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 . .

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Zero shape parameters
1 Uniform Distribution
Uniform . . . . . .
Special cases . . . . . . .
Half uniform . . . .
Unbounded uniform
Degenerate . . . . .
Interrelations . . . . . .

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2 Exponential Distribution
Exponential . . . . . .
Special cases . . . . . . . .
Anchored exponential
Standard exponential
Interrelations . . . . . . .

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3 Laplace Distribution
Laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Standard Laplace . . . . . . . . . . . . . . . . . . . . . . . .

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8

Contents

Interrelations

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4 Normal Distribution
Normal . . . . .
Special cases . . . . .
Error function .
Standard normal
Interrelations . . . .

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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 . . . . . . . . .

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6 Log-Normal Distribution
Log-normal . . . . . .
Special cases . . . . . . . .
Anchored log-normal
Gibrat . . . . . . . . .
Interrelations . . . . . . .

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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 . . . . .

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8 Gamma-Exponential Distribution
Gamma-exponential . . . . .
Special cases . . . . . . . . . . . .
Standard gamma-exponential
Chi-square-exponential . . .
Generalized Gumbel . . . . .
Gumbel . . . . . . . . . . . .
Standard Gumbel . . . . . .
BHP . . . . . . . . . . . . . .
Moyal . . . . . . . . . . . . .
Interrelations . . . . . . . . . . .

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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 . . . . . . . . .

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Two shape parameters

10

G. E. Crooks – Field Guide to Probability Distributions

Contents

10 Unit Gamma Distribution
Unit gamma . . . .
Special cases . . . . . . .
Uniform product . .
Interrelations . . . . . .

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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 . . . . . . . . . . . . . . .

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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 . . .

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13 Beta Prime Distribution
Beta prime . . . . .
Special cases . . . . . . .
Standard beta prime
F . . . . . . . . . . .
Inverse Lomax . . .
Interrelations . . . . . .

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14 Beta-Exponential Distribution
Beta-exponential . . . . . .
Standard beta-exponential
Special cases . . . . . . . . . . .
Exponentiated exponential
Hyperbolic sine . . . . . .
Nadarajah-Kotz . . . . . . .
Interrelations . . . . . . . . . .

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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 . . . . . . . . .

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16 Pearson IV Distribution
113
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Interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Three (or more) shape parameters
17 Generalized Beta Distribution
Generalized beta . . . .
Special Cases . . . . . . . . .
Kumaraswamy . . . . .
Interrelations . . . . . . . .

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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 . . . . . . . . . . .

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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 . . . . . . . . . . . . . . . .

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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 . . . . . . . . . .

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G. E. Crooks – Field Guide to Probability Distributions

Contents

Stable . . . . . . . .
Suzuki . . . . . . .
Triangular . . . . .
Uniform difference
Voigt . . . . . . . .

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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 . . .

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161
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E Algebra of Random Variables
Transformations . . . . . .
Combinations . . . . . . .
Transmutations . . . . . .
Generation . . . . . . . . .

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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 . . . . . . . . . . . . . . . .

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25
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57
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64
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102
102
103
108
111
119
125

List of Figures

37
38
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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 . . . . . . . . . .

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41
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55
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62
66
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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.

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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].

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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.

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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.

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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)

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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

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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 )

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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)

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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

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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]

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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

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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)

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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).

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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)

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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 .

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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

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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

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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

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(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 ;
=

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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

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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

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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.

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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

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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|

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[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)

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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-

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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

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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)

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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

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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.

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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

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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→∞

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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

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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

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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

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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

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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

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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

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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
)

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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→∞

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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-

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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)

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[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)

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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

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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

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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).

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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

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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

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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) ,

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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

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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

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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]

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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

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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

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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

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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).

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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 ,

0 0, 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

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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 → ∞.

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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)

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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

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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].

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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.

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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→∞

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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

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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 ) =

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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-

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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).

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(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

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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.

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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.

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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).

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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.

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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).

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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

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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

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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

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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)

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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

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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

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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

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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

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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

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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)

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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
−∞

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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|

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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.

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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.

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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].

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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

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G. E. Crooks – Field Guide to Probability Distributions

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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)

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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

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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

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[125]

[179]

[180]

[181]
[3]
[87]
[87]
[87]

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[87]

[2]

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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

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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)

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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

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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

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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)

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[1]
[1]

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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

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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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