# Field Guide to Continuous Probability Distributions

Field Guide to Continuous Probability Distributions Gavin E. Crooks

v1.0.0 (2019)

Over 170 continuous univariate probability distributions (and at least as many synonyms) organized into 20 families.

### 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.

### Index of Distributions

β

β′

χ

χ2

Γ

Λ

Φ

Amaroso (11.1)

anchored Amaroso

anchored exponential

anchored log-normal

anti-log-normal

arcsine (12.6)

Appell Beta (20.17)

ascending wedge

ballasted Pareto

Bates (21.1)

bell curve

beta (12.1)

beta, J shaped

beta, U shaped

beta-exponential (14.1)

beta-Fisher-Tippett (21.2)

beta-k

beta-kappa

beta-logistic (15.1)

beta-log-logistic

beta type I

beta type II

beta-P

beta-pert

beta-power

beta-prime (13.1)

beta-prime exponential

biexponential

bilateral exponential

Birnbaum-Saunders (21.3)

biweight (12.10)

BHP (8.7)

Box-Tiao

Bramwell-Holdsworth-Pinton

Breit-Wigner

Brody

Burr (18.3)

Burr type I

Burr type II (15.2)

Burr type III

Burr type XII

Cauchy(9.6)

Cauchy-LorentzCauchy

centered arcsine(12.7)

central-beta(12.5)

central-logistic(15.4)

Champernowne

chi(11.8)

chi-square(7.3)

chi-square-exponential (8.3)

circular normal

Coale-McNeil

Cobb-Douglas

compound gamma

confluent hypergeometric (20.12)

Dagum(18.4)

Dagum type I

de Moivre

degenerate

delta

descending wedgeSee wedge (5.4)

Diracde

double exponential

doubly exponential

doubly non-central F

Epanechnikov (12.9)

Erlang

error

error function

exponential(2.1)

exponential Burr

exponential gamma

exponential generalized beta type I

exponential generalized beta type II

exponential generalized beta prime

exponential power (21.4)

exponential ratio (5.7)

exponentiated exponential (14.2)

exponentiated Weibull

extended Pearson (20.2)

extreme valueGumbel

extreme value type N

F (13.3)

F-ratio

fatigue life distribution

Feller-Pareto

Fisher-F

Fisher-Snedecor

Fisher-Tippett (11.25)

Fisher-Tippett type I

Fisher-Tippett type II

Fisher-Tippett type III

Fisher-Tippett-Gumbel

Fisher-z

Fisk

flat

folded normal

Fréchet (11.29)

FTG

Galton

Galton-McAlister

gamma (7.1)

gamma-exponential (8.1)

gamma ratio

Gaussian

Gauss

Gauss hypergeometric (20.11)

generalized arcsin

generalized beta (17.1)

generalized beta-exponential

generalized beta-prime (18.1)

generalized beta type II

generalized Cauchy

generalized error

generalized exponential

generalized extreme value

generalized F

generalized Feller-Pareto

generalized Fisher-Tippett (11.24)

generalized Fréchet (11.28)

generalized gamma

generalized gamma ratio

generalized generalized inverse Gaussian

generalized Gompertz

generalized Gompertz-Verhulst type I

generalized Gompertz-Verhulst type II

generalized Gompertz-Verhulst type III

generalized Gumbel (8.4)

generalized Halphen (20.13)

generalized inverse gamma

generalized inverse Gaussian

generalized K (21.5)

generalized log-logistic

generalized logistic type I

generalized logistic type II

generalized logistic type III

generalized logistic type IV

generalized normal

generalized Pareto (5.2)

generalized Pearson type I

generalized Pearson type II

generalized Pearson type III

generalized Pearson type VII (21.6)

generalized Rayleigh

generalized Sichel (20.14)

generalized semi-normal

generalized-t

generalized Weibull (11.26)

GEV

Gibrat

Gompertz

Gompertz-Verhulst

grand unified distribution

Grassiaunit gamma

greater grand unified distribution (20.1)

GUDgrand unified distribution

Gumbel (8.5)

Gumbel-Fisher-Tippett

Gumbel type N

half-Cauchy (18.9)

half-exponential power (11.4)

half generalized Pearson VII (18.10)

half-LahaSee half generalized Pearson VII (18.10)

half-normal (11.7)

half-Pearson type VII (18.8)

half-Subbotin

half-t

half-uniform

Halphen (20.5)

Halphen A

Halphen B (20.7)

harmonic

Hohlfeld (11.5)

Holtsmark (21.7)

hyperbola (20.6)

hyperbolic secant (15.6)

hyperbolic secant square

hyperbolic sine (14.3)

hydrograph Stacy

hyper gamma Stacy

inverse beta

inverse beta exponential

inverse Burr

inverse chi (11.19)

inverse chi-square (11.17)

inverse cosh

inverse exponential (11.14)

inverse gamma (11.13)

inverse Gaussian (20.3)

inverse half-normal (11.22)

inverse Halphen B (20.8)

inverse hyperbolic cosine

inverse Lomax (13.4)

inverse Nakagami (11.23)

inverse normal

inverse Maxwell (11.21)

inverse Rayleigh (11.20)

inverse paralogistic (18.6)

inverse Pareto

inverse Weibull

Irwin-Hall (21.9)

Johnson

Johnson SB

Johnson SL

Johnson SN

Johnson SU (21.10)

K (21.8)

Kumaraswamy (17.2)

Laha (20.18)

Landau (21.11)

Laplace (3.1)

Laplace’s first law of error

Laplace’s second law of error

Laplace-Gauss

Laplacian

law of error

left triangular

Leonard hydrograph

Lévy (11.15)

Lévy skew alpha-stable

Lévy stable

Lévy symmetric alpha-stable

Libby-Novick (20.10)

log-beta

log-Cauchy (21.12)

log-chi-square

log-F

log-gamma

log-Gaussian

log-Gumbel

log-logistic (18.7)

log-normal (6.1)

log-normal, two parameter

log-Pearson III

log-stable

log-Weibull

logarithmic-normal

logarithmico-normal

logistic (15.5)

logit

Lomax (5.6)

Lorentz

Lorentzian

m

m-Erlang

Majumder-Chakravart

March

max stable

Maxwell (11.11)

Maxwell-Boltzmann

Maxwell speed

Meridian

Mielke

min stable

minimax

modified Lorentzian

modified pert

Moyal (8.8)

Nakagami (11.6)

Nakagami-m

negative exponential

noncentral chi (21.14)

noncentral chi-square (21.15)

noncentral F (21.16)

normal(4.1)

normal ratio

Nukiyama-Tanasawa

one-sided normal

parabolic

paralogistic (18.5)

Pareto (5.5)

Pareto type I

Pareto type II

Pareto type III

Pareto type IV

Pearson (19.1)

Pearson type I

Pearson type II

Pearson type III

Pearson type IV (16.1)

Pearson type V

Pearson type VI

Pearson type VII (9.1)

Pearson type VIII

Pearson type IX

Pearson type X

Pearson type XI

Pearson type XII (12.4)

Pearson exponential (20.15)

Perks (20.16)

Pert (12.3)

Poisson’s first law of error

Porter-Thomas (7.5)

positive definite normal

power

power function(5.1)

power prime

Prentice

pseudo-Voigt (21.17)

pseudo-Weibull (11.3)

q-exponential (5.3)

q-Gaussian (19.2)

quartic

Rayleigh (11.10)

Rayleigh-Rice

reciprocal inverse Gaussian (20.4)

rectangular

relativistic Breit-Wigner (9.8)

reversed Burr type II (15.3)

reversed Weibull

Rice (21.18)

Rician

right triangular

Rosin-Rammler

Rosin-Rammler-Weibull

Sato-Tate

scaled chi (11.9)

scaled chi-square (7.4)

scaled inverse chi (11.18)

scaled inverse chi-square (11.16)

sech-square

semicircle (12.8)

semi-normal

Sichel (20.9)

singly non-central F

skew-t

skew logistic

Slash (21.19)

Snedecor’s F

spherical normal

stable (21.20)

stable Paretian

Stacy (11.2)

Stacy-Mihram

standard Amoroso

standard beta (12.2)

standard beta exponential

standard beta logistic

standard beta prime (13.2)

standard Cauchy (9.7)

standard exponential

standard gamma (7.2)

standard Gumbel (8.6)

standard gamma exponential (8.2)

standard Laplace

standard log-normal

standard normal

standard uniform (1.2)

standardized normal

standardized uniform

stretched exponential

Student

Student-Fisher

Student’s t (9.2)

Student’s t2 (9.3)

Student’s t3 (9.4)

Student’s z (9.5)

Subbotin

Suzuki (21.21)

symmetric beta

symmetric beta-logistic

symmetric Pearson

t

t2

t3

tine

transformed beta (18.2)

transformed gamma

triangular (21.22)

triweight (12.11)

truncated normal

two-tailed exponential

uniform (1.1)

uniform difference (21.23)

uniform prime (5.8)

uniform product (10.2)

uniform sum

unbounded uniform

unit gamma (10.1)

unit normal

van der Waals profile

variance ratio

Verhulst

Vienna

Vinci

Voigt (21.24)

Voigtian

Voigt profile

von Mises

von Mises-Jenkinson

waiting time

Wald

wedge (5.4)

Weibull (11.27)

Weibull-exponential

Weibull-gamma

Weibull-Gnedenko

Wien

Wigner semicircle

Wilson-Hilferty (11.12)

Witch of Agnesi

z