# 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

 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) asc 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 desc 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