distribution Table 17.1: Special cases of generalized beta (17.1) generalized beta a s α γ β (17.2) Kumaraswamy . . 1 . . (12.1) beta . . . . 1 (12.2) standard beta 0 1 . . 1 (12.1) beta, U shaped . . <1 <1 1 (12.1) beta, J shaped . . . . 1 (α-1)(γ-1) ⩽ 0 (12.5) Pearson II . . α α 1 (12.6) arcsine . . central arcsine -b 2b (12.8) semicircle -b 2b 1 2 1 2 1 12 1 (12.7) 1 2 1 2 1 12 (12.9) Epanechnikov . . 2 2 1 (12.10) Biweight . . 3 3 1 (12.11) Triweight . . 4 4 1 (12.4) Pearson XII . . . 2-α (13.1) beta prime . . . . -1 (5.1) power function . . 1 1 . (1.1) uniform . . 1 1 1 (1.1) standard uniform 0 1 1 1 1 1 1 1 α<2 G. E. Crooks – Field Guide to Probability Distributions 117 17 Generalized Beta Distribution Table 17.2: Properties of the generalized beta distribution Properties name GenBeta(x ; a, s, α, γ, β) )αβ−1 ( )β )γ−1 ( ( 1 β x−a x−a PDF 1− B(α, γ) s s s ) ( x−a β / B α, γ; ( s ) β > 0 βs < 0 CDF / CCDF s B(α, γ) ( ) β = I α, γ; ( x−a s ) parameters a, s, α, γ, β, in R, α, γ ⩾ 0 support mean variance x ∈ [a, a + s], 0 < s, 0 < β x ∈ [a + s, a], s < 0, 0 < β x ∈ [a + s, +∞], 0 < s, β < 0 x ∈ [−∞, a + s], s < 0, β < 0 a+ sB(α + 1 β , γ) B(α, γ) 2 s B(α + β2 , γ) B(α, γ) skew not simple ex. kurtosis not simple − α+ 1 β >0 1 2 β , γ) B(α, γ)2 s2 B(α + MGF none E(Xh ) 118 sh B(α + h β , γ) B(α, γ) a = 0, α + h β G. E. Crooks – Field Guide to Probability Distributions > 0 [51] 17 Generalized Beta Distribution 3 2.5 2 1.5 1 0.5 0 0 1 Figure 35: A Kumaraswamy distribution, Kumaraswamy(0, 1, 2, 4) form, KumaraswamyCDF(x ; 0, 1, γ, β) = 1 − (1 − xβ )γ . Interrelations The generalized beta distribution describes the order statistics of a power function distribution (5.1). OrderStatisticPowerFn(a,s,β) (x ; α, γ) = GenBeta(x ; a, s, α, γ, β) Conversely, the power function (5.1) distribution is a special case of the generalized beta distribution. GenBeta(x ; a, s, 1, 1, β) = PowerFn(x ; a, s, β) Setting β = 1 yields the beta distribution (12.1), GenBeta(x ; a, s, α, γ, 1) = Beta(x ; a, s, α, γ) , G. E. Crooks – Field Guide to Probability Distributions 119 17 Generalized Beta Distribution and setting β = −1 yields the beta prime (or inverse beta) distribution (13.1), GenBeta(x ; a, s, α, γ, −1) = BetaPrime(x ; a + s, s, γ, α) . The beta (§12) and beta prime (§13) distributions have many named special cases, see tables 17.1 and 18.1. The unit gamma distribution (10.1) arises in the limit limβ→0 with αβ = constant, δ , γ, β) = UnitGamma(x ; a, s, γ, δ) . lim GenBeta(x ; a, s, β β→0 In the limit γ → ∞ (or equivalently α → ∞) we obtain the Amoroso distribution (11.1) with semi-infinite support, the parent of the gamma distribution family [51], 1 lim GenBeta(x ; a, θγ β , α, γ, β) = Amoroso(x ; a, θ, α, β) . γ→∞ The limit limβ→+∞ yields the beta-exponential distribution (14.1) lim GenBeta(x ; ζ + βλ, −βλ, α, γ, β) = BetaExp(x ; ζ, λ, α, γ) . β→+∞ 120 G. E. Crooks – Field Guide to Probability Distributions G. E. Crooks – Field Guide to Probability Distributions 18 Generalized Beta Prime Distribution The Generalized beta prime (Feller-Pareto, beta-log-logistic, generalized gamma ratio, Majumder-Chakravart, generalized beta type II, generalized Feller-Pareto) distribution [67, 51, 104] is a five parameter, continuous, univariate, unimodal probability density, with semi-infinite support. The functional form in the most straightforward parameterization is GenBetaPrime(x ; a, s, α, γ, β) (18.1) )αβ−1 ( ( )β )−α−γ ( 1 x−a β x−a = 1+ B(α, γ) s s s a, s, α, γ, β in R, α, γ > 0 The five real parameters of the generalized beta prime distribution consist of a location parameter a, scale parameter s, two shape parameters, α and γ, and the Weibull power parameter β. The shape parameters, α and γ, are positive. The generalized beta prime arises as the Weibull transform of the standard beta prime distribution (13.2), and as order statistics of the log-logistic distribution. The Amoroso distribution is a limiting form, and a variety of other distributions occur as special cases. (See Table 18.1). These distributions are most often encountered as parametric models for survival statistics developed by economists and actuaries. Special cases Transformed beta distribution [51, 105]: TransformedBeta(x ; s, α, γ, β) ( ( x )β )−α−γ 1 β ( x )αβ−1 1+ = B(α, γ) s s s (18.2) = GenBetaPrime(x ; 0, s, α, γ, β) A generalized beta prime distribution without a location parameter, a = 0. Burr (Burr type XII, Pareto type IV, beta-P, Singh-Maddala, generalized log- 121 18 Gen. Beta Prime Distribution Table 18.1: Special cases of generalized beta prime (18.1) generalized beta prime (18.3) Burr (18.4) Dagum (18.5) paralogistic a s α γ β . . 1 . . 0 1 . 1 . 0 1 1 β . (18.6) inverse paralogistic 0 1 β 1 . (18.7) log-logistic 0 . 1 1 . (18.1) transformed beta 0 . . . . . 1 β m- β1 . (18.10) half gen. Pearson VII . (13.1) beta prime . . . . 1 (5.6) Lomax . . 1 . 1 (13.4) inverse Lomax . . . 1 1 (13.2) std. beta prime 0 1 . . 1 k1 2 k2 2 1 (13.3) F 0 k2 k1 (5.8) uniform-prime . . 1 1 1 (5.7) exponential ratio 0 . 1 1 1 . 1 2 1 2 . 2 1 2 2 (18.8) (18.9) half-Pearson VII half-Cauchy . . . logistic, exponential-gamma,Weibull-gamma) distribution [95, 106, 66]: ( ( )β−1 ( )β )−γ−1 x−a βγ x − a 1+ Burr(x ; a, s, γ, β) = |s| s s (18.3) = GenBetaPrime(x ; a, s, 1, γ, β) Most commonly encountered as a model of income distribution. Dagum (Inverse Burr, Burr type III, Dagum type I, beta-kappa, beta-k, 122 G. E. Crooks – Field Guide to Probability Distributions 18 Gen. Beta Prime Distribution Table 18.2: Properties of the generalized beta prime distribution Properties notation GenBetaPrime(x ; a, s, α, γ, β) PDF CDF / CCDF parameters )αβ−1 ( ( )β )−α−γ ( 1 x−a β x−a 1+ B(α, γ) s s s ( ) −β −1 /β B α, γ; (1 + ( x−a ) β s ) s >0 s <0 B(α, γ) ( ) −β −1 = I α, γ; (1 + ( x−a ) s ) a, s, α, γ, β in R α > 0, γ > 0 support x⩾a s>0 x⩽a mean a + variance s 2 s<0 sB(α + − 1 β) B(α, γ) 2 β, γ B(α + − 2 β) B(α, γ) skew not simple ex. kurtosis not simple E[Xh ] 1 β, γ |s|h B(α + h β, γ B(α, γ) − h β) ( − B(α + 1 β, γ − 1 β) −α < )2 1 β <γ −α < 2 β <γ B(α, γ) a = 0, −α < h β < γ [51] G. E. Crooks – Field Guide to Probability Distributions 123 18 Gen. Beta Prime Distribution Mielke) distribution [95, 107, 106]: ( )γβ−1 ( ( )β )−γ−1 βγ x − a x−a Dagum(x ; γ, β) = 1+ |s| s s (18.4) = GenBetaPrime(x ; a, s, 1, γ, −β) = GenBetaPrime(x ; a, s, γ, 1, +β) Paralogistic distribution [66]: ( x−a )β−1 β2 s Paralogistic(x ; a, s, β) = ( ) |s| (1 + x−a β )β+1 (18.5) s = GenBetaPrime(x ; a, s, 1, β, β) Inverse paralogistic distribution [105]: ( x−a )β2 −1 β2 s InvParalogistic(x ; a, s, β) = ( ) |s| (1 + x−a β )β+1 (18.6) s = GenBetaPrime(x ; a, s, β, 1, β) Log-logistic (Fisk, Weibull-exponential, Pareto type III, power prime) distribution [108, 3]: ( x−a )β−1 β s LogLogistic(x ; a, s, β) = ( ( x−a )β )2 s 1+ s (18.7) = Burr(x ; a, s, 1, β) = GenBetaPrime(x ; 0, s, 1, 1, β) Used as a parametric model for survival analysis and, in economics, as a model for the distribution of wealth or income. The logistic and log-logistic distributions are related by an exponential transform. ( ) LogLogistic(0, s, β) ∼ exp − Logistic(− ln s, β1 ) 124 G. E. Crooks – Field Guide to Probability Distributions 18 Gen. Beta Prime Distribution 2 1.5 1 0.5 0 0 1 2 Figure 36: Log-logistic distributions, LogLogistic(x ; 0, 1, β). Half-Pearson VII (half-t) distribution [109]: HalfPearsonVII(x ; a, s, m) = ( 1 B( 12 , m − 1 2) 2 1+ |s| ( x−a s )2 )−m (18.8) = GenBetaPrime(x ; a, s, 12 , m − 12 , 2) The Pearson type VII (9.1) distribution truncated at the center of symmetry. Investigated as a prior for variance parameters in hierarchal models [109]. Half-Cauchy distribution [109]: ( ( )2 )−1 x−a 2 1+ HalfCauchy(x ; a, s) = π|s| s (18.9) = HalfPearsonVII(x ; a, s, 1) = GenBetaPrime(x ; a, s, 12 , 12 , 2) A notable subclass of the Half-Pearson type VII, the Cauchy distribution G. E. Crooks – Field Guide to Probability Distributions 125 18 Gen. Beta Prime Distribution (9.6) truncated at the center of symmetry. Half generalized Pearson VII distribution [1]: HalfGenPearsonVII(x ; a, s, m, β) = β |s|B(m − 1 1 β, β) ( ( 1+ x−a s )β )−m = GenBetaPrime(x ; a, s, β1 , m − (18.10) 1 β , β) One half of a Generalized Pearson VII distribution (21.6). Special cases include half Pearson VII (18.8), half Cauchy (18.9), half Laha (See (20.14)), and uniform prime (5.8) distributions. HalfGenPearsonVII(x ; a, s, m, 2) = HalfPearsonVII(x ; a, s, m) HalfGenPearsonVII(x ; a, s, 1, 2) = HalfCauchy(x ; a, s) HalfGenPearsonVII(x ; a, s, 1, 4) = HalfLaha(x ; a, s) HalfGenPearsonVII(x ; a, s, 2, 1) = UniPrime(x ; a, s) The half exponential power (11.4) distribution occurs in the large m limit. 1 lim HalfGenPearsonVII(x ; a, θm β , m, β) = HalfExpPower(x ; a, θ, β) m→∞ Interrelations Negating the Weibull parameter of the generalized beta prime distribution is equivalent to exchanging the shape parameters α and γ. GenBetaPrime(x ; a, s, α, γ, β) = GenBetaPrime(x ; a, s, γ, α, −β) The distribution is related to ratios of gamma distributions. ( StdGamma1 (α) GenBetaPrime(a, s, α, γ, β) ∼ a + s StdGamma2 (γ) )1 β Limit of the generalized beta prime distribution include the Amoroso 126 G. E. Crooks – Field Guide to Probability Distributions 18 Gen. Beta Prime Distribution (11.1) [51] and beta-logistic (15.1) distributions. 1 lim GenBetaPrime(x ; a, θγ β , α, γ, β) = Amoroso(x ; a, θ, α, β) γ→∞ lim GenBetaPrime(x ; ζ + βλ, −βλ, α, γ, β) = BetaLogistic(x ; ζ, λ, γ, α) β→∞ Therefore, the generalized beta prime also indirectly limits to the normal (4.1), log-normal (6.1), gamma-exponential (8.1), Laplace (3.1) and powerfunction (5.1) distributions, among others. Generalized beta prime describes the order statistics (§C) of the loglogistic distribution (18.7)). OrderStatisticLogLogistic(a,s,β) (x ; γ, α) = GenBetaPrime(x ; a, s, α, γ, β) Despite occasional claims to the contrary, the log-Cauchy distribution is not a special case of the generalized beta prime distribution (generalized beta prime is mono-modal, log-Cauchy is not). G. E. Crooks – Field Guide to Probability Distributions 127 G. E. Crooks – Field Guide to Probability Distributions 19 Pearson Distribution The Pearson distributions [5, 6, 7, 110, 2] are a family of continuous, univariate, unimodal probability densities with distribution function Pearson(x ; a, s; a1 , a2 ; b0 , b1 , b2 ) )e0 ( ( = N1P 1 − r10 x−a 1− s 1 x−a r1 s )e1 (19.1) a, s, a1 , a2 , b0 , b1 , b2 , x in R √ −b1 + b21 −4b2 b0 2 r0 e0 = −ar11−a r0 = 2b −r0 √ 22 −b1 − b1 −4b2 b0 2 r1 r1 = e1 = a1r1+a 2b2 −r0 Here NP is the normalization constant. Note that the parameter a2 is redundant, and can be absorbed into the scale. Thus the Pearson distribution effectively has 4 shape parameters. We retain a2 in the general definition since this makes parameterization of subtypes easier. Pearson constructed his family of distributions by requiring that they satisfy the differential equation a1 + a2 x d ln Pearson(x ; 0, 1; a1 , a2 ; b0 , b1 , b2 ) = − , dx b0 + b1 x + b2 x2 1 a1 x + a2 x2 =− , x b0 + b1 x + b2 x2 e0 e1 = + . x − r0 x − r1 Pearson’s original motivation was that the discrete hypergeometric distribution obeys an analogous finite difference relation [110], and that at the time very few continuous, univariate, unimodal probability distributions had been described. The numbering of the a1 , a2 coefficients is chosen to be consistent with Weibull transformed generalization of the Pearson distribution (20.1), where an a0 parameter naturally arises. The Pearson distribution has three main subtypes determined by r0 and r1 , the roots of the quadratic denominator. First, we can have two roots located on the real line, at the minimum and maximum of the distribution. This is commonly known as the beta distribution (12.1). (The 128 19 Pearson Distribution parameterization is based on standard conventions.) p(x) ∝ xα−1 (1 − x)γ−1 , 00, mean µ > 0, and shape λ > 0. The name ‘inverse Gaussian’ is misleading, since this is not in any direct sense the inverse of a Gaussian distribution. The Wald distribution is a special case with µ = 1. The inverse Gaussian distribution describes first passage time in one dimensional Brownian diffusion with drift [116]. The displacement x of a diffusing particle√after a time t, with diffusion constant D and drift velocity v, is Normal(vt, 2Dt). The ‘inverse’ problem is to ask for the first passage time, the time taken to first reach a particular position y > 0, which is y2 distributed as InvGaussian( yv , 2D ). In the limit that µ goes to infinity we recover the Lévy distribution (11.15), the first passage time distribution for Brownian diffusion without drift. lim InvGaussian(x ; µ, λ) = Lévy(x ; 0, λ) µ→∞ The sum of independent inverse Gaussian random variables is also inverse Gaussian, provided that µ2 /λ is a constant. ∑ InvGaussiani (x ; µ ′ wi , λ ′ w2i ) i ( ∑ (∑ )2 ) wi , λ ′ wi ∼ InvGaussian x ; µ ′ i i Scaling an inverse Gaussian scales both µ and λ. c InvGaussian(µ, λ) ∼ InvGaussian(cµ, cλ) It follows from the previous two relations the sample mean of an inverse Gaussian is inverse Gaussian. 1 ∑ InvGaussiani (µ, λ) ∼ InvGaussian(µ, Nλ) N N i=1 134 G. E. Crooks – Field Guide to Probability Distributions 20 Grand Unified Distribution Halphen (Halphen A) distribution [117]: Halphen(x ; a, s, α, κ) (20.4) { ( ( )α−1 ) ( )−1 } 1 x−a x−a x−a = exp −κ −κ , 2|s|Kα (2κ) s s s = GUD(x ; a, s ; −κ, 1 − α, κ ; 0, 1, 0 ; 1) 0⩽ x−a s Developed by Étienne Halphen for the frequency analysis of river flows. Limits to gamma, inverse gamma, and normal. Hyperbola (harmonic) distribution [117, 118]: Hyperbola(x ; a, s, κ) (20.5) { ( ( )−1 ) ( )−1 } 1 x−a x−a x−a = exp −κ −κ , 2|s|K0 (2κ) s s s = Halphen(x ; a, s, 0, κ) = GUD(x ; a, s ; −κ, 1, κ ; 0, 1, 0 ; 1) 0⩽ x−a s Halphen B distribution [117, 118]: HalphenB(x ; a, s, α, κ) (20.6) { ( ( )α−1 )2 ( )} x−a x−a x−a 2 exp − +κ , = |s|H2α (κ) s s s = GUD(x ; a, s ; 1 − α, −κ, 2 ; 1, 0, 0 ; 1) 0⩽ x−a s The normalizing function H2α (κ) was called the exponential factorial function by Halphen [119, 118]. Limits to gamma distribution (7.1) as κ → ∞. G. E. Crooks – Field Guide to Probability Distributions 135 20 Grand Unified Distribution Inverse Halphen B distribution [120, 118]: InvHalphenB(x ; a, s, α, κ) (20.7) { ( ( )−α+1 )−2 ( )−1 } 2 x−a x−a x−a = exp − +κ , |s|H2α (κ) s s s = GUD(x; a, s; 1 − α, κ, −2; 0, 0, 1; 1) 0⩽ x−a s Limits to inverse gamma distribution (11.13) as κ → ∞. Sichel (generalized inverse Gaussian) distribution [121, 122, 123]: Sichel(x ; a, s, α, κ, λ) (20.8) { ( ( )α−1 ) ( )−1 } α/2 (κ/λ) x−a x−a x−a √ = −λ exp −κ , s s s 2|s|Kα (2 κλ) = GUD(x ; a, s ; −λ, 1 − α, κ ; 0, 1, 0 ; 1) 0⩽ x−a s Special cases include Halphen (20.4) λ = κ, and inverse Gaussian (20.3) α = 31 . Libby-Novick distribution [124, 125, 126, 127] LibbyNovick(x ; a, s, c, α, γ) )γ−1 ( )−α−γ ( x−a )α−1 ( 1 1 − x−a 1 − (1 − c) x−a = s s |s|B(α, γ) s (20.9) = GUD(x|a, s; α − 1, 3 − α − c − cγ, 2c − 2; 1, c − 2, 1 − c; 1) for a, s, c, α, γ in R, α, γ > 0 0⩽ x−a s ⩽1 A generalized three-parameter beta distribution that arises naturally as a beta distribution style ratio of gamma distributions [126]. LibbyNovick(0, ss12 , α, γ) ∼ 136 Gamma1 (0, s1 , α) Gamma1 (0, s1 , α) + Gamma2 (0, s2 , γ) G. E. Crooks – Field Guide to Probability Distributions 20 Grand Unified Distribution Limits to both the beta (u = 1) and beta-prime (u → ∞ ) distributions. Gauss hypergeometric distribution [128, 126] GaussHypergeometric(x ; a, s, u, α, γ, δ) (20.10) ( )α−1 ( )γ−1 ( )−δ 1 x−a x−a x−a = 1− 1 − (1 − u) |s|N s s s N = B(α, γ) 2 F1 (α, δ; α + γ, 1 − u) for a, s, u, α, γ, δ in R, α, γ, δ > 0 = GUD(x ; a, s ; α − 1, 2 − α − γ + (1 − u)(1 + ρ + α), u(α + γ − ρ − 2) ; 1, −1 − c, −u ; 1) 0⩽ x−a s ⩽1 A natural generalization of the three-beta distribution. Motivated by the Euler integral formula for the Gauss hypergeometric function (§F). Confluent hypergeometric distribution [129, 130, 127] Confluent(x ; α, γ, δ) (20.11) ( )α−1 ( ( ))γ−1 { ( )} x−a x−a 1 x−a 1− exp −κ = N s s s N = B(α, γ) 1 F1 (α; α + γ; −κ) = GUD(x ; 0, 1; 1 − α, α + γ + κ − 2; −κ; 1, −1, 0; 1) 0⩽ x−a s ⩽1 This distribution was introduced by Gordy [129] for applications to auction theory. Generalized Halphen [1] : GenHalphen(x ; a, s, α, κ, β) (20.12) { ( ( )βα−1 )β ( )−β } |β| x−a x−a x−a = exp −κ −κ 2|s|Kα (2κ) s s s = GUD(x ; a, s; −κ, 1 − α, κ; 0, 1, 0; β) 0⩽ x−a s ⩽1 G. E. Crooks – Field Guide to Probability Distributions 137 20 Grand Unified Distribution Greater Grand Unified Distributions There are only a few interesting specials cases of the Grand Unified Distribution with order greater than 2. Appell Beta distribution [130]: AppellBeta(x ; a, s, α, γ, ρ, δ) ( x−a )α−1 ( )γ−1 1 − x−a 1 s s = ( ) ( ) N |s| 1 − u x−a ρ 1 − v x−a δ s (20.13) s N = B(α, γ) F1 (α, ρ, δ, α + γ; u, v) = GUD(3) (x ; a, s ; a0 , a1 , a2 , a3 ; b0 , b1 , b2 , b3 ; 1) b0 = −1, b1 = 1 + u + v, b2 = −u − v − uv, b3 = uv Here F1 is the Appell hypergeometric function of the first kind. Laha distribution [131, 132, 133]: √ 1 2 ( ) Laha(x ; a, s) = 4 |s| π 1 + ( x−a s ) (20.14) = GUD(4) (x ; a, s ; 0, −4, 0, 0, 0 ; 1, 0, 2, 0, 1 ; 1) A symmetric, continuous, univariate, unimodal probability density, with infinite support. Originally introduced to disprove the belief that the ratio of two independent and identically distributed random variables is distributed as Cauchy (9.6) if, and only if, the distribution is normal. A 4th order Grand Unified Distribution (§20), and a special case of the generalized Pearson VII distribution (21.6). In contradiction to the literature [133], Laha random variates can be easily generated by noting that the distribution is symmetric, and that the half-Laha distribution (18.10) is a special case of the generalized beta prime distribution, which can itself be generated as the ratio of two gamma distributions [1]. 138 G. E. Crooks – Field Guide to Probability Distributions G. E. Crooks – Field Guide to Probability Distributions 21 Miscellaneous Distributions In this section we detail various related distributions that do not fall into the previously discussed families; either because they are not continuous, not univariate, not unimodal, or simply not simple. The notation is less uniform in this section and we do not provide detailed properties for each distribution, but instead list a few pertinent citations. Bates distribution [134, 3]: 1∑ Uniformi (0, 1) n n Bates(n) ∼ (21.1) i=1 ∼ 1 IrwinHall(n) n The mean of n independent standard uniform variates. Beta-Fisher-Tippett (generalized beta-exponential) distribution [1]: BetaFisherTippett(x ; ζ, λ, α, γ, β) ( )β−1 ( ) x−ζ β x−ζ β γ−1 1 β x−ζ e−α( λ ) 1 − e−( λ ) = B(α, γ) λ λ (21.2) for x, ζ, λ, α, γ, β in R, α, γ > 0, x−ζ λ >0 A five parameter, continuous, univariate probability density, with semiinfinite support. The Beta-Fisher-Tippett occurs as the weibullization of the beta-exponential distribution (14.1), and as the order statistics of the Fisher-Tippett distribution (11.22). OrderStatisticFisherTippett(a,s,β) (x ; α, γ) = BetaFisherTippett(x ; a, s, α, γ, β) The order statistics of the Weibull (11.24) and Fréchet (11.26) distributions are therefore also Beta-Fisher-Tippett. 139 21 Miscellaneous Distributions With β = 1 we recover the beta-exponential distribution (14.1). Other special cases include the inverse beta-exponential, β = −1 [1] (The order statistics of the inverse exponential distribution, (11.14) ), and the exponentiated Weibull (Weibull-exponential) distribution, α = 1 [135, 136]. Birnbaum-Saunders (fatigue life distribution) distribution [137, 3]: BirnbaumSaunders(x ; a, s, γ) s 1 √ ( = 2γ 2πs2 x − a √ x−a + s √ (21.3) √ √ s 2 ( x−a s − x−a ) s ) exp x−a 2γ2 Models physical fatigue failure due to crack growth. Exponential power (Box-Tiao, generalized normal, generalized error, Subbotin) distribution [138, 139]: ExpPower(x ; ζ, θ, β) = β β −| x−ζ θ | 1 e 2|θ|Γ ( β ) (21.4) A generalization of the normal distribution. Special cases include the normal, Laplace and uniform distributions. ExpPower(x ; ζ, θ, 1) = Laplace(x ; ζ, θ) √ ExpPower(x ; ζ, θ, 2) = Normal(x ; ζ, θ/ 2) lim ExpPower(x ; ζ, θ, β) = Uniform(x ; ζ − θ, 2θ) β→∞ 140 G. E. Crooks – Field Guide to Probability Distributions 21 Miscellaneous Distributions Generalized K distribution [140]: GenK(x ; s, α1 , α2 , β) = ( x ) 12 (α1 +α2 )β−1 ( (x)β ) 2|β| 2 Kα1 −α2 2 |s|Γ (α1 )Γ (α2 ) s s (21.5) x ⩾ 0, α1 > 0, α2 > 0 The Weibull transform of the K-distribution (21.8). Arises as the product of anchored Amoroso distributions with common Weibull parameters. GenK(s1 s2 , α1 , α2 , β) ∼ Amoroso1 (0, s1 , α1 , β) Amoroso2 (0, s2 , α2 , β) 1 1 ∼ s1 Gamma1 (0, α1 ) β s2 Gamma2 (0, α2 ) β ( )1 ∼ s1 s2 Gamma1 (1, α1 ) Gamma2 (1, α2 ) β 1 ∼ s1 s2 K(1, α1 , α2 ) β Generalized Pearson VII (generalized Cauchy, generalized-t) distribution [131, 141, 142, 93, 143, 144]: GenPearsonVII(x ; a, s, m, β) = β 2|s|B(m − 1 1 β, β) ( 1+ x−a s β )−m (21.6) x, a, s, m, β in R β > 0, m > 0, βm > 1 A generalization of the Pearson type VII distribution (9.1). Special cases include Pearson VII (9.1), Cauchy (9.6), Laha (20.14), Meridian (21.13) and G. E. Crooks – Field Guide to Probability Distributions 141 21 Miscellaneous Distributions exponential power (21.4) distributions, GenPearsonVII(x ; a, s, m, 2) = PearsonVII(x ; a, s, m) GenPearsonVII(x ; a, s, 1, 2) = Cauchy(x ; a, s) GenPearsonVII(x ; a, s, 1, 4) = Laha(x ; a, s) GenPearsonVII(x ; a, s, 2, 1) = Meridian(x ; a, s) lim GenPearsonVII(x ; a, m1/β θ, m, β) = ExpPower(x ; a, θ, β) m→∞ A related distribution is the half generalized Pearson VII (18.10), a special case of generalized beta prime (18.1). Holtsmark distribution [145]: Holtsmark(x ; µ, c) = Stable(x ; µ, c, 32 , 0) (21.7) A symmetric stable distribution (21.20). Although the Holtsmark distribution cannot be expressed with elementary functions, it does have an analytic form in terms of hypergeometric functions [146]. − + ( ) 5 11 1 1 5 4 x−µ 6 12 , 12 ; 3 , 2 , 6 ; − 729 ( c ) ( ) 3 5 2 5 7 4 4 x−µ 6 1 x−µ 2 3π ( c ) 3 F4 4 , 1, 4 ; 3 , 6 , 6 , 3 ; − 729 ( c ) ( 13 19 7 3 5 ) 4 x−µ 4 4 x−µ 6 7 81π Γ ( 3 )( c ) 2 F3 12 , 12 ; 6 , 2 , 3 ; − 729 ( c ) Holtsmark(x ; µ, c) = π1 Γ ( 53 ) 2 F3 K distribution [140, 147, 148, 149]: ( x ) 12 (α1 +α2 )−1 ( √x) 2 K(x ; s, α1 , α2 ) = Kα1 −α2 2 |s|Γ (α1 )Γ (α2 ) s s (21.8) x ⩾ 0, α1 > 0, α2 > 0 Note that modified Bessel function of the second kind (p.173) is symmetric with respect to its argument, Kv (+z) = Kv (−z). Thus the K-distribution is symmetric with respect to the two shape parameters, K(x ; s, α1 , α2 ) = 142 G. E. Crooks – Field Guide to Probability Distributions 21 Miscellaneous Distributions K(x ; s, α2 , α1 ). The K-distribution arises as the product of Gamma distributions [140, 148, 149]. K(s1 s2 , α1 , α2 ) ∼ Gamma1 (0, s1 , α1 ) Gamma2 (0, s2 , α2 ) The K-distribution has applications to radar scattering [147, 148] and superstatistical thermodynamics [150, Eq. 21]. Irwin-Hall (uniform sum) distribution [151, 152, 3]: IrwinHall(x ; n) = ( ) n ∑ n 1 (−1)k (x − k)n−1 sgn(x − k) 2(n − 1)! k (21.9) k=0 The sum of n independent standard uniform variates. IrwinHall(n) ∼ n ∑ Uniformi (0, 1) i=1 Related to the Bates distribution (21.1). For n = 1 we recover the uniform distribution (1.1), and with n = 2 the triangular distribution (21.22). Johnson SU distributions [153, 2]: 2 δ 1 − 12 (γ+δ sinh−1 ( x−ξ λ )) JohnsonSU(x ; µ, σ, γ, δ) = √ √ e λ 2π 1 + ( x−ξ )2 λ (21.10) Johnson’s distributions are transforms of the normal distribution, Johnsong (µ, σ, γ, δ) ∼ σg( StdNormal()−γ) )+µ δ Where for Johnson SU the function is g(x) = sinh(x). For Johnson SB the function is g(x) = 1/(1 + exp(x)), for Johnson SL , g(x) = exp(x)) (i.e. log- G. E. Crooks – Field Guide to Probability Distributions 143 21 Miscellaneous Distributions normal), and for Johnson SN the function is constant, recapitulating the normal distribution. Landau distribution [154]: Landau(x ; µ, c) = Stable(x ; µ, c, 1, 1) (21.11) A stable distribution (21.20). Describes the average energy loss of a charged particles traveling through a thin layer of matter [154]. Log-Cauchy distribution [155]: LogCauchy(x ; a, s, β) = ( )−1 |β| x − a |s|π s 1 ( ( )β )2 1 + ln x−a s (21.12) A logstable distribution with very heavy tails. The anti-log transform of the Cauchy distribution (9.6). ( ) LogCauchy(0, s, β) ∼ exp − Cauchy(− ln s, β1 ) Meridian distribution [144, Eq. 18] : Meridian(x ; a, s) = 1 1 ( ) 2|s| 1 + | x−a | 2 s The Laplace ratio distribution [144]. Meridian(x ; 0, ss12 ) ∼ Laplace1 (0, s1 ) Laplace2 (0, s2 ) A special case of the generalized Pearson VII distribution (21.6). 144 G. E. Crooks – Field Guide to Probability Distributions (21.13) 21 Miscellaneous Distributions Noncentral chi-square (Noncentral χ2 , χ ′ ) distribution [32, 3]: 2 √ 1 −(x+λ)/2 ( x ) k4 − 21 e I k −1 ( λx) 2 2 λ k, λ, x in R, > 0 NoncentralChiSqr(x ; k, λ) = (21.14) Here, Iv (z) is a modified Bessel function of the first kind (p.173). A generalization of the chi-square distribution. The distribution of the sum of k squared, independent, normal random variables with means µi and standard deviations σi , NoncentralChiSqr(k, λ) ∼ k ∑ (1 )2 Normali (µi , σi ) σi (21.15) i=1 where the non-centrality parameter λ = ∑k i=1 (µi /σi ) 2 . Non-central F distribution [32, 3] : NoncentralF(k1 , k2 , λ1 , λ2 ) ∼ NoncentralChiSqr1 (k1 , λ1 )/k1 NoncentralChiSqr2 (k2 , λ2 )/k2 for k1 , k2 , λ1 , λ2 > 0 support x > 0 (21.16) The ratio distribution of non-central chi square distributions. If both centrality parameters λ1 , λ2 are non zero, then we have a doubly non-central F distribution; if one is zero then we have a singly non-central F distribution; and if both are zero we recover the standard F distribution (13.3). Pseudo Voigt distribution [156]: PseudoVoigt(x ; a, σ, s, η) = (1 − η) Normal(x ; a, σ) + η Cauchy(x ; a, s) for 0 ⩽ η ⩽ 1 (21.17) A linear mixture of Cauchy (Lorentzian) and normal distributions. Used G. E. Crooks – Field Guide to Probability Distributions 145 21 Miscellaneous Distributions as a more analytically tractable approximation to the Voigt distribution (21.24). Rice (Rician, Rayleigh-Rice, generalized Rayleigh, noncentral-chi) distribution [157, 158]: Rice(x ; ν, σ) = ( 2 ) x x + ν2 x|ν| exp − I0 ( 2 ) σ2 2σ2 σ (21.18) x>0 Here, I0 (z) is a modified Bessel function of the first kind (p.173). The absolute value of a circular bivariate normal distribution, with nonzero mean, Rice(ν, σ) ∼ √ Normal21 (ν cos θ, σ) + Normal22 (ν sin θ, σ) thus directly related to a special case of the noncentral chi-square distribution (21.14). Rice(ν, 1)2 ∼ NoncentralChiSqr(2, ν2 ) Slash distribution [159, 2]: Slash(x) = StdNormal(x) − StdNormal(x) x2 The standard normal – standard uniform ratio distribution, Slash() ∼ StdNormal() StdUniform() √ Note that limx→0 Slash(x) = 1/ 8π. 146 G. E. Crooks – Field Guide to Probability Distributions (21.19) 21 Miscellaneous Distributions Stable (Lévy skew alpha-stable, Lévy stable) distribution [160]: The PDF of the stable distribution does not have a closed form in general. Instead, the stable distribution can be defined via the characteristic function ( ) StableCF(t ; µ, c, α, β) = exp itµ − |ct|α (1 − iβ sgn(t)Φ(α) (21.20) where Φ(α) = tan(πα/2) if α ̸= 1, else Φ(1) = −(2/π) log |t|. Location parameter µ, scale c, and two shape parameters, the index of stability or characteristic exponent α ∈ (0, 2] and a skewness parameter β ∈ [−1, 1]. This distribution is continuous and unimodal [161], symmetric if β = 0 (Lévy symmetric alpha-stable), and indefinite support, unless β = ±1 and 0 < α ⩽ 1, in which case the support is semi-infinite. If c or α is zero, the distribution limits to the degenerate distribution, (§1). Non-normal stable distributions (α < 2) are called stable Paretian distributions, since they all have long, Pareto tails. Table 21.1: Special cases of the stable family (21.20) stable µ c α β (9.6) Cauchy . . 1 0 0 0 (21.7) Holtsmark . . 3 2 (4.1) normal . . 2 1 1 (11.15) Lévy . . 1 2 (21.11) Landau . . 1 A distribution is stable if it is closed under scaling and addition, a1 Stable1 (µ, c, α, β) + a2 Stable2 (µ, c, α, β) ∼ a3 Stable3 (µ, c, α, β) + b for real constants a1 , a2 , a3 , b. The anti-log transform of a stable distribution is logstable: it is stable under multiplication instead of addition. There are three special cases of the stable distribution where the probability density functions can be expressed with elementary functions: The normal (4.1), Cauchy (9.6), and Lévy (11.15) distributions, all of which are simple. G. E. Crooks – Field Guide to Probability Distributions 147 21 Miscellaneous Distributions Suzuki distribution [162]. A compounded mixture of Rayleigh and lognormal distributions Suzuki(ϑ, σ) ∼ Rayleigh(σ ′ ) ∧′ LogNormal(0, ϑ, σ) σ (21.21) Introduced to model radio propagation in cluttered urban environments. Triangular (tine) distribution [68]: { Triangular(x ; a, b, c) = 2(x−a) (b−a)(c−a) 2(b−x) (b−a)(b−c) a⩽x⩽c c⩽x⩽b (21.22) Support x ∈ [a, b] and mode c. The wedge distribution (5.4) is a special case. Uniform difference distribution [48]: { UniformDiff(x) = (1 + x) −1 ⩾ x ⩾ 0 (1 − x) 0 ⩾ x ⩾ 1 (21.23) = Triangular(x ; −1, 1, 0) The difference of two independent standard uniform distributions (1.2). Voigt (Voigt profile, Voigtian) distribution [163]: Voigt(a, σ, s) = Normal(0, σ) + Cauchy(a, s) (21.24) The convolution of a Cauchy (Lorentzian) distribution with a normal distribution. Models the broadening of spectral lines in spectroscopy [163]. See also Pseudo Voigt distribution (21.17). 148 G. E. Crooks – Field Guide to Probability Distributions G. E. Crooks – Field Guide to Probability Distributions A Notation and Nomenclature Notation We write Amoroso(x ; a, θ, α, β) for a density function, AmorosoCDF(x ; a, θ, α, β) for the cumulative distribution function, Amoroso(a, θ, α, β) for the corresponding random variable, and X ∼ Amoroso(a, θ, α, β) to indicate that two random variables have the same probability distribution [65]. The semicolon, which we verbalize as “given” or “parameterized by“, separates the arguments from the parameters. parameter a b ζ µ ν ζ s λ σ ϑ† θ ω β α γ n k m v type location location location location location location scale scale scale scale scale scale power shape shape shape shape shape shape notes power-function arcsine, b = a + s exponential eta normal mu gamma-exponential nu beta-exponential zeta power function exponential lambda normal sigma log-normal theta Amoroso theta gen. Fisher Tippett omega power function beta > 0, beta and beta prime families alpha > 0, beta and beta prime families gamma integer > 0, number of samples or events integer > 0, degrees of freedom > 21 , Pearson IV > 0, Pearson IV † A curly theta, or “vartheta”. Throughout I have endeavored to use consistent parameterization, both within families, and between subfamilies and superfamilies. For instance, β is always the Weibull parameter. Location (or translation) parameters: a, b, ν, µ. Scale parameters: s, θ, σ. Shape parameters: α, γ, m, v. All parameters are real and the shape parameters α, γ and m are positive. The 149 A Notation and Nomenclature negation of a standard parameter is indicated by a bar, e.g. β = −β̄. In tables of special cases, for clarity we use a dot ‘.’ to indicate repetition of the base distribution’s parameters. Nomenclature interesting Informally, an “interesting distribution” is one that has acquired a name, which generally indicates that the distribution is the solution to one or more interesting problems. generalized-X The only consistent meaning is that distribution “X” is a special case of the distribution “generalized-X”. In practice, often means “add another parameter”. We use alternative nomenclature whenever practical, and generally reserve “generalized” for the power (Weibull) transformed distribution. standard-X The distribution “X” with the location parameter set to 0 and scale to 1. Not to be confused with standardized which generally indicates zero mean and unit variance. shifted-X (or translated-X) A distribution with an additional location parameter. anchored-X (or ballasted-X) A distribution with a fixed location (typically with a lower bound set to zero). scaled-X (or scale-X) A distribution with an additional scale parameter. inverse-X (Occasionally inverted-X, reciprocal-X, or negative-X) Generally labels the transformed distribution with x 7→ x1 , or more generally the distribution with the Weibull shape parameter negated, β 7→ −β. An exception is the inverse Gaussian distribution (20.3) [2]. log-X Either the anti-logarithmic or logarithmic transform of the random variable X, i.e. either exp − X() ∼ log-X() (e.g. log-normal) or − ln X() ∼ log-X(). This ambiguity arises because although the second convention may seem more logical, the log-normal convention has historical precedence. Herein, we follow the log-normal convention. 150 G. E. Crooks – Field Guide to Probability Distributions A Notation and Nomenclature X-exponential The logarithmic transform of distribution X, i.e. ln X() ∼ X-exponential(). This naming convention, which arises from the betaexponential distribution (14.1), sidesteps the confusion surrounding the log-X naming convention. reversed-X (Occasionally negative-X) The scale is negated. X of the Nth kind See “X type N”. folded-X The distribution of the absolute value of random variable X. beta-X A distribution formed by inserting the cumulative distribution function of X into the CDF of the standard beta distribution (12.2). Distributions of this form arise naturally in the study of order statistics (§C). G. E. Crooks – Field Guide to Probability Distributions 151 G. E. Crooks – Field Guide to Probability Distributions B Properties of Distributions notation The multi-letter, camel-cased function name, arguments and parameters used for the probability density of the family in this text. probability density function (PDF) The probability density fX (x) of a continuous random variable is the relative likelihood that the random variable will occur at a particular point. The probability to occur within a particular interval is given by the integral P[a ⩽ X ⩽ b] = ∫b fX (x)dx . a cumulative density function (CDF) The probability that a random variable has a value equal or less than x, typically denoted by FX (x), and also called the distribution function for short. ∫x FX (x) = fX (z)dz −∞ The probability density is equal to the derivative of the distribution function, assuming that the distribution function is continuous. fX (x) = d FX (x) dx Negating a scale parameter gives a reversed distribution with the cumulative distribution function replaced by the complimentary cumulative distribution function (CCDF = 1 − CDF). complimentary cumulative density function (CCDF) (survival function, reliability function) One minus the cumulative distribution function, 1 − FX (x).The probability that a random variable has a value greater than x. In lifetime analysis the complimentary cumulative distribution function is also called the survival function or reliability function. support The support of a probability density function are the set of values that have non-zero probability. The compliment of the support has zero probability. The range (or image) of a random variable (the set of values that can be generated) is the support of the corresponding probability density. 152 B Properties of Distributions mode The point where the distribution reaches its maximum value. An anti-mode is the point where the distribution reaches its minimum value. A distribution is called unimodal if there is only one local extremum away from the boundaries of the distribution. In other words, the distribution can have one mode ⌢ or one anti-mode ⌣, or be monotonically increasing / or decreasing \. mean The expectation value of the random variable. ∫ E[X] = x fX (x) dx Not all interesting distributions have finite means, notably the Cauchy family (9.6). Often denoted by the symbol µ. variance The variance measures the spread of a distribution. [ ] [ ] [ ]2 var[X] = E (X − E[X])2 = E X2 − E X The variance is also know as the second central moment, or second cumulant, and commonly denoted by the symbol σ. The standard deviation is the square root of the variance. central moment [( )n ] µn [X] = E X − E[X] (2.1) The nth moment about the mean. The first central moment is zero, and the second is the variance. skew A distribution is skewed if it is not symmetric. A positively skewed distribution tends to have a majority of the probability density above the mean; a negatively skewed distribution tends to have a majority of density below the mean. The standard measure of skew is the third cumulant (third central moment) normalized by the 32 power of the second cumulant. [( )3 ] X − E[X] κ3 skew[X] = E = 3 σ[X] κ2 2 G. E. Crooks – Field Guide to Probability Distributions 153 B Properties of Distributions kurtosis Kurtosis measures the peakedness of a distribution. The normal distribution has zero excess kurtosis. A positive kurtosis distribution has a sharper peak and longer tails, while a negative kurtosis distribution has a more rounded peak and shorter tails. The standard measure of kurtosis is the forth cumulant normalized by the square of the second cumulant. ExKurtosis[X] = κ4 κ2 2 This measure is called the excess kurtosis to distinguish it from an older definition of kurtosis that used the forth central moment µ4 instead of the forth cumulant. (Note that κκ242 = κµ242 − 3). entropy The differential (or continuous) entropy of a continuous probability distribution is ∫ entropy[X] = − f(x) ln f(x) dx Note that unlike the entropy of a discrete variable, the differential entropy is not invariant under a change of variables, and can be negative. moment generating function (MGF) The expectation MGFX (t) = E[etX ] . The nth derivative of the moment generating function, evaluated at 0, is equal to the nth moment of the distribution. dn MGFX (t) dtn 0 = E[Xn ] If two random variables have identical moment generating functions, then they have identical probability densities. cumulant generating function (CGF) erating function. The logarithm of the moment gen- CGFX (t) = ln E[etX ] 154 G. E. Crooks – Field Guide to Probability Distributions B Properties of Distributions Note that some authors define the cumulant generating function as the logarithm of the characteristic function. The nth derivative of the cumulant generating function, evaluated at 0, is equal to the nth cumulant of the distribution. dn CGFX (t) dtn 0 (2.2) = κn (X) The nth cumulant is a function of the first n moments of the distribution, and the second and third are equal to the second and third central moments. κ1 = E[X] [ ] κ2 = E (X − E[X])2 ] [ κ3 = E (X − E[X])3 [ ] [ ] κ4 = E (X − E[X])4 − 3 E (X − E[X])2 The cumulant expansion, if it exists, either terminates at second order (normal distribution), or continues to infinite order. Cumulants are often more useful than central moments, since cumulants are additive under summation of independent random variables. CGFX+Y (t) = CGFX (t) + CGFY (t) characteristic function (CF) Neither the moment nor cumulant generating functions need exist for a given distribution. An alternative that always exists is the characteristic function ϕX (t) = E[eitX ] , essentially the Fourier transform of the probability density function. The characteristic function for a sum of independent random variables is the product of the respective characteristic functions. ϕX+Y (t) = ϕX (t) ϕY (t) More generally, the characteristic function of any linear sum of independent random variables is ϕZ (t) = ∏ i ϕXi (ci t), Z= ∑ ci Xi . i G. E. Crooks – Field Guide to Probability Distributions 155 B Properties of Distributions quantile function The inverse of the cumulative distribution function, typically denoted F−1 (p) (or occasionally Q(p)). The median is the middle value of the inverse cumulative distribution function. 1 median[X] = F−1 X (2) Half the probability density is above the median, half below. The quantile and median rarely have simple forms. hazard function The ratio of the probability density function to the complimentary cumulative distribution function hazardX (x) = 156 fX (x) 1 − FX (x) G. E. Crooks – Field Guide to Probability Distributions G. E. Crooks – Field Guide to Probability Distributions C Order statistics Order statistics Order statistics [164]: If we draw m+n−1 independent samples from a distribution, then the distribution of the mth smallest value (or equivalently the nth largest) is OrderStatisticX (x ; m, n) = (m + n − 1)! F(x)m−1 f(x) (1 − F(x))n−1 (m − 1)!(n − 1)! Here X is a random variable, f(x) is the corresponding probability density and F(x) is the cumulative distribution function. The first term is the number of ways to separate m+n−1 things into three groups containing 1,m−1 and n − 1 things; the second is the probability of drawing m − 1 samples smaller than the sample of interest; the third term is the distribution of the mth sample; and the fourth term is the probability of drawing n − 1 larger samples. Note that the smallest value is obtained if m = 1, the largest value if n = 1, and the median value if m = n. The cumulative distribution function (CDF) for order statistics can be written in terms of the regularized beta function, I(p, q; z). ( ) OrderStatisticCDFX (x ; m, n) = I m, n; F(x) ( ) Conversely, if a CDF for a distribution has the form I m, n; F(x) , then F(x) is the cumulative distribution function of the corresponding ordering ( ) distribution. Since I α, γ; x is the CDF of the beta distribution (12.1), beta( ) generalized distributions of the form I α, γ; FX (x) (with arbitrary positive α and γ) are often referred to as ‘beta-X‘ [165], e.g. the beta-exponential distribution (14.1). The order statistic of the uniform distribution (1.1) is the beta distribution (12.1), that of the exponential distribution (2.1) is the beta-exponential distribution (14.1), and that of the power function distribution (5.1) is the 157 C Order statistics generalized beta distribution (17.1). OrderStatisticUniform(a,s) (x ; α, γ) = Beta(x ; a, s, α, γ) OrderStatisticExp(ζ,λ) (x ; γ, α) = BetaExp(x ; ζ, λ, α, γ) OrderStatisticPowerFn(a,s,β) (x ; α, γ) = GenBeta(x ; a, s, α, γ, β) OrderStatisticUniPrime(a,s) (x ; α, γ) = BetaPrime(x ; a, s, α, γ) OrderStatisticLogistic(ζ,λ) (x ; γ, α) = BetaLogistic(x ; ζ, λ, α, γ) OrderStatisticLogLogistic(a,s,β) (x ; α, γ) = GenBetaPrime(x ; a, s, α, γ, β) Extreme order statistics In the limit that n ≫ m (or equivalently m ≫ n) we obtain the distributions of extreme order statistics. Extreme order statistics depends only on the tail behavior of the sampled distribution; whether the tail is finite, exponential or power-law. This explains the central importance of the generalized beta distribution (17.1) to order statistics, since the power function distribution (5.1) displays all three classes of tail behavior, depending on the parameter β. Consequentially, the generalized beta distribution limits to the generalized Fisher-Tippett distribution (11.21), which is the parent of the other, specialized extreme order statistics. See also extreme order statistics, (§11). Median statistics If we draw N independent samples from a distribution (Where N is odd), then the distribution of the statistical median value is MedianStatisticX (x ; N) = OrderStatisticX (x ; N−1 N−1 2 , 2 ) Notable examples of median statistic distributions include MedianStatisticsUniform(a,s) (x ; 2α + 1) = PearsonII(x ; a + s, 2s, α) MedianStatisticsLogistic(a,s) (x ; 2α + 1) = SymBetaLogistic(x ; a, s, α) The median statistics of symmetric distributions are also symmetric. 158 G. E. Crooks – Field Guide to Probability Distributions C Order statistics 4 3 shape parameters Figure 38: Order Statistics Gen. Beta 1 = − 1 Beta Exp. Beta β Beta Prime = ± 1 β→∞ = β→∞ β Gen. Beta Prime β Beta-Logistic 2 1 Log-Logistic Power Func. Uniform 1 Exponential = − 1 β Uni. Prime ± 1 β→∞ 0 = β→∞ β β Logistic G. E. Crooks – Field Guide to Probability Distributions 159 G. E. Crooks – Field Guide to Probability Distributions D Limits Exponential function limit A common and important limit is ( x )ac 1+ = eax . c→+∞ c lim In particular, the X-exponential distributions are the exponential limit of Weibullized distributions. [( [ x−ζ ] [( x − a ) ] 1 x − ζ )β ] β = lim f 1 − = f e− λ lim f β→∞ β→∞ s β λ (a = ζ + βλ, s = −βλ) Exp(x ; a, θ) = lim PowerFn(x ; a + βθ, −βθ, β) β→∞ GammaExp(x ; ν, λ, α) = lim Amoroso(x ; ν + βλ, −βλ, α, β) β→∞ Gamma(x ; a, s, α) = lim UnitGamma(x ; a + βs, −βs, α, β) β→∞ BetaExp(x ; ζ, λ, α, γ) = lim GenBeta(x ; ζ + βλ, −βλ, α, γ, β) β→∞ BetaLogistic(x ; ζ, λ, α, γ) = lim GenBetaPrime(x ; ζ + βλ, −βλ, α, γ, β) β→∞ Normal(x ; µ, σ) = lim LogNormal(x ; µ + βσ, −βσ, β) β→∞ We can play the same trick with the γ shape parameter in the beta and beta prime families. ( )β ( )β [( [( x − a )γ−1 ] 1 x − a )γ−1 ] lim f 1 − = lim f 1 − γ→∞ γ→∞ s γ θ ] [ 1 x−a β s = θγ β = f e−( θ ) 1 Amoroso(x ; a, θ, α, β) = lim GenBeta(x ; a, θγ β , α, γ, β) γ→∞ Gamma(x ; a, θ, α) = lim Beta(x ; a, θγ, α, γ) γ→∞ 160 D Limits ( )β ( )β [( [( x − a )−α−γ ] 1 x − a )−α−γ ] lim f 1 + = lim f 1 + γ→∞ γ→∞ s γ θ [ ] 1 β −( x−a ) θ =f e s = θγ β 1 Amoroso(x ; a, θ, α, β) = lim GenBetaPrime(x ; a, θγ β , α, γ, β) γ→∞ Gamma(x ; 0, θ, α) = lim BetaPrime(x ; 0, θγ, α, γ) γ→∞ InvGamma(x ; θ, α) = lim BetaPrime(x ; 0, θ/γ, α, γ) γ→∞ Essentially the same limit takes the beta-exponential and beta-logistic distributions to the Gamma-Exponential distribution. GammaExp(x ; ν, λ, α) = lim BetaExp(x ; ν + λ/ ln γ, λ, α, γ) γ→∞ GammaExp(x ; ν, λ, α) = lim BetaLogistic(x ; ν + λ/ ln γ, λ, α, γ) γ→∞ Logarithmic function limit xc − 1 = ln x c→0 c lim UnitGamma(x ; a, s, γ, β) = lim GenBeta(x ; a, s, α, γ, β/α) α→∞ Gaussian function limit √ ( c lim e−z c→∞ 1 2 z )c 1+ √ = e− 2 z c G. E. Crooks – Field Guide to Probability Distributions 161 D Limits √ LogNormal(x ; a, ϑ, σ) = lim UnitGamma(x ; a, ϑeσ γ , α, γ→∞ √ Normal(x ; µ, σ) = lim Gamma(x ; µ − σ α, √σα , α) √ γ σ ) α→∞ 3 √ Normal(x ; µ, σ) = lim InvGamma(x ; µ − σ α, σα 2 , α) α→∞ lim ec+c z √ −ce c c→∞ z √ c = e− z2 2 LogNormal(x ; a, ϑ, σ) = lim Amoroso(x ; a, ϑα−σ √ α 1 ) , α, σ√ α √ √ Normal(x ; µ, σ) = lim GammaExp(x ; µ + σ α ln α, σ α, α) α→∞ α→∞ Miscellaneous limits θ α+1 InvGamma(x ; θ, α) = lim PearsonIV(x ; 0, − 2v , 2 , v) v→∞ See (§16) √ Normal(x ; µ, σ) = lim PearsonVII(x ; µ, σ 2m, m) m→∞ √ Normal(x ; µ, σ) = lim PearsonII(x ; µ, σ 8α, α) α→∞ Laplace(x ; η, θ) = lim BetaLogistic(x ; η, θα, α, α) α→0 162 G. E. Crooks – Field Guide to Probability Distributions D Limits β=1 Pearson Gen. Beta ∞ γ ← ∞ v ← ∞ γ ← ∞ 1 = β = − 1 ← Gamma-Exp. Inv. gamma Pearson VII 1 ∞ α= α= α→∞ α→ α ∞ Gamma 1 ∞ ∞ ← m m=1 Exponential β α γ→ β→∞ 1 Pearson IV 0 ∞ α=1 Uniform = Beta-Logistic v= → γ=1 ∞ Power Func. β 0 → 1 ∞ Amoroso β→∞ β α 1 ± γ→ Unit Gamma Pearson II = γ Beta Prime ∞ α β ∞ ∞ γ= 1 γ→ γ→ 2 − ← ← Beta Exp. ∞ Beta = γ→ α 1 Gen. Beta Prime β→∞ = β→∞ β β ← 3 GUD ∞ 4 shape parameters Figure 39: Limits and special cases of principal distributions Log Normal β → ∞ Normal Inv. Exponential Cauchy G. E. Crooks – Field Guide to Probability Distributions 163 G. E. Crooks – Field Guide to Probability Distributions E Algebra of Random Variables Various operations can be applied to combine or transform random variables, providing a rich tapestry of interrelations between different distributions [48, 41]. Transformations Given a continuous random variable X, with distribution function FX and density fX , and a monotonic function h(x) (either strictly increasing or strictly decreasing) on the range of X, we can create a new random variable Y , Y ∼ h(X) { ( ) FX h−1 (y) h(x) is increasing function FY (y) = ( −1 ) 1 − FX h (y) h(x) is decreasing function ( ) d −1 fY (y) = dy h (y) fX h−1 (y) In the last line above, the prefactor is the Jacobian of the transformation. For h (And h−1 ) increasing we have ( ( ) ( ) ( ) ( ) FY y) = P Y ⩽ y = P h(X) ⩽ y = P X ⩽ h−1 (y) = FX h−1 (y) and decreasing ( ( ) ( ) ( ) ( ) FY y) = P Y ⩽ y = P h(X) ⩽ y = P X ⩾ h−1 (y) = 1 − FX h−1 (y) . Linear transformation h(x) = a + sx A linear transform creates a location-scale family of distributions. Weibull transformation 1 h(x) = a + sx β 164 E Algebra of Random Variables The Weibull transform only applies to distributions with non-negative support. 1 PowerFn(a, s, β) ∼ a + s StdUniform() β 1 Weibull(a, θ, β) ∼ a + θ StdExp() β 1 LogNormal(a, ϑ, β) ∼ a + ϑ StdLogNormal() β 1 Amoroso(a, θ, α, β) ∼ a + θ StdGamma(α) β 1 GenBeta(a, s, α, γ, β) ∼ a + s StdBeta(α, γ) β 1 GenBetaPrime(a, s, α, γ, β) ∼ a + s StdBetaPrime(α, γ) β The Weibull transform is increasing if s β > 0, and decreasing if s β < 0. Inverse (reciprocal) transformation h(x) = x−1 The Weibull transform with a = 0, s = 1, and β = −1. Gamma(0, 1, α) ∼ InvGamma(0, 1, α)−1 Exp(0, 1) ∼ InvExp(0, 1)−1 Cauchy(0, 1) ∼ Cauchy(0, 1)−1 Log and anti-log transformations h(x) = − ln(x) h(x) = exp(−x) The log and anti-log transforms are inverses of one another. See p.150 for a discussion of transformed distribution naming conventions. ( ) StdUniform() ∼ exp − StdExp() ( ) StdLogNormal() ∼ exp − StdNormal() ( ) StdGamma(α) ∼ exp − StdGammaExp(α) ( ) StdBeta(α, γ) ∼ exp − StdBetaExp(α, γ) ( ) StdBetaPrime(α, γ) ∼ exp − StdBetaLogistic(α, γ) G. E. Crooks – Field Guide to Probability Distributions 165 E Algebra of Random Variables The anti-log transform converts a location parameter into a scale parameter, and a scale parameter into a Weibull shape parameter. ( ) PowerFn(0, s, β) ∼ exp − Exp(− ln s, β1 ) ( ) LogLogistic(0, s, β) ∼ exp − Logistic(− ln s, β1 ) ( ) FisherTippett(0, s, β) ∼ exp − Gumbel(− ln s, β1 ) ( ) Amoroso(0, s, α, β) ∼ exp − GammaExp(− ln s, β1 , α) ( ) LogNormal(0, ϑ, β) ∼ exp − Normal(− ln ϑ, β1 ) ) ( UnitGamma(0, s, α, β) ∼ exp − Gamma(− ln s, β1 , α) ) ( GenBeta(0, s, α, γ, β) ∼ exp − BetaExp(− ln s, β1 , α, γ) ( ) GenBetaPrime(0, s, α, γ, β) ∼ exp − BetaLogistic(− ln s, β1 , α, γ) Prime transformation [1] prime(x) = 1 x 1 , −1 prime−1 (y) = 1 y 1 +1 This transformation relates the beta and beta-prime distributions. ( ) StdUniPrime() ∼ prime StdUniform() ( ) StdBetaPrime(α, γ) ∼ prime StdBeta(α, γ) Combinations Sum The sum of two random variables is Z∼X+Y The resultant probability distribution function is the convolution of the component distribution functions. fZ (z) = (fX ∗ fY )(z) = ∫ +∞ fX (x) fY (z − x) dx −∞ 166 G. E. Crooks – Field Guide to Probability Distributions E Algebra of Random Variables The characteristic function for a sum of independent random variables is the product of the respective characteristic functions (p155). ϕX+Y (t) = ϕX (t)ϕY (t) Examples: Normal1 (µ1 , σ1 ) + Normal2 (µ2 , σ2 ) ∼ Normal3 (µ1 + µ2 , √ σ21 + σ22 ) Exp1 (a1 , θ) + Exp(a2 , θ) ∼ Gamma(a1 , a2 , θ, 2) Gamma1 (a1 , θ, α1 ) + Gamma2 (a2 , θ, α2 ) ∼ Gamma3 (a1 + a2 , θ, α1 + α2 ) Stable distributions (21.20) are those that are invariant under summation, changing only location and scale. Difference The difference of two random variables. Z∼X−Y ϕX−Y (t) = ϕX (t)ϕY (−t) Examples: UniformDiff(x) ∼ StdUniform1 (x) − StdUniform2 (x) BetaLogistic(x ; ζ1 − ζ2 , λ, α, γ) ∼ GammaExp1 (x ; ζ1 , λ, α) − GammaExp2 (x ; ζ2 , λ, γ) Product A product distribution is the product of two independent random variables. Z ∼ XY The probability distribution of Z is ∫ (z) 1 dx fZ (z) = fX (x) fY x |x| G. E. Crooks – Field Guide to Probability Distributions 167 E Algebra of Random Variables Examples: n ∏ Uniformi (0, 1) ∼ UniformProduct(n) i=1 n ∏ PowerFni (0, si , β) ∼ UnitGamma(0, i=1 n ∏ UnitGammai (0, si , αi , β) ∼ UnitGamma(0, i=1 n ∏ LogNormali (0, ϑi , βi ) ∼ LogNormali (0, i=1 n ∏ i=1 n ∏ i=1 n ∏ si , n, β) si , n ∑ αi , β) i=1 n ∑ ϑi , ( i=1 1 −2 β−2 ) i ) i=0 Ratio The ratio (or quotient) distribution is the ratio of two random variables. R∼ X Y Examples: StdGamma1 (α) StdGamma2 (γ) StdNormal1 () StdCauchy() ∼ StdNormal2 () StdBetaPrime(α, γ) ∼ Mixture A mixture (or compound) of two distributions is formed by selecting a parameter of one distribution from the probability distribution of the other. ∫ Z(x ; α) = X(x ; β)Y(β ; α) dβ For random variables this can be notated as ( ) Z(α) ∼ X Y(α) or Z(α) ∼ X(β) ∧ Y(α) . β The name ‘X-Y’ is sometimes assigned to a compound of distributions ‘X’ and ‘Y’, although this is ambiguous when there are multiple parameters that could be compounded. 168 G. E. Crooks – Field Guide to Probability Distributions E Algebra of Random Variables Transmutations Fold Folded distributions arise when only magnitude, and not the sign, of a random variable is observed. FoldedX (ζ) ∼ |X − ζ| An important example is the folded normal distribution FoldedNormal(x ; µ, σ) = 12 Normal(x ; +µ, σ) + 1 2 Normal(x ; −µ, σ) for x, µ, σ in R, x ⩾ 0 If we fold about the center of a symmetric distribution we obtain a ‘halved’ distribution. Examples already encountered are the half normal (11.7), halfPearson type VII (18.8), and half-Cauchy (18.9) distributions. A halved Laplace (3.1) distribution is exponential (2.1). Truncate A truncated distribution arises from restricting the support of a distribution. TruncatedX (x ; a, b) = f(x) |F(a) − F(b)| The truncation of a continuous, univariate, unimodal distribution is also continuous, univariate and unimodal. Examples include the Gompertz distribution (a left-truncated Gumbel (8.5) distribution) and the truncated normal distribution. Dual We create a dual distribution by interchanging the role of a variable and parameter in the probability density function. Z(z ; x) = ∫ X(x ; z) dz X(x ; z) The integral (or sum, if z takes discrete values) in the denominator ensures that the dual distribution is normalized. G. E. Crooks – Field Guide to Probability Distributions 169 E Algebra of Random Variables Tilt (exponential tilt, Esscher transform, exponential change of measure (ECM), twist) [166, 167] ( ) f(x)eθx Tiltedθ f(x) = ∫ = f(x)eθx−κ(θ) f(x)eθx dx ∫ Here κ(θ) = ln f(x)eθx dx is the cumulant generating function (p.154). Generation For an introduction to uniform random generation see Knuth [168], and for generating non-uniform variates from uniform random numbers see Devroye (1986) [41]. Fast, high quality algorithms are widely available for uniform random variables (e.g. the Mersenne Twister [169]), for the gamma distribution (e.g. the Marsaglia-Tsang fast gamma method [170]) and normal distributions (e.g. the ziggurat algorithm of Marsaglia and Tsang (2000) [171]). The exponential (§2), Laplace (§3) and power function (§5) distributions can be obtained from straightforward transformations of the uniform distribution. The remaining simple distributions can be obtained from transforms of 1 or 2 gamma random variables [41] (See gamma distribution interrelations, (§7), p53), with the exception of the Pearson IV distribution, which can be sampled with a rejection method [41, 101]. 170 G. E. Crooks – Field Guide to Probability Distributions G. E. Crooks – Field Guide to Probability Distributions F Miscellaneous mathematics Special functions Gamma function [62]: ∫∞ Γ (a) = ta−1 e−t dt 0 = (a − 1)! = (a − 1)Γ (a − 1) Γ ( 21 ) = √ π Γ (1) = 1 √ π 3 Γ(2) = 2 Γ (2) = 1 Incomplete gamma function [62]: ∫∞ Γ (a, z) = ta−1 e−t dt z Γ (a, 0) = Γ (a) Γ (1, z) = exp(−x) √ √ Γ ( 12 , z) = π erfc( z) Regularized gamma function [62]: Q(a; z) = Γ (a; z) Γ (a) √ Q( 21 ; z) = erfc( z) Q(1; z) = exp(−z) d dz Q(a; z) 1 = − Γ (a) za−1 e−z 171 F Miscellaneous mathematics Beta function [62]: ∫1 ta−1 (1 − t)b−1 dt B(a, b) = 0 Γ (a)Γ (b) = Γ (a + b) B(a, b) = B(b, a) B(1, b) = 1 b B( 12 , 12 ) = π Incomplete beta function [62]: ∫z ta−1 (1 − t)b−1 dt B(a, b; z) = 0 d dz B(a, b; z) = za−1 (1 − z)b−1 B(1, 1; z) = z Regularized beta function [62]: I(a, b; z) = B(a, b; z) B(a, b) I(a, b; 0) = 0 I(a, b; 1) = 1 I(a, b; z) = 1 − I(b, a; 1 − z) Error function [62]: 2 erf(z) = √ π 172 ∫z 2 e−t dt 0 G. E. Crooks – Field Guide to Probability Distributions F Miscellaneous mathematics Complimentary error function [62]: erfc(z) = 1 − erf(z) ∫ 2 ∞ −t2 e dt. =√ π z Gudermannian function [62]: ∫z sech(t) dt gd(z) = 0 = 2 arctan(ex ) − π 2 A sinusoidal function. Modified Bessel function of the first kind [62]: Iv (z) = ∞ ( 1 )v ∑ z 2 k=0 ( 14 z2 )k k! Γ (v + k + 1) A monotonic, exponentially growing function. Modified Bessel function of the second kind [62]: Kv (z) = π I−v (z) − Iv (z) 2 sin(vπ) Another monotonic, exponentially growing function. Arcsine function : ∫z arcsin(z) = 0 1 √ dx 1 − x2 arcsin(sin(z)) = z d dz 1 arcsin(z) = √ 1 − z2 The functional inverse of the sin function. G. E. Crooks – Field Guide to Probability Distributions 173 F Miscellaneous mathematics Arctangent function : 1 − iz arctan(z) = 21 i ln 1 + iz ∫z 1 arctan(z) = dx 1 + x2 0 arctan(tan(z)) = z d dz 1 1 + z2 arctan(z) = − arctan(−z) arctan(z) = The functional inverse of the tangent function. Hyperbolic sine function : sinh(z) = e+x − e−x 2 Hyperbolic cosine function : cosh(z) = e+x + e−x 2 Hyperbolic secant function : sech(z) = 1 2 = e+x + e−x cosh(z) Hyperbolic cosecant function : csch(z) = 1 2 = e+x − e−x sinh(z) Hypergeometric function [62, 172]: All of the preceding functions can be expressed in terms of the hypergeometric function: p Fq (a1 , a2 , . . . , ap ; b1 , b2 , . . . , bq ; z) = 174 ∞ n̄ n ∑ an̄ 1 , . . . , ap z bn̄ , . . . , bn̄ q n! n=0 1 G. E. Crooks – Field Guide to Probability Distributions F Miscellaneous mathematics where xn̄ are rising factorial powers [62, 172] xn̄ = x(x + 1) · · · (x + n − 1) = (x + n − 1)! . (x − 1)! (6.1) The most common variant is 2 F1 (a, b; c; z), the Gauss hypergeometric function, which can also be defined using an integral formula due to Euler, 1 2 F1 (a, b; c; z) = B(b, c − b) ∫1 0 tb−1 (1 − t)c−b−1 dt (1 − zt)a |z| ⩽ 1 . (6.2) The variant 1 F1 (a; c; z) is called the confluent hypergeometric function, and 0 F1 (c; z) the confluent hypergeometric limit function. Special cases include, za 2 F1 (a, 1 − b; a + 1; z) a 1 B(a, b) = 2 F1 (a, 1 − b; a + 1; 1) a za Γ (a; z) = Γ (a) − 1 F1 (a; a + 1; −z) a 2z erfc(z) = √ 1 F1 ( 12 ; 23 ; −z2 ) π B(a, b; z) = 2 sinh(z) = z0 F1 (; 32 ; z4 ) 2 cosh(z) = 0 F1 (; 21 ; z4 ) arctan(z) = z 2 F1 ( 12 , 1; 32 ; −z2 ) arcsin(z) = z 2 F1 ( 12 , 12 ; 32 ; z2 ) Iv (z) = d dz 2 F1 (a, b; c; z) = ( 12 v)v Γ (v+1) 0 F1 (; v ab c 2 F1 (a 2 + 1; z4 ) + 1, b + 1; c + 1; z) Sign function : The sign of the argument. For real arguments, the sign function is defined as −1 if x < 0 sgn(x) = 0 if x = 0 , +1 if x > 0 G. E. Crooks – Field Guide to Probability Distributions 175 F Miscellaneous mathematics and for complex arguments the sign function can be defined as { sgn(z) = z |z| if z ̸= 0 0 if z = 0 . Polygamma function [62]: The (n + 1)th logarithmic derivative of the gamma function. The first derivative is called the the digamma function (or psi function) ψ(x) ≡ ψ0 (x), and the second the trigamma function ψ1 (x). ψn (x) = = dn+1 dzn+1 ln Γ (x) dn dzn ψ(x) q-exponential and q-logarithmic functions [173, 174]: Two common and important limits are xc − 1 = ln x c→0 c lim and ( x )ac 1+ = eax . c→+∞ c lim It is sometimes useful to introduce ‘q-deformed’ exponential and logarithmic functions that extrapolate across these limits [173, 174]. exp(x) 1 (1 + (1 − q)x) 1−q expq (x) = 0 +∞ { 1−q x −1 q ̸= 1 1−q lnq (x) = ln(x) q=1 q=1 q ̸= 1, 1 + (1 − q)x > 0 q < 1, 1 + (1 − q)x ⩽ 0 q > 1, 1 + (1 − q)x ⩽ 0 Note that these q-functions are unrelated to the q-exponential function defined in combinatorial mathematics. 176 G. E. Crooks – Field Guide to Probability Distributions G. E. Crooks – Field Guide to Probability Distributions References (Recursive citations mark neologisms and other innovations [1]. ) [1] Gavin E. Crooks. 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(page 197). [183] S. Nukiyama and Y. Tanasawa. Experiments on the atomization of liquids in an air stream. Report 3 : On the droplet-size distribution in an atomized jet. Trans. SOC. Mech. Eng. Jpn., 5:62–67 (1939). Translated by E. Hope, Defence Research Board, Ottawa, Canada. (page 197). [184] P. Rosin and E. Rammler. The laws governing the fineness of powdered coal. J. Inst. Fuel, 7:29–36 (1933). (page 198). [185] J. Laherrère and D. Sornette. Stretched exponential distributions in nature and economy: “fat tails” with characteristic scales. Eur. Phys. J. B, 2:525–539 (1998). doi:10.1007/s100510050276. (page 199). 190 G. E. Crooks – Field Guide to Probability Distributions G. E. Crooks – Field Guide to Probability Distributions Index of distributions invert, inverted, or reciprocal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See inverse squared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See square of the first kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See type I of the second kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .See type II Distribution Synonym or Equation β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta β ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta prime χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . chi χ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . chi-square Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gamma Λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . log-normal Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . standard normal anchored exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . See exponential (2.1) anti-log-normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . log-normal arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.6) Amaroso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.1) Appell Beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20.13) ascending wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See wedge (5.4) ballasted Pareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lomax bell curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . normal beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.1) beta, J shaped . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See beta (12.1) beta, U shaped . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See beta (12.1) beta-exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14.1) beta-k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dagum beta-kappa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dagum beta-logistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15.1) beta log-logistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . generalized beta prime beta type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta beta type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta prime beta-P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burr beta-pert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pert beta power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . generalized beta [175] [130] [51] [51] [1] [1] [51] ** Citations in this table document the origin (or early usage) of the distribution name. 191 Index of distributions beta prime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13.1) beta prime exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta-logistic biexponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laplace bilateral exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laplace biweight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.10) BHP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.7) Box-Tiao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . exponential power Bramwell-Holdsworth-Pinton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BHP Breit-Wigner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy Brody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fisher-Tippett Burr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18.3) Burr type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . uniform Burr type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15.2) Burr type III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dagum Burr type XII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burr Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(9.6) Cauchy-Lorentz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy central arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.7) chi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.8) chi-square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.3) chi-square-exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.3) circular normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Rayleigh Coale-McNeil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gamma-exponential Cobb-Douglas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . log-normal confluent hypergeometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20.11) compound gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta prime Dagum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18.4) Dagum type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dagum de Moivre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . normal degenerate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See uniform (1.1) delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . degenerate descending wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See wedge (5.4) Dirac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . degenerate double exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gumbel or Laplace doubly exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gumbel doubly non-central F . . . . . . . . . . . . . . . . . . . . . . . . See non-central F (21.16) Epanechnikov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.9) Erlang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See gamma (7.1) 192 G. E. Crooks – Field Guide to Probability Distributions [1] [176] [1] [177] [143] Index of distributions error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . normal error function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .See normal (4.1) exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.1) exponential-Burr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burr type II exponential-gamma . . . . . . . . . . . . . . . . . . . . . . Burr or gamma-exponential exponential generalized beta type I . . . . . . . . . . . . . . . . . . beta-exponential exponential generalized beta type II . . . . . . . . . . . . . . . . . . . . . . beta-logistic exponential generalized beta prime . . . . . . . . . . . . . . . . . . . . . . beta-logistic exponential power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21.4) exponential ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.7) exponentiated exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14.2) extreme value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Gumbel extreme value type N . . . . . . . . . . . . . . . . . . . . . . . . . . .Fisher-Tippett type N F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13.3) F-ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F Feller-Pareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . generalized beta prime Fisher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F or Student’s t Fisher-F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F Fisher-Snedecor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F Fisher-Tippett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.22) Fisher-Tippett type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gumbel Fisher-Tippett type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fréchet Fisher-Tippett type III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weibull Fisher-Tippett-Gumbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gumbel Fisher-z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta-logistic Fisk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . log-logistic flat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . uniform Fréchet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.26) FTG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fisher-Tippett-Gumbel Galton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . log-normal Galton-McAlister . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . log-normal gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.1) gamma-exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.1) gamma ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .beta prime Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . normal Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . normal Gauss hypergeometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20.10) generalized arcsin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pearson type II G. E. Crooks – Field Guide to Probability Distributions [86] [125] [125] [3] 193 Index of distributions generalized beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (17.1) generalized beta prime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18.1) generalized beta type II . . . . . . . . . . . . . . . . . . . . . . . . generalized beta prime generalized error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . exponential power generalized exponential . . . . . . . . . . . . . . . . . . . exponentiated exponential generalized extreme value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fisher-Tippett generalized F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta-logistic generalized Feller-Pareto . . . . . . . . . . . . . . . . . . . . . . generalized beta prime generalized Fisher-Tippett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.21) generalized Fréchet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.25) generalized gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stacy or Amoroso generalized gamma ratio . . . . . . . . . . . . . . . . . . . . . . .generalized beta prime generalized Gompertz . . . . . . . . . . . . . . . . . . . . . . . . . . . . gamma-exponential generalized Gompertz-Verhulst type I . . . . . . . . . . . . gamma-exponential generalized Gompertz-Verhulst type II . . . . . . . . . . . . . . . . . . . beta-logistic generalized Gompertz-Verhulst type III . . . . . . . . . . . . . .beta-exponential generalized Gumbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(8.4) generalized Halphen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20.12) generalized inverse gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . See Stacy (11.2) generalized inverse Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sichel generalized K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21.5) generalized log-logistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burr generalized logistic type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burr type II generalized logistic type II . . . . . . . . . . . . . . . . . . . . . . . reversed Burr type II generalized logistic type III . . . . . . . . . . . . . . . . . . . symmetric beta-logistic generalized logistic type IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta-logistic generalized normal . . . . . . . . . . . . . . . . . . Nakagami or exponential power generalized Pareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.2) generalized Rayleigh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scaled-chi generalized semi-normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stacy generalized Weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.23) or Stacy GEV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . generalized extreme value Gibrat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . standard log-normal Gompertz-Verhulst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta-exponential grand unified distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20.1) GUD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . grand unified distribution Gumbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.5) Gumbel-Fisher-Tippett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gumbel 194 G. E. Crooks – Field Guide to Probability Distributions [178] [125] [179] [180] [181] [3] [87] [87] [87] [1] [87] [2] [179] [1] [1] Index of distributions Gumbel type N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fisher-Tippett type N half Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18.9) half exponential power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.4) half generalized Pearson VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18.10) half Laha . . . . . . . . . . . . . . . . . . . . . See half generalized Pearson VII (18.10) half normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.7) half Pearson Type VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18.8) half Subbotin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . half exponential power half-t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . half-Pearson Type VII half-uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See uniform (1.1) Halphen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20.4) Halphen A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Halphen Halphen B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20.6) harmonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .hyperbola Hohlfeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.5) hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20.5) hyperbolic secant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15.6) hyperbolic secant square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . logistic hyperbolic sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14.3) hydrograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Stacy hyper gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stacy inverse beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta prime inverse beta-exponential . . . . . . . . . . . . . . . . See Beta-Fisher-Tippett (21.2) inverse Burr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Dagum inverse chi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.19) inverse chi-square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.17) inverse exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.14) or exponential inverse gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.13) inverse Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20.3) inverse Halphen B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20.7) inverse hyperbolic cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . hyperbolic secant inverse Lomax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13.4) inverse normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inverse Gaussian inverse Rayleigh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.20) inverse paralogistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18.6) inverse Pareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inverse Lomax inverse Weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fréchet Johnson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Johnson SU G. E. Crooks – Field Guide to Probability Distributions [1] [1] [1] 195 Index of distributions Johnson SB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see Johnson SU , (21.10) Johnson SL . . . . . . . . . . . . . . . . . . . . . . . log-normal, see Johnson SU , (21.10) Johnson SN . . . . . . . . . . . . . . . . . . . . . . . . . . . normal, see Johnson SU , (21.10) Johnson SU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21.10) K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21.8) Kumaraswamy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (17.2) Laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.1) Laplace’s first law of error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laplace Laplace’s second law of error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . normal Laplace-Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . normal Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laplace law of error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . normal left triangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . descending wedge Leonard hydrograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stacy Lévy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.15) Libby-Novick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20.9) log-beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta-exponential log-Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21.12) log-chi-square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . chi-square-exponential log-F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta-logistic log-gamma . . . . . . . . . . . . . . . . . . . . . . . gamma-exponential or unit-gamma log-Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . log-normal log-Gumbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fisher-Tippett log-logistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18.7) log-normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.1) log-normal, two parameter . . . . . . . . . . . . . . . . . . . . . . anchored log-normal log-Weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gumbel logarithmic-normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . log-normal logarithmico-normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . log-normal logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .logistic logistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15.5) Lomax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.6) Lorentz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy Lorentzian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nakagami Majumder-Chakravart . . . . . . . . . . . . . . . . . . . . . . . . . generalized beta prime March . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inverse gamma max stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See Fisher-Tippett (11.22) 196 G. E. Crooks – Field Guide to Probability Distributions [20] [57] [125] Index of distributions Maxwell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.11) Maxwell-Boltzmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maxwell Maxwell speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maxwell Mielke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dagum min stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See Fisher-Tippett (11.22) minimax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kumaraswamy modified Lorentzian . . . . . . . . . . . . . . . . . . . . . . . . . relativistic Breit-Wigner modified pert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See pert (12.3) Moyal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.8) m-Erlang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Erlang Nadarajah-Kotz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14.4) Nakagami . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(11.6) Nakagami-m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nakagami negative exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . exponential non-central F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21.16) normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.1) normal ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy Nukiyama-Tanasawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stacy one-sided normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .half normal parabolic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Epanechnikov paralogistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18.5) Pareto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.5) Pareto type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pareto Pareto type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lomax Pareto type III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . log-logistic Pareto type IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burr Pearson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19.1) Pearson type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta Pearson type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.5) Pearson type III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gamma Pearson type IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16.1) Pearson type V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inverse gamma Pearson type VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta prime Pearson type VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.1) Pearson type VIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See power function (5.1) Pearson type IX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See power function (5.1) Pearson type X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . exponential Pearson type XI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pareto G. E. Crooks – Field Guide to Probability Distributions [8] [182] [1] [183] [7] 197 Index of distributions Pearson type XII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.4) Perks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . hyperbolic secant Pert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.3) Poisson’s first law of error . . . . . . . . . . . . . . . . . . . . . . . . . . . standard Laplace Porter-Thomas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.5) positive definite normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . half normal power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . power function power function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.1) power prime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18.7) Prentice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta-logistic pseudo-Voigt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21.17) pseudo-Weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.3) q-exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.3) q-Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19.2) quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . biweight Rayleigh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.10) rectangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .uniform relativistic Breit-Wigner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.8) reversed Burr type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15.3) reversed Weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See Weibull (11.24) right triangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ascending wedge Rosin-Rammler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weibull Rosin-Rammler-Weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weibull Sato-Tate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . semicircle sech-square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . logistic Sichel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sichel Singh-Maddala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burr singly non-central F . . . . . . . . . . . . . . . . . . . . . . . . . See non-central F (21.16) scaled chi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.9) scaled chi-square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.4) scaled inverse chi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(11.18) scaled inverse chi-square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.16) semicircle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(12.8) semi-normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . half normal skew-t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pearson Type IV skew logistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burr type II Snedecor’s F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F spherical normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maxwell 198 G. E. Crooks – Field Guide to Probability Distributions [1] [94] [184] [65] Index of distributions Stacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.2) Stacy-Mihram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amoroso standard Amoroso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .standard gamma standard beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.2) standard beta-exponential . . . . . . . . . . . . . . . . . See beta-exponential (14.1) standard beta-logistic . . . . . . . . . . . . . . . . . . . . . . . . . . See beta-logistic (15.1) standard beta-prime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13.2) standard Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.7) standard exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . See exponential (2.1) standard gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.2) standard Gumbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.6) standard gamma-exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.2) standard Laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See Laplace (3.1) standard log-normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See log-normal (6.1) standard normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See normal (4.1) standard uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.2) standardized normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . standard normal standardized uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See uniform (1.1) stretched exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weibull Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Student’s-t Student’s-t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.2) Student’s-t2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.3) Student’s-t3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.4) Student’s-z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.5) Student-Fisher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Student’s-t Subbotin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . exponential power symmetric beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pearson II symmetric Pearson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q-Gaussian symmetric beta-logistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(15.4) t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Student’s-t t2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Student’s-t2 t3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Student’s-t3 transformed beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(18.2) transformed gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stacy triweight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.11) two-tailed exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laplace uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.1) uniform prime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.8) G. E. Crooks – Field Guide to Probability Distributions [185] [131] [1] [1] 199 Index of distributions uniform product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.2) unbounded uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See uniform (1.1) unit gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.1) unit normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . standard normal van der Waals profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lévy variance ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . beta prime Verhulst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . exponentiated exponential Vienna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wien Vinci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .inverse gamma von Mises extreme value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fisher-Tippett von Mises-Jenkinson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fisher-Tippett waiting time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . exponential Wald . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See inverse Gaussian (20.3) wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.4) Weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.24) Weibull-exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . log-logistic Weibull-gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burr Weibull-Gnedenko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weibull Wien . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . See gamma (7.1) Wigner semicircle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . semicircle Wilson-Hilferty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.12) Witch of Agnesi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . standard normal 200 G. E. Crooks – Field Guide to Probability Distributions [155] [176] G. E. Crooks – Field Guide to Probability Distributions Subject Index B(a, b), see beta function B(a, b; z), see incomplete beta function F−1 (p), see quantile function p Fq , see hypergeometric function F(x), see cumulative distribution function I(a, b; z), see regularized beta function Iv (z), see modified Bessel function of the first kind Kv (z), see modified Bessel function of the second kind Q(a; z), see regularized gamma function Γ (a), see gamma function Γ (a, z), see incomplete gamma function arcsin(z), see arcsine function arctan(z), see arctangent function csch(z), see hyperbolic cosecant function E, see mean cosh(z), see hyperbolic cosine function erfc(z), see complimentary error function erf(z), see error function gd(z), see Gudermannian function sgn(x), see sign function ϕ(t), seecharacteristic function155 ψ(x), see digamma function ψ1 (x), see trigamma function ψn (x), see polygamma function sech(z), see hyperbolic secant function sinh(z), see hyperbolic sine function ∧, see mixture distributions anchored, 150 anti-log transform, 150, 165 anti-mode, 153 arcsine function, 173 arctangent function, 174 ballasted, 150 beta function, 172 beta-generalized distributions, 157 CCDF, see complimentary cumulative distribution function CDF, see cumulative distribution function central limit theorem, 34 central moment, 153 CF, see characteristic function CGF, see cumulant generating function characteristic function, 155, 167 complimentary cumulative distribution function, 152 complimentary error function, 173 compound distributions, 168 confluent hypergeometric function, 175 confluent hypergeometric limit function, 175 convolution, 166 cumulant generating function, 154 cumulants, 154 cumulative distribution function, 152 density, 152 difference distribution, 167 diffusion, 80, 93, 134 digamma function, 176 Dirchlet distribution, 95 201 Subject Index distribution function, see cumulative distribution function dual distributions, 169 entropy, 154 error function, 172 Esscher transform, 170 excess kurtosis, 154 exponential change of measure, 170 exponential factorial function, 135 exponential tilt, 170 extreme order statistics, 83, 158 first passage time, 80, 134 fold, 169 folded, 151 folded distributions, 169 gamma function, 171 Gauss hypergeometric function, 175 Gaussian function limit, 68, 161 generalized, 150 geometric distribution, 28 given, 149 Gudermannian function, 110, 173 half, 169 halved-distribution, 169 hazard function, 156 hyperbolic cosecant function, 174 hyperbolic cosine function, 174 hyperbolic secant function, 174 hyperbolic sine function, 174 hypergeometric function, 174 inverse cumulative distribution function, see quantile function inverse probability integral transform, 25 inverse transform, 165 inverse transform sampling, 26 inverted, 150 Jacobian, 164 kurtosis, 154 limits, 160, 176 linear transformation, 164 location parameter, 149, 164 location parameters, 150 location-scale family, 164 log transform, 150, 151, 165 Logarithmic function limit, 161 logstable, 147 mean, 153 median, 156, 158 median statistics, 158 memoryless, 28 MGF, see moment generating function mixture distributions, 168 mode, 153 modified Bessel function of the first kind, 173 modified Bessel function of the second kind, 173 moment generating function, 154 moments, 154 order statistics, 157 image, 152 incomplete beta function, 172 incomplete gamma function, 171 interesting, 150 inverse, 150 202 PDF, see probability density function polygamma function, 176 prime transform, 166 G. E. Crooks – Field Guide to Probability Distributions Subject Index probability density function , 152 product distributions, 167 psi function, see digamma function q-deformed functions, 176 q-exponential function, 176 q-logarithm function, 176 quantile function, 156 quotient distributions, see ratio distributions Rademacher distribution (discrete), see sign distribution random number generation, 170 range, 152 ratio distributions, 168 reciprocal, 150 reciprocal transform, 165 recursion, 177, 202 regularized beta function, 172 regularized gamma function, 171 reliability function, 152 reversed, 151 shape parameter, 149 shifted, 150 sign distribution (discrete), 53 sign function, 175 skew, 153 Smirnov transform, 25 stable, 147 stable distributions, 35, 63, 79 standard, 150 standard deviation, 153 standardized, 150 sum distributions, 166 support, 152 survival function, 152, 156 tilt, 170 transforms, 164 trigamma function, 176 truncate, 169 unimodal, 153 variance, 153 Weibull transform, 149, 165 scale parameter, 149, 150, 164 scaled, 150 202 Zipf distribution, 40 G. E. Crooks – Field Guide to Probability Distributions Subject Index This guide is inevitably incomplete, inaccurate, and otherwise imperfect — caveat emptor. 204 G. E. Crooks – Field Guide to Probability Distributions
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