Field Guide to Continuous Probability Distributions


Field Guide
Probability Distributions

Gavin E. Crooks

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

Distribution Synonym or Equation
β beta
β′ beta prime
χ chi
χ2 chi-square
Γ gamma
Λ log-normal
Φ standard normal
Amaroso (11.1)
anchored Amaroso Stacy
anchored exponential See exponential (2.1)
anchored log-normal See log-normal (6.1)
anti-log-normal log-normal
arcsine (12.6)
Appell Beta (20.17)
ing wedge See wedge (5.4)
ballasted Pareto Lomax
Bates (21.1)
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-Fisher-Tippett (21.2)
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
beta-prime (13.1)
beta-prime exponential beta-logistic
biexponential Laplace
bilateral exponential Laplace
Birnbaum-Saunders (21.3)
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
centered arcsine (12.7)
central-beta (12.5)
central-logistic (15.4)
Champernowne Perks
chi (11.8)
chi-square (7.3)
chi-square-exponential (8.3)
circular normal Rayleigh
Coale-McNeil gamma-exponential
Cobb-Douglas log-normal
compound gamma beta prime
confluent hypergeometric (20.12)
Dagum (18.4)
Dagum type I Dagum
de Moivre normal
degenerate See uniform (1.1)
delta degenerate
ing 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)
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)
exponentiated Weibull See Beta-Fisher-Tippett (21.2)
extended Pearson (20.2)
extreme value Gumbel
extreme value type N Fisher-Tippett type N
F (13.3)
F-ratio F
fatigue life distribution Birnbaum-Saunders
Feller-Pareto generalized beta prime
Fisher F or Student’s t
Fisher-F F
Fisher-Snedecor F
Fisher-Tippett (11.25)
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
folded normal See pp. 173
Fréchet (11.29)
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 (20.11)
generalized arcsin central-beta
generalized beta (17.1)
generalized beta-exponential beta-Fisher-Tippett
generalized beta-prime (18.1)
generalized beta type II generalized beta prime
generalized Cauchy generalized Pearson type VII
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 (11.24)
generalized Fréchet (11.28)
generalized gamma Stacy or Amoroso
generalized gamma ratio generalized beta prime
generalized generalized inverse Gaussian generalized Sichel
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 (20.13)
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 central logistic
generalized logistic type IV beta-logistic
generalized normal Nakagami or exponential power
generalized Pareto (5.2)
generalized Pearson type I Nakagami
generalized Pearson type II generalized Sichel
generalized Pearson type III generalized beta prime
generalized Pearson type VII (21.6)
generalized Rayleigh scaled-chi or Rice
generalized Sichel (20.14)
generalized semi-normal Stacy
generalized-t generalized Pearson type VII
generalized Weibull (11.26) or Stacy
GEV generalized extreme value
Gibrat standard log-normal
Gompertz See pp. 173
Gompertz-Verhulst beta-exponential
grand unified distribution See (20.1)
Grassia unit gamma
greater grand unified distribution (20.1)
GUD grand unified distribution
Gumbel (8.5)
Gumbel-Fisher-Tippett Gumbel
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 (20.5)
Halphen A Halphen
Halphen B (20.7)
harmonic hyperbola
Hohlfeld (11.5)
Holtsmark (21.7)
hyperbola (20.6)
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 cosh hyperbolic secant
inverse exponential (11.14) or exponential
inverse gamma (11.13)
inverse Gaussian (20.3)
inverse half-normal (11.22)
inverse Halphen B (20.8)
inverse hyperbolic cosine hyperbolic secant
inverse Lomax (13.4)
inverse Nakagami (11.23)
inverse normal inverse Gaussian
inverse Maxwell (11.21)
inverse Rayleigh (11.20)
inverse paralogistic (18.6)
inverse Pareto inverse Lomax
inverse Weibull Fréchet
Irwin-Hall (21.9)
Johnson Johnson SU
K (21.8)
Kumaraswamy (17.2)
Laha (20.18)
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)
Landau (21.11)
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 desc
ing wedge
Leonard hydrograph Stacy
Lévy (11.15)
Lévy skew alpha-stable stable
Lévy stable stable
Lévy symmetric alpha-stable See stable (21.20)
Libby-Novick (20.10)
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-Pearson III unit gamma
log-stable See stable (21.20)
log-Weibull Gumbel
logarithmic-normal log-normal
logarithmico-normal log-normal
logistic (15.5)
logit logistic
Lomax (5.6)
Lorentz Cauchy
Lorentzian Cauchy
m Nakagami
m-Erlang Erlang
Majumder-Chakravart generalized beta prime
March inverse gamma
max stable See Fisher-Tippett (11.25)
Maxwell (11.11)
Maxwell-Boltzmann Maxwell
Maxwell speed Maxwell
Meridian Meridian
Mielke Dagum
min stable See Fisher-Tippett (11.25)
minimax Kumaraswamy
modified Lorentzian relativistic Breit-Wigner
modified pert See pert (12.3)
Moyal (8.8)
Nadarajah-Kotz (14.4)
Nakagami (11.6)
Nakagami-m Nakagami
negative exponential exponential
noncentral chi (21.14)
noncentral chi-square (21.15)
noncentral 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 central beta
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
Pearson type XII (12.4)
Pearson exponential (20.15)
Perks (20.16)
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 log-logistic
Prentice beta-logistic
pseudo-Voigt (21.17)
pseudo-Weibull (11.3)
q-exponential (5.3)
q-Gaussian (19.2)
quartic biweight
Rayleigh (11.10)
Rayleigh-Rice Rice
reciprocal inverse Gaussian (20.4)
rectangular uniform
relativistic Breit-Wigner (9.8)
reversed Burr type II (15.3)
reversed Weibull See Weibull (11.27)
Rice (21.18)
Rician Rice
right triangular asc
ing wedge
Rosin-Rammler Weibull
Rosin-Rammler-Weibull Weibull
Sato-Tate semicircle
scaled chi (11.9)
scaled chi-square (7.4)
scaled inverse chi (11.18)
scaled inverse chi-square (11.16)
sech-square logistic
semicircle (12.8)
semi-normal half normal
Sichel (20.9)
Singh-Maddala Burr
singly non-central F See non-central F (21.16)
skew-t Pearson type IV
skew logistic Burr type II
Slash (21.19)
Snedecor’s F F
spherical normal Maxwell
stable (21.20)
stable Paretian See stable (21.20)
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-Fisher Student’s-t
Student’s t (9.2)
Student’s t2 (9.3)
Student’s t3 (9.4)
Student’s z (9.5)
Subbotin exponential power
Suzuki (21.21)
symmetric beta central-beta
symmetric beta-logistic central-logistic
symmetric Pearson q-Gaussian
t Student’s-t
t2 Student’s-t2
t3 Student’s-t3
tine triangular
transformed beta (18.2)
transformed gamma Stacy
triangular (21.22)
triweight (12.11)
truncated normal See pp. 173
two-tailed exponential Laplace
uniform (1.1)
uniform difference (21.23)
uniform prime (5.8)
uniform product (10.2)
uniform sum Irwin-Hall
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
Voigt (21.24)
Voigtian Voigt
Voigt profile Voigt
von Mises extreme value Fisher-Tippett
von Mises-Jenkinson Fisher-Tippett
waiting time exponential
Wald See inverse Gaussian (20.3)
wedge (5.4)
Weibull (11.27)
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