For most of the classical distributions, base R provides
probability distribution functions (p), density functions (d), quantile functions
(q), and random number generation (r).
Beyond this basic functionality, many CRAN packages provide
additional useful distributions. In particular, multivariate distributions as well as copulas
are available in contributed packages.
Ultimate bibles on probability distributions are:

different volumes of N. L. Johnson, S. Kotz and N. Balakrishnan books, e.g. Continuous Univariate Distributions, Vol. 1,

Thesaurus of univariate discrete probability distributions by G. Wimmer and G. Altmann.

Statistical Distributions by M. Evans, N. Hastings, B. Peacock.

Distributional Analysis with Lmoment Statistics using the R Environment for Statistical Computing, Asquith (2011).
The maintainer greatfully acknowledges Achim Zeileis, David Luethi, Tobias Verbeke, Robin
Hankin, Mathias Kohl, G. Jay Kerns, Kjetil Halvorsen, William Asquith for their useful comments/suggestions.
If you think information is not accurate or not complete, please let me know.
Base functionality:

Base R provides probability distribution functions
p
foo
()
density functions
d
foo
(), quantile functions
q
foo
(), and random number generation
r
foo
()
where
foo
indicates the type
of distribution:
beta (
foo
=
beta),
binomial
binom,
Cauchy
cauchy,
chisquared
chisq,
exponential
exp,
Fisher F
f,
gamma
gamma,
geometric
geom,
hypergeometric
hyper,
logistic
logis,
lognormal
lnorm,
negative binomial
nbinom,
normal
norm,
Poisson
pois,
Student t
t,
uniform
unif,
Weibull
weibull.
Following the same naming scheme, but somewhat less standard are the following distributions
in base R: probabilities of coincidences (also known as "birthday paradox")
birthday
(only p and q), studentized range distribution
tukey
(only p and q), Wilcoxon signed rank distribution
signrank, Wilcoxon
rank sum distribution
wilcox.

On base R distributions,
prob
provides the characteristic function, while
graphical methods for illustrating probability distributions can be found
in
denstrip.
Following abbreviations are used: m for moments (both raw and centered) computation,
lm for Lmoment, tlmr for trimmed Lmoment ratio,
lev for limited expected value, mgf for moment generating function, pgf for probability
generating functions, h for hazard function, i for integrated hazard function.

Probability generating function:
Compounding
provides pgf for
xxx
distribution,
inverse
xxx
distribution,
first derivative of the
xxx
distribution,
where
xxx
belongs to binomial, binomialPoisson, geometric, hypergeometric,
hyperPoisson, Katti type H1/H2, logarithmic, logarithmicbinomial,
logarithmicPoisson, negative binomial, Neyman type A/B/C, PascalPoisson, Poisson,
Poissonbinomial, PoissonLindley, PoissonPascal, Polya Aeppli, Thomas,
Waring, Yule.
Discrete univariate distributions:

Betabinomial distribution
: provided in
VGAM.
ZI/ZM beta binomial distributions are
implemented in
gamlss.dist.

Betageometric distribution
: provided in
VGAM.

Binomial (including Bernoulli) distribution
: provided in
stats
.
Zeromodified, zeroinflated, truncated versions are provided
in
gamlss.dist
and in
VGAM.

Benford distribution
: provided in
VGAM.

BorelTanner distribution
: provided in
VGAM.

ConwayMaxwellPoisson distribution
: provided in
compoisson
and
CompGLM.

Delaporte distribution
: provided in
gamlss.dist
and
Delaporte.

Dirac distribution
: provided in
distr.

Discrete exponential distribution
: provided in
poweRlaw.

Discrete inverse Weibull distribution
:
DiscreteInverseWeibull
provides d, p, q, r functions for the
inverse Weibull as well as hazard rate function and moments.

Discrete Laplace distribution
:
The skew discrete Laplace distribution has two parametrization (DSL and ADSL),
both provided in
DiscreteLaplace
and DSL in
disclap.

Discrete lognormal distribution
: provided in
poweRlaw.

Discrete uniform distribution
: can be easily obtained with the functions
sum,cumsum,sample.

Discrete Weibull distribution
: provided in
DiscreteWeibull:
d, p, q, r, m for disc. Weib. type 1, d, p, q, r, m, h for disc. Weib. type 3.

Felix distribution
: provided in
VGAM.

Lindley distribution
: provided in
gambin.

Geometric distribution
: provided in
stats
.
Zeromodified, zeroinflated, truncated versions are provided
in
gamlss.dist
and in
VGAM.

Geometric (compound) Poisson distribution
(also known PolyaAeppli distribution)
: provided in
polyaAeppli.

Generalized binomial distribution
: provided in
GenBinomApps.

Generalized Hermite distribution
: provided in
hermite.

Hypergeometric distribution
:
provided in
stats
.
Extented hypergeometric distribution
can be found in
BiasedUrn
package, which provides
not only p, d, q, r functions but also mean, variance, mode functions.
Generalized hypergeometric distribution is implemented in
SuppDists.
Negative hypergeometric distribution is provided in
tolerance.

Lagrangian Poisson distribution
:
RMKdiscrete
provides d, p, q, r functions for the
univariate and the bivariate Lagrangian Poisson distribution.

Lindley distribution
: provided in
VGAM.

Logarithmic distribution
: This can be found in
VGAM
and
gamlss.dist. A fast random generator is
available for the logarithmic distribution
is implemented in
Runuran
as well as the 'density' function.

Poisson distribution
: provided in
stats
and in
poweRlaw.
Zeromodified, zeroinflated, truncated versions are provided
in
gamlss.dist
and in
VGAM.

PoissonLindley distribution
: provided in
tolerance.

Power law distribution
: provided in
poweRlaw.

Mana Clash distribution
: provided in
RMKdiscrete.

Negative binomial distribution
: provided in
stats
.
Zeromodified, zeroinflated, truncated versions are provided
in
gamlss.dist
and in
VGAM.
New parametrization of the negative binomial distribution is available in
RMKdiscrete.

Sichel distribution
: provided in
gamlss.dist.

Skellam distribution
: provided in
VGAM
and
skellam.

Waring distribution
: sampling in
degreenet.

YuleSimon distribution
: provided in
VGAM
and sampling in
degreenet.

Zeta and Haight's Zeta distribution
: provided in
VGAM,
tolerance.

Zipf law
:
d, p, q, r functions of the Zipf and the ZipfMandelbrot distributions
are provided in
tolerance,
VGAM.
Package
zipfR
provides tools for distribution of word frequency,
such as the Zipf distribution.
Discrete multivariate distributions:

Multinomial distribution
:
stats,
mc2d
packages provide d, r functions.

Negative multinomial distribution
:
A bivariate distribution with negativebinomial marginals is available in
RMKdiscrete.
The multiplicative multinomial distribution is implemented in
MM.

Multivariate Poisson distribution
:
not yet implemented?

Multivariate hypergeometric distribution
:
not yet implemented?

Multivariate PolyaEggenberger distribution
:
not yet implemented?

Multivariate Ewens distribution
:
not yet implemented?

Permutation distribution
:
Mallows, generalized Mallows and weighted Mallows distributions are implemented in
PerMallows
(functions d, r).
Continuous univariate distributions:

Arcsine distribution
: implemented in package
distr.

Beard distribution
:
ActuDistns
provide d, h, i, q functions
for the Beard and the MakehamBeard distributions.

Beta distribution and its extensions
:
Base R and
ExtDist
provide
the d, p, q, r functions for this distribution (see above).
actuar
provides moments and limited expected values,
ActuDistns
provides hazard and integrated hazard rate functions.
sadists
implements Gram Charlier, Edgeworth and CornishFisher approximations
for doubly non central beta distribution for computing d, p, q, r functions.
The fourparameter beta (with location and scale param), the symmetric reflected truncated
beta (both for twoparameter and fourparameter beta) are available in
ExtDist.
The beta prime (or beta of the second kind), which is the distribution of X/(1X) when
X follows a beta distribution, is provided in
VGAM
and
mc2d.
The zero and one inflated beta distribution can be found in
gamlss.dist.
The generalized beta of the first kind (GB1) is provided in
gamlss.dist
The generalized beta of the second kind (GB2) is provided in
gamlss.dist,
GB2.
Several special cases of
the generalized beta distribution are also implemented in
VGAM:
Lomax, inverse Lomax, Dagum and SinghMaddala distribution and in
mc2d: Pert.
Summary for Betarelated distributions
Distribution name

Packages

Functions

Distribution suffix

Beta (1st kind)

stats

d, p, q, r

beta

Beta

actuar

m, mgf, lev

beta

Beta

ExtDist

d, p, q, r

Beta

Doubly non central beta

sadists

d, p, q, r

nbeta

4param beta

ExtDist

d, p, q, r

Beta_ab

zeroinfl beta

gamlss.dist

d, p, q, r

BEZI

oneinfl beta

gamlss.dist

d, p, q, r

BEOI

GB1

gamlss.dist

d, p, q, r

GB1

Summary for Beta2related distributions
Distribution name

Packages

Functions

Distribution suffix

Beta (2nd kind)

VGAM

d, p, q, r

beta

GB2

VGAM

d, p, q, r

genbetaII

GB2

gamlss.dist

d, p, q, r

GB2

GB2

GB2

d, p, q, r

gb2


Benini distribution
: provided in
VGAM.

BirnbaumSaunders distribution
: provided in package
VGAM.

Box Cox distribution
:
gamlss.dist
provides the BoxCox normal,
the BoxCox power exponential and the BoxCox t distributions.

Burr distribution
: see Pareto.

Cardioid distribution
: provided in
VGAM.

Cauchy distribution
: Base R provides
the d, p, q, r functions for this distribution (see above). Other
implementations are available in
lmomco
and
sgt.
The skew Cauchy distribution is provided in
sgt.

Chen distribution
: provided in
reliaR
and
ActuDistns.

Chi(squared or not) distribution
: Base R provides
the d, p, q, r functions for the chisquared distribution, both central
and noncentral (see above).
Moments, limited
expected values and the moment generating function are provided in
actuar.
Only d,r functions are available for the inverse chisquared distribution
in package
geoR. A fast random generator is available for the Chi distribution
is implemented in
Runuran
as well as the density function.
The noncentral Chi distribution is not yet implemented. The chibarsquared distribution
is implemented in
emdbook.
sadists
implements Gram Charlier, Edgeworth and CornishFisher approximations
for sums of non central chisquared raised to powers distribution and sums of log of
non central chisquared for computing d, p, q, r functions.
Summary for Chirelated distributions
Distribution name

Packages

Functions

Distribution suffix

Chisquared

stats

d, p, q, r

chisq

Chisquared

actuar

m, mgf, lev

chisq

Chisquared

Runuran

d, r

chisq

Chibarsquared

emdbook

d, p, q, r

chibarsq

Chi

Runuran

d, r

chi

Inverse Chisquared

geoR

d, r

invchisq

Sum of power Chisquared

sadists

d, p, q, r

sumchisqpow

Sum of log Chisquared

sadists

d, p, q, r

sumlogchisq


Dagum distribution
: see beta.

Davies distribution
: The Davies distribution is provided
in
Davies
package.

Dirichlet distribution
: functions d, r
are provided in
MCMCpack,
mc2d,
hyperdirichlet,
dirmult
and
bayesm. The standard and generalized
hyperdirichlet distributions are provided in
hyperdirichlet.

(noncentral) Dunnett's test distribution
: provided
in
nCDunnett.

Etamu distribution
: provided in
lmomco.
sadists
implements Gram Charlier, Edgeworth and CornishFisher approximations
for doubly non central eta distribution for computing d, p, q, r functions.

Exponential distribution and its extensions
: Base R provides the d, p, q, r functions
for this distribution (see above).
actuar
provides additional functions
such as the moment generating function, moments and limited expected values. It also has
the d, p, q, r for the inverse exponential distribution.
ActuDistns
provides hazard and integrated hazard rate functions.
The shifted (or twoparameter exponential) and the
truncated exponential distributions are implemented in
lmomco
and
tolerance
packages with d, p, q, r functions.
Exponential Power distribution is also known as General Error Distribution:
d, p, q, r functions for the power and the skew power exponential type 14 distributions are implemented
in
gamlss.dist
and
lmomco.
The power exponential distribution is also provided in
normalp
and
sgt.
The skew power exponential is provided
sgt.
reliaR
and
ActuDistns
provide
the generalized exponential, the inverse generalized exponential,
the logistic exponential, the MarshallOlkin Extended Exponential and
the exponential extension distributions.
ExtDist
also provides d, p, q, r functions for the exponential distribution.
A fast random generator is available for the power Exponential distribution
is implemented in
Runuran
as well as the density function.
Summary for exponentialrelated distributions
Distribution name

Packages

Functions

Distribution suffix

Exponential

stats

d, p, q, r

exp

Exponential

actuar

m, mgf, lev

exp

Exponential

ActuDistns

h, i

exponential

Exponential

ExtDist

d, p, q, r

Exp

Exponential

gamlss.dist

d, p, q, r

EXP

Exponential

poweRlaw

d, p, q, r

exp

Inverse exponential

actuar

d, p, q, r, m, lev

invexp

Shifted exponential

lmomco

d, p, q, r, lm, tlmr

exp

Shifted exponential

tolerance

d, p, q, r

2exp

Truncated exponential

lmomco

d, p, q, r, lm, tlmr

texp

Power exponential

normalp

d, p, q, r

normp

Power exponential

Runuran

d, r

exp

Skew power exp.

lmomco

d, p, q, r, lm, tlmr

aep4

Power and skew power exp.

gamlss.dist

d, p, q, r

PE, SEP

Generalized and inverse gen. exp.

ActuDistns

d, p, q, r, h, i

ige

Generalized and inverse gen. exp.

reliaR

d, p, q, r

gen.exp, inv.genexp

Logistic, MarshallOlkin Ext. exp. and exp. ext.

ActuDistns

d, p, q, r, h, i

le, moe, expext

Logistic, MarshallOlkin Ext. exp. and exp. ext.

reliaR

d, p, q, r

logis.exp, moee, exp.ext


Externally studentized midrange distribution
:
Package
SMR
computes the studentized midrange distribution (d, p, q, r).

FisherSnedecor (or F) distribution
: Base R provides the d, p, q, r functions
for the F distribution, possibly with a noncentral parameter.
sadists
implements Gram Charlier, Edgeworth and CornishFisher approximations
for doubly non central Fisher distribution (and product of multiple
doubly non central Fisher distribution) for computing d, p, q, r functions.
flexsurv
provides d, p, q, r functions as well as
hazard (h) and integrated hazard rate (i) functions for
the generalized F distribution.
ActuDistns
provides integrated hazard rate function
for generalized F distribution.
fpow
returns the noncentrality parameter of the noncentral F distribution
if probability of type I and type II error, degrees of freedom of the numerator
and the denominator are given.

Frechet distribution
: provided in
VGAM,
RTDE
and
evd. A fast random generator is available for the Frechet distribution
is implemented in
Runuran
as well as the density function.

Friedman's Chi distribution
: provided in
SuppDists.

Gamma distribution and its extensions
: Base R provides
the d, p, q, r functions for this distribution (see above).
ExtDist
provides d, p, q, r functions for exponential distribution.
actuar
provides d, p, q, r functions for the inverse, the inverse
transformed and the log gamma distributions
while
ghyp
provides those functions for the variance gamma distribution.
VarianceGamma
provides d, p, q, r functions for the variance gamma distribution
as well as moments (skewness, kurtosis, ...).
ActuDistns
provides hazard (h) and integrated hazard rate (i) functions for gamma distribution,
q, h, i, q for the generalized gamma, the loggamma distributions.
VGAM
provides d, p, q, r functions of the log
gamma and the generalized gamma distribution. The generalized gamma distribution
can also be found in
gamlss.dist.
reliaR
provides
the log gamma distribution. See Pearson III for a threeparameter
gamma distribution with a location parameter.
flexsurv
provides d, p, q, r functions as well as
hazard (h) and integrated hazard rate (i) functions for
the generalized gamma distribution.
Summary for gammarelated distributions
Distribution name

Packages

Functions

Distribution suffix

Gamma

stats

d, p, q, r

gamma

Gamma

actuar

m, mgf, lev

gamma

Gamma

ActuDistns

h, i

gamma

Gamma

ExtDist

d, p, q, r

Gamma

Inverse gamma

actuar

d, p, q, r, m, lev, mgf

invgamma

Loggamma

actuar

d, p, q, r, m, lev

lgamma

Loggamma

ActuDistns

d, p, q, r, h, i

lgammad

Loggamma

VGAM

d, p, q, r

lgamma

Variance gamma

ghyp

d, p, q, r

VG

Variance gamma

VarianceGamma

d, p, q, r, m

vg

Generalized gamma

ActuDistns

d, p, q, r, h, i

gengammad

Generalized gamma

flexsurv

d, p, q, r, h, i

gengamma

Generalized gamma

gamlss.dist

d, p, q, r

GG

Generalized gamma

VGAM

d, p, q, r

gengamma.stacy


Gaussian (or normal) distribution and its extensions
: Base R provides
the d, p, q, r functions for this distribution (see above).
actuar
provides the moment generating function and moments.
ActuDistns
provides hazard (h) and integrated hazard rate (i) functions.
The
truncnorm
package provides d, p, q, r functions for the truncated
univariate gaussian distribution as well as functions for the first two moments.
ExtDist
provides the normal, the truncated and the symmetric truncated distributions.
mvrtn
provides random variates for left/right truncated normal distributions.
lmomco
implements the generalized normal distribution.
The Exponentially modified Gaussian is available in
emg,
gamlss.dist
and
retimes.
sgt
implements the skew normal distribution.
VGAM
implements the folded and the skewed normal distribution, and
csn
provides d, r functions for the closed skew normal distribution.
CompQuadForm
provides the distribution function of quadratic forms
in normal variates.
NormalGamma
provides the density of the sum of a gaussian and a gamma random variables.
NormalLaplace
provides d, p, q, r functions for the sum of a normal and a Laplace random variables.
Summary for Gaussianrelated distributions
Distribution name

Packages

Functions

Distribution suffix

Normal

stats

d, p, q, r

norm

Normal

actuar

m, mgf

norm

Normal

ActuDistns

h, i

norm

Normal

ExtDist

d, p, q, r

Normal

Truncated normal

truncnorm

d, p, q, r, m

truncnorm

Truncated normal

ExtDist

d, p, q, r

Normal_trunc_ab

Truncated normal

mvrtn

r, m

tn

Generalized normal

lmomco

d, p, q, r

gno

Exponentially modified Gaussian

emg

d, p, q, r

emg

Exponentially modified Gaussian

gamlss.dist

d, p, q, r

exGAUSS

Exponentially modified Gaussian

retimes

d, p, q, r

exgauss

Folded and skew normal

gamlss.dist

d, p, q, r

SN1, SN2

Closed skew normal

csn

d, p, q, r

csn

Skew normal

sgt

d, p, q, r

sn


General error distribution (also known as exponential power distribution)
:
see
exponential
item.

Generalized extreme value distribution
: provided in
lmomco
(d, p, q) ,
VGAM,
evd,
evir,
FAdist
and
fExtremes
(d, p, q, r).
evdbayes
provides some
additional functions for GEV distribution using MCMC.

Gompertz distribution
: provided in
reliaR,
ActuDistns,
flexsurv.
ActuDistns
and
flexsurv
also
provide hazard (h) and integrated hazard rate (i) functions.

Govindarajulu distribution
: provided in
lmomco.

Gumbel distribution
: provided in packages
lmomco,
VGAM,
gamlss.dist,
FAdist,
reliaR,
ActuDistns,
ExtDist
and
evd.
ActuDistns
also provides hazard (h) and integrated hazard rate (i) functions.
A fast random generator is available for the Gumbel distribution
is implemented in
Runuran
as well as the density function.
The reverse Gumbel distribution is implemented in
lmomco
and
gamlss.dist.
The Gumbel II distribution is implemented in
ActuDistns.

Hjorth distribution
: provided in
ActuDistns.

Huber's least favourable distribution
: d, r in package
smoothmest,
d, p, q, r in
VGAM.

Hyperbolic distribuion
:
fBasics,
ghyp,
GeneralizedHyperbolic
and
HyperbolicDist
packages
provide d, p, q, r functions for the generalized hyperbolic distributions.
SkewHyperbolic
provides the skewed Hyperbolic Student tDistribution.
fBasics
also implements the standardized generalized Hyperbolic distribution.
A fast random generator is available for the hyperbolic distribution
is implemented in
Runuran
as well as the density function.

Hyperbolic sine distribution and extension
:
gamlss.dist
provides the sinh and the asinh distributions.
ihs
provides the asinh distribution.
Generalized Power Hyperbolic sine distributions are provided in
FatTailsR.

Inverse Gaussian distribution
:
d, p, q, and r functions of the inverse Gaussian are provided in
statmod,
SuppDists
and
STAR.
SuppDists
also provides a function that returns moments, skewness, kurtosis.
ActuDistns
provides d, q functions as well as (integrated) hazard function.
fBasics
the normal inverse Gaussian and standardized
normal inverse Gaussian distributions.
The generalized inverse gaussian distribution can be found in
gamlss.dist
and
HyperbolicDist. A random generator is available for the (generalized)
Inverse Gaussian distribution is implemented in
Runuran
as well as the density function.
GIGrvg
generates random variables from the generalized inverse Gaussian distribution.
frmqa
computes p function of the generalized inverse Gaussian distribution.

Jshaped distribution
: provided in
ActuDistns.

Johnson distribution
: provided in
SuppDists.
JohnsonDistribution
provides functions to convert normal variate
to Johnson distribution and viceversa.

Kprime distribution
:
sadists
implements Gram Charlier, Edgeworth and CornishFisher approximations
for Kprime distribution for computing d, p, q, r functions.

Kappa distribution
: A 4parameter Kappa distribution is provided in
lmomco
and
FAdist.

Kappamu distribution
: provided in
lmomco.

Kendall's tau distribution
: provided in
SuppDists.

Kiener distribution
:
a family of distributions generalizing hyperbolic sine distributions (see hyperbolic sine section),
d, p, q, r, m provided in
FatTailsR.

Kolmogorov distribution
: p function provided in
kolmim.

Kruskal Wallis distribution
: provided in
SuppDists.

Kumaraswamy distribution
: provided in packages
VGAM,
ActuDistns
and
lmomco.

(Tukey) Lambda distribution and its extensions
: The generalized Lambda
distribution (GLD) is well known for its wide range of shapes.
The original Tukey Lambda distribution can be obtained as a special
case of the generalized Lambda distribution.
There exists different parametrization of GLD in the literature:
RS (RambergSchmeiser or tailindex param),
FMKL (FreimerMudholkarKolliaLin),
FM5 (Fiveparameter version of FKML by Gilchrist),
GPD (gen. Pareto dist.)
and AS (Asymmetrysteepness). The following packages implement such distributions
(with d, p, q, r functions):
gld
(RS, FKML, FM5, GPD),
Davies
(RS),
gb
(RS),
lmomco
(FMKL),
gldist
(AS).

Tukey's H distribution
: provided as a special case of
Lambert W x F distribution.

Lai distribution
: provided in
ActuDistns.

Lambdaprime distribution
:
sadists
implements Gram Charlier, Edgeworth and CornishFisher approximations
for Kprime distribution for computing d, p, q, r functions.

Lambert W x F distribution
:
LambertW
package provides
d, p, q, r functions as well as the first 4 central moments and a qqplot.

Laplace (also called double exponential distribution)
and asymetric Laplace distribution
:
provided in
distr,
lmomco,
VGAM,
ExtDist,
sgt
and
HyperbolicDist
packages.
Asymetric Laplace distribution is implemented in
ald.
A fast random generator is available for the Laplace distribution
is implemented in
Runuran
as well as the density function.
smoothmest
implements the density and the random generator.
The skew Laplace distribution is available in
sgt.

Linear failure rate distribution
: provided in
reliaR
and
ActuDistns.
ActuDistns
also provides hazard (h) and integrated hazard rate (i) functions.

Loglog distribution
: provided in
reliaR
and
ActuDistns.
ActuDistns
also provides hazard (h) and integrated hazard rate (i) functions.

Lomax distribution
: see beta.

Logistic distribution and its extensions
: Base R provides
the d, p, q, r functions for this distribution (see above).
ActuDistns
also provides hazard (h) and integrated hazard rate (i) functions.
actuar
and
VGAM
provide d, p, q, r functions for
the log logistic (also called Fisk), the paralogistic and the inverse paralogistic distributions.
FAdist
the loglogistic distribution with two and three parameters.
The generalized logistic distribution (Type I, also known as skewlogistic distribution) is provided
in
lmomco,
sld,
SCI
and
glogis. Finally,
ActuDistns
implements the loglogistic distribution.
ExtDist
implements the logistic distribution.
Summary for Logisticrelated distributions
Distribution name

Packages

Functions

Distribution suffix

Logistic

stats

d, p, q, r

logis

Logistic

actuar

m, mgf

logis

Logistic

ActuDistns

d, p, q, r, h, i

logis

Logistic

ExtDist

d, p, q, r

Logistic

Log logistic

actuar

d, p, q, r, m, lev

llogis

Log logistic

VGAM

d, p, q, r

fisk

Log logistic

FAdist

d, p, q, r

llog, llog3

Log logistic

ActuDistns

d, p, q, r, h, i

loglogistic

Paralogistic

actuar

d, p, q, r, m, lev

paralogis

Paralogistic

VGAM

d, p, q, r

paralogistic

Inv. paralogistic

actuar

d, p, q, r, m, lev

invparalogis

Inv. paralogistic

VGAM

d, p, q, r

inv.paralogistic

Generalized logistic

glogis

d, p, q, r

glogis

Generalized logistic

SCI

d, p, q

genlog

Generalized logistic

lmomco

d, p, q, r

glo

Generalized logistic

sld

d, p, q, r

sl


Logitnormal distribution
:
provided in
logitnorm.

Lognormal distribution and its extensions
:
The log normal distribution is implemented in Base R (see above)
and
poweRlaw, but the 3parameter lognormal
distribution is available in
lmomco
and
FAdist.
ActuDistns
provides hazard (h) and integrated hazard rate (i) functions
for the lognormal distribution.
The package
loglognorm
implements d, p, q, r functions for the double lognormal distribution,
as well as the raw moment, the expected value and the variance functions.

Makeham distribution
: provided in
VGAM
and
ActuDistns
provides Makeham, MakehamBeard and MakehamPerks distributions.

Maxwell distribution
: provided in
VGAM.

Minimax distribution
: provided in
minimax.

Nakagami distribution
: provided in
VGAM.

Pareto distribution
: d, p, q, r functions are implemented in
VGAM
for the Pareto distribution type IV (which includes Burr's distribution,
Pareto type III, Pareto type II (also called the lomax distribution) and
Pareto type I) and the (upper/lower) truncated Pareto distribution.
In an actuarial context,
actuar
provides d, p, q, r functions as well as moments and
limited expected values for the Pareto I and II, the inverse Pareto, the 'generalized pareto'
distributions, the Burr and the inverse Burr distributions. A fast random generator for the
Burr and the Pareto II distribution is implemented in
Runuran
as well as the density.
ActuDistns
also provides d, p, q, h functions of the Pareto II distribution and
the BurrX distribution.
ExtDist
provides the Burr distribution.
lmomco,
evd,
fExtremes,
FAdist
and
evir
packages
implement the Generalized Pareto Distribution (from Extreme Value Theory), which is
depending the shape parameter's value a Pareto II distribution,
a shifted exponential distribution or a generalized beta I distribution.
ParetoPosStable
implements the Pareto positive stable distribution.
Finally, the extended Pareto distribution is implemented in
RTDE.
Summary for Paretorelated distributions
Distribution name

Packages

Functions

Distribution suffix

Pareto I

VGAM

d, p, q, r

paretoI

Pareto I

actuar

d, p, q, r, m, lev

pareto1

Pareto II

VGAM

d, p, q, r

paretoII

Pareto II

actuar

d, p, q, r, m, lev

pareto, pareto2

Pareto II

Runuran

d, r

pareto

Pareto II

ActuDistns

d, p, q, h, i

pareto

Pareto III

VGAM

d, p, q, r

paretoIII

Pareto IV

VGAM

d, p, q, r

paretoIV

Inverse Pareto

actuar

d, p, q, r, m, lev

invpareto

Extended Pareto

RTDE

d, p, q, r

EPD

Gen. Pareto (actuarial)

actuar

d, p, q, r, m, lev

genpareto

Gen. Pareto (EVT)

lmomco

d, p, q, r

gpa

Gen. Pareto (EVT)

evd

d, p, q, r

gpd

Gen. Pareto (EVT)

fExtremes

d, p, q, r

gpd

Gen. Pareto (EVT)

evir

d, p, q, r

gpd

Burr

actuar

d, p, q, r, m, lev

burr

Burr

ActuDistns

d, p, q, h, i

burrx

Inverse Burr

actuar

d, p, q, r, m, lev

invburr


Pearson's distribution
: Pearson type III available
in
lmomco
and
FAdist.
A logPearson type III distribution is also available in
FAdist.
PearsonDS
provides the d, p, q, r functions as well as the first
four moments for the Pearson distributions: types I, II, III, IV, V, VI, VII.

Pearson's Rho distribution
: provided in
SuppDists.

Perks distribution
: provided in
VGAM
and in
ActuDistns
as well as
the MakehamPerks distribution.

Planck's distribution
: a random generator is available in
Runuran.

Phasetype distribution
: provided in
actuar, see also
PhaseType
for inference.

Poisson subordinated distributions
: provided in
LIHNPSD
(d, p, q, r, m functions).

Power distribution
:
reliaR,
poweRlaw
and
ActuDistns
implement the
exponential power distribution.

Proportion distribution
: this is the distribution for the difference
between two independent beta distributions. d, p, q, r functions in
tolerance.

Rayleigh distribution
: provided in packages
VGAM
and
lmomco. Generalized and logistic Rayleigh distributions are available in
reliaR
and
ActuDistns.

Response time distribution
:
rtdists
provides d, p, r functions
for Ratcliff distribution and linear ballistic accumulator (LBA) with different
underlying driftdistributions (Normal, Gamma, Frechet, and lognormal).

Rice distribution
: provided in
VGAM
and
lmomco.

Schabe distribution
: provided in
ActuDistns.

SinghMaddala distribution
: see beta.

Slash distribution
: provided in
lmomco
and
VGAM.

Spearman's Rho distribution
: provided in
SuppDists.

Stable distribution
: d, p, q, r functions are available in
fBasics
and
stabledist, the functions use the approach of J.P. Nolan
for general stable distributions.
MixedTS
provides mixed tempered
stable distribution (d, p, q, r).
FMStable
provides (d, p, q) the extremal or maximally skew stable and
the finite moment log stable distributions.

Student distribution and its extensions
: Base R provides
the d, p, q, r functions for Student and non central Student distribution (see above).
sadists
implements Gram Charlier, Edgeworth and CornishFisher approximations
for doubly non central Student distribution for computing d, p, q, r functions.
The skewed Student distribution is provided in
skewt,
sn
and
gamlss.dist
packages.
The generalized skew distribution is provided in
sgt.
d, p, q, r functions for the generalized tdistribution
can be found in
gamlss.dist.
fBasics
provides d, p, q, r functions
for the skew and the generalized hyperbolic tdistribution.
The Lmoments of the Student t (3parameter) are provided in
lmomco.

Triangle/trapezoidal distribution
: packages
triangle,
mc2d,
ExtDist
and
VGAM
provide d, p, q, r functions for the triangle distribution, while the package
trapezoid
provides d, p, q, r functions for the Generalized Trapezoidal Distribution.
A fast random generator is available for the triangle distribution
is implemented in
Runuran
as well as the density function.

Tsallis or qExponential distribution
:
tsallisqexp
provides d, p, q, r functions for two parametrizations of
the Tsallis distribution and also implements a leftcensored version.

Tweedie distribution
: the Tweedie distribution is implemented
in package
tweedie. Let us note that the Tweedie distribution
is not necessarily continuous, a special case of it is the Poisson distribution.

Uniform distribution
: d, p, q, r functions are of course provided
in R. See section RNG for random number generation topics.
ActuDistns
provides hazard and integrated hazard rate functions.
HI
generates
uniformly random points on a bounded convex set, in particular the unit ball.

Upsilon distribution
:
sadists
implements Gram Charlier, Edgeworth and CornishFisher approximations
for Upsilon distribution for computing d, p, q, r functions.

Wakeby distribution
: A 5parameter Wakeby is provided in
lmomco.

Weibull distribution and its extensions
: Base R provides
the d, p, q, r functions for this distribution (see above).
The inverse Weibull is provided in
actuar
package and also the moments
and the limited expected value for both the raw and the inverse Weibull distribution.
FAdist
implements the threeparameter Weibull distribution, while
reliaR
and
ActuDistns
implement the exponential Weibull, the flexible Weibull,
the generalized power weibull, the MarshallOlkin Extended Weibull and
the Weibull extension distributions.
Finally,
lmomco,
ExtDist
implements the Weibull distribution while
evd
implements the reverse Weibull distribution. d, p, q, r functions for
the reverse generalized extreme value distribution are provided in
gamlss.dist.

Xie distribution
: provided in
ActuDistns.
Continuous multivariate distributions:

Dirichlet distribution
:
not yet implemented?

Multivariate exponential distribution
:
not yet implemented?

Multivariate Gaussian (or normal) distribution
:
The multivariate Gaussian distribution is provided in the packages
mvtnorm
(d, r),
mnormt
(d, p, r),
EMMIXskew
(d, r).
mvprpb
computes the orthant probability of the multivariate Gaussian distribution.
symmoments
computes central and noncentral moments of the multivariate Gaussian distribution.
Futhermore,
tmvtnorm
implements the truncated
multivariate normal distribution;
sparseMVN
implements very
fast algorithms to compute the density and generate random variates of
a multivariate normal distribution for which the covariance matrix or precision matrix is sparse.
cmvnorm
implements the complex multivariate normal distribution (d, r).
Finally,
condMVNorm
implements d, p, r functions for the conditional multivariate
normal distribution.
EMMIXskew
provide the multivariate skew normal distribution (d, r).
mnormpow
computes the expected product of the components of a multivariate Gaussian vector.

Multivariate generalized extreme value distribution
:
Both bivariate and multivariate Extreme Value distributions as well as order/maxima/minima distributions
are implemented in
evd
(d, p, r).

Multivariate logistic distribution
:
VGAM
package implements the bivariate logistic distribution.

Multivariate Pareto distribution
:
mgpd
provides the density for the multivariate generalized Pareto distribution
of type II, while
evd
provides the density for type I.

Multivariate Stable distribution
:
not yet implemented?

Multivariate Student distribution
:
The multivariate Student distribution is provided in the packages
mvtnorm
(d, r),
mnormt
(d, p, r),
EMMIXskew
(d, r).
First two moments (m) and sampling (r) of the Truncated Multivariate t Distribution
are provided in
TTmoment.
EMMIXskew
provide the multivariate skew t distribution (d, r).
Mixedtype distributions:

MaxwellBoltzmannBoseEinsteinFermieDirac (MBBEFD) distribution
:
provided in
mbbefd.

Mixed ordinal and normal distribution
:
provided in
OrdNor.
Mixture of probability laws (and composite):

Bernoullidist mixture
: d, p, q, r functions
for Bernoulliexponential, BernoulliGamma, Bernoullilognormal,
BernoulliWeibull distributions are provided in
qmap.

Cauchypolynomial quantile mixture
: d, p, q, r functions are
provided in
Lmoments.

composite lognormal distribution
: d, p, q, r functions are
provided in
CompLognormal.

Gaussian mixture
: Functions d, r are provided in
mixtools
package when dealing with finite mixture models.
nor1mix
provides
d, p, r functions for Gaussian mixture.

Gamma mixture
: Gamma shape mixtures are implemented (d, p, r)
in the
GSM
package.

Generic mixtures
: there is an implementation via S4class
UnivarMixingDistribution in package
distr.
gamlss.mx
uses the
gamlss.dist
package.

Normalpolynomial quantile mixture
: d, p, q, r functions are
provided in
Lmoments.

Poisson Binomial distribution
:
poibin
implements
the Poisson Binomial distribution.

Poisson lognormal distribution
:
poilog
implements
the Poisson lognormal distribution.

PoissonTweedie exponential family models
: provided in
poistweedie.

Student mixture
: The
AdMit
package provides d, r functions
for Student mixtures in the context of Adaptive Mixture of Studentt distributions.
MitISEM
package also provides d, r functions for mixture of Studentt distributions.

von Mises Fisher (or Langevin) mixture
: The
movMF
package provides d, r functions
for finite von Mises Fisher mixtures.
Compound, exponentiated and tranformation of distributions:

Compound distribution
: d, p, q, r, m functions are implemented by
Compounding
where the parent distribution is any continuous distribution and the compound distribution
is any distribution among the list: binomial, binomialPoisson, geometric, hypergeometric,
hyperPoisson, Katti type H1/H2, logarithmic, logarithmicbinomial,
logarithmicPoisson, negative binomial, Neyman type A/B/C, PascalPoisson, Poisson,
Poissonbinomial, PoissonLindley, PoissonPascal, Polya Aeppli, Thomas,
Waring, Yule.

Exponentiated distribution
:
Exponentiated exponential, exponentiated logistic, exponentiated Weibull,
inverse exponentiated exponential
are implemented in
ActuDistns.

G transformed distribution
: implemented in
Newdistns
which includes Marshall Olkin G distribution, exponentiated G distribution,
beta G distribution, gamma G distribution, Kumaraswamy G distribution,
generalized beta G distribution, beta extended G distribution, gamma G distribution,
gamma uniform G distribution, beta exponential G distribution, Weibull G distribution,
log gamma G1/G2 distribution, exponentiated generalized G distribution,
exponentiated Kumaraswamy G distributions, geometric exponential Poisson
G distribution, truncatedexponential skewsymmetric G distribution, modified beta
G distribution, and exponentiated exponential Poisson G distribution.
Moments, skewness, kurtosis and etc:

Empirical moments
: base R provides
mean(),
sd(),
var()
functions to compute
the mean, standard deviation and variance, respectively.
The skewness is available in
agricolae,
e1071,
GLDEX,
HyperbolicDist,
modeest,
moments,
npde,
TSA,
s20x,
DistributionUtils
packages.
The kurtosis is available in
agricolae,
DistributionUtils,
e1071,
GLDEX,
HyperbolicDist,
moments,
npde,
TSA
packages.
The raw or centered moments are provided in
e1071,
moments.
Lmoments are available in
lmom,
lmomco,
Lmoments,
GLDEX,
trimmed Lmoments are available in
lmomco, and
Lmoments,
rightcensored Lmoments are available in
lmomco,
and cumulants in
GLDEX.

Mode estimation
:
Package
modeest
provides mode estimation for various distributions.

Order statistics
:
Distribution function of the jth order statistic can be obtained with base R functions.
ORDER2PARENT
transforms distribution function of order statistics to its parent
distribution function.

Theoretical moments
:
The
actuar
package implements raw moments, limited expected values
and moment generating function for base R distributions.
HyperbolicDist
provides the mean, variance, skewness, kurtosis, mode,
raw and centered moments for the hyperbolic, the generalized hyperbolic and the
generalized inverse Gaussian distributions.
GLDEX
also provides the mean, variance, skewness, kurtosis of generalized Lambda
distribution.
mvrtn
provides mean, variance for left/right truncated normal distributions.
lmomco
provides Lmoments (L), trimmed Lmoments (TL), and rightcensored [RC] for
the following distributions: Asymmetric Exponential Power (L), Cauchy (TL), EtaMu (L),
Exponential (L), Gamma (L), Generalized Extreme Value (L), Generalized Lambda (L and TL),
Generalized Logistic (L), Generalized Normal (L), Generalized Pareto (L[RC] and TL),
Govindarajulu (L), Gumbel (L), Kappa (L), KappaMu (L), Kumaraswamy (L), Laplace (L),
Normal (L), 3parameter logNormal (L), Pearson Type III (L), Rayleigh (L), Reverse Gumbel
(L[RC]), Rice/Rician (L), Slash (TL), 3parameter Student T (L), Truncated Exponential
(L), Wakeby (L), and Weibull (L). Multivariate Lmoments (Lcomoments).
Random matrices:

Wishart and inverse Wishart distributions
:
Base R provides the r function for the Wishart distribution.
the d, r functions are provided in
MCMCpack
and
bayesm.

MarcenkoPastur distribution
:
provided in
RMTstat,
MCMCpack
and
bayesm.

TracyWidom distribution
:
provided in
RMTstat,
MCMCpack
and
bayesm: supported beta values
are 1 (Gaussian Orthogonal Ensemble), 2 (Gaussian Unitary Ensemble),
and 4 (Gaussian Symplectic Ensemble).

Wishart distribution
:
provided in
RMTstat.

Spiked Wishart Maximum Eigenvalue Distribution
:
provided in
RMTstat,
MCMCpack
and
bayesm.

White Wishart Maximum Eigenvalue Distribution
:
provided in
RMTstat,
MCMCpack
and
bayesm.
Copulas:

Unified approaches
:
The packages
fCopulae,
copula, and
copBasic
provide a lot of
general functionality for copulas. Although lacking support for
many existing copulas themselves,
copBasic
is primarily oriented
around utility functions for the general mathematics of copulas
as described in the well known introduction to copulas by Nelsen.

Archimedean copulas
:
gumbel
is a standalone package for the Gumbel copula
fCopulae
implements the 22 Archimedean
copulas of Nelsen (1998,
Introduction to Copulas
, SpringerVerlag) including
Gumbel, Frank, Clayton, and AliMikhailHaq.
VGAM
provides AliMikhailHaq, Clayton, Frank, Frechet copulas.
copula
provides AliMikhailHaq, Clayton, Frank, Gumbel and Joe copulas.
The Frank bivariate distribution is available in
RTDE.
CDVine
and
VineCopula
provide
Clayton, Gumbel, Frank, Joe, BB1, BB6, BB7 and BB8 copulas.
Nested Archimedean copulas are available in the
HAC
package.
Generalized Archimedean copulas are implemented in the
fgac
package.
BivarP
provides cdf, pdf and survival function for Clayton, Gumbel and Frank copula.

Blomqvist copula
: provided in
copBasic.

Composition of copula
:
copBasic
provides functions
for composition of a single symmetric copula and composition of two copulas.

Cubic copula
: Not yet implemented?

Dirichlet copula
: Not yet implemented?

Empirical copula
: provided in
copBasic
and in
HAC.
GenOrd
provides sampling function for multivariate discrete random vectors
with a specified correlation matrix.

Elliptical copulas
: Gaussian, Student and Cauchy copulas are implemented in
fCopulae
for the bivariate cases.
copula,
CDVine,
VGAM,
VineCopula
provide the Gaussian and the Student copula.

Extreme value copulas
:
fCopulae
provides the following copulas
Gumbel, Galambos, HuslerReiss, Tawn, or BB5.
copula
implements
Gumbel, Galambos and HuslerReiss.

EyraudFarlieGumbelMorgenstern copula
: provided in
VGAM,
RTDE,
and
copula.

Mardia copula
: Not yet implemented?

Nested copulas
: arbitrary nested versions of copulas can
be implemented in
copula.

Plackett
: provided in
VGAM,
copBasic
and
copula.

Vine copulas
: Packages
CDVine,
vines
provide functions for C and Dvine copulas and
VineCopula
for general Rvine copulas.
Random Number Generators:

Basic functionality
: R provides several random number generators (RNGs).
The random seed can be provided via
set.seed
and the kind of RNG can be specified
using
RNGkind. The default RNG is the MersenneTwister algorithm.
Other generators include WichmannHill, MarsagliaMulticarry, SuperDuper,
KnuthTAOCP, KnuthTAOCP2002, as well as usersupplied RNGs. For normal
random numbers, the following algorithms are available: KindermanRamage,
AhrensDieter, BoxMuller, Inversion (default). In addition to the tools above,
setRNG
provides an easy way to set, retain information about the setting, and reset the RNG.

Pseudorandomness
:
RDieHarder
offers several dozen new RNGs
from the GNU GSL.
randtoolbox
provides
more recent RNGs such as SF MersenneTwister and WELL,
which are generators of Mersenne Twister type, but with
improved quality parameters.
rngwell19937
provides one of the WELL generators with 53 bit
resolution of the output and allows seeding by a vector
of integers of arbitrary length.
randaes
provides the deterministic part of the Fortuna cryptographic
pseudorandom number generator (AES).
SuppDists
implements two RNGs of G. Marsaglia.

Support for several independent streams:
rstream
focuses on multiple independent streams of random numbers
from different sources (in an object oriented approach).

For nonuniform generation, the
Runuran
package interfaces to the UNU.RAN library for universal nonuniform generation
as well as customised distributions based on polynomial interpolation of the inverse
cumulative distribution function.

Quasirandomness
: The
randtoolbox
provides
the following quasi random sequences: the Sobol sequence, the Halton
(hence Van Der Corput) sequence and the Torus sequence (also known as Kronecker
sequence).
lhs
and
mc2d
packages implement the latin hypercube sampling,
an hybrid quasi/pseudo random method.
sfsmisc
also provides
the Halton sequences.

True randomness
: The
random
package provides several functions that access the
true random number service at
random.org
.

RNG tests
:
RDieHarder
offers
numerous tests of RNGs based on a
reimplementation and extension of Marsaglia's DieHarder battery.
randtoolbox
provides basic RNG tests.

Parallel computing
:
Randomnumber generators for parallel computing are available via
the
rlecuyer
package.
See the
HighPerformanceComputing
task view for more details.
Miscellaneous:

Computation
:

Approximation of d, p, q, r functions
:
PDQutils
provides tools for computing the density, cumulative distribution,
and quantile functions of a distribution when the cumulants or moments are given,
using the classical Gram Charlier, Edgeworth and CornishFisher approximations.
sadists
is a showcase for PDQutils, providing density, cumulative distribution,
quantile, and random generation for the doubly noncentral t, doubly noncentral F,
Kprime, Lambdaprime, Upsilon, and sum of (noncentral) chisquares to powers
distributions.

For nonuniform generation, see the
Runuran
above.

Benchmark
: A set of 28 densities suitable for comparing
nonparametric density estimators in simulation studies can be found
in the
benchden
package. The densities vary greatly in degree of
smoothness, number of modes and other properties. The package
provides d,p,q and r functions.

Non parametric models
:

Binned Empirical distributions
: The
HistogramTools
package
provides a number of methods for manipulating empirical data that has
been binned into histogram form, including: (1) the empirical
cumulative distribution function, (2) the empirical quantile, and (3)
information loss metrics associated with binning.

Empirical distribution
: Base R provides functions for univariate analysis:
(1) the empirical density (see density()), (2) the empirical cumulative
distribution function (see ecdf()), (3) the empirical quantile (see quantile())
and (4) random sampling (see sample()).

Non Parametric distributions
:
spd
provides the Semi Parametric Piecewise Distribution, while
fBasics
implements spline smoothed distributions.

Hierarchical models
: Distributions whose some parameters
are no longer constant but random according to a particular distribution.
VGAM
provides a lot of hierarchical models: beta/binomial,
beta/geometric and beta/normal distributions.
bayesm
implements: binary logit, linear, multivariate logit and negative binomial
models. Furthermore
LearnBayes
and
MCMCpack
provides poisson/gamma, beta/binomial,
normal/normal and multinomial/Dirichlet models.

Distribution handling
:

Objectorientation
:
General discrete and continuous distributions are implemented in package
distr
respectively via S4class DiscreteDistribution and
AbscontDistribution providing the classic d, p, q and r functions.
distrEx
extends available distributions to multivariate and conditional distributions as well
as methods to compute useful statistics (expectation, variance,...) and distances
between distributions (Hellinger, Kolmogorov,... distance). Finally
package
distrMod
provides functions for the computation of minimum
criterion estimators (maximum likelihood and minimum distance estimators). See other
packages of the distrfamily (distrSim,
distrTEst,
distrTeach,
distrDoc,
distrEllipse).

Transformation
: Lebesgue decomposition are implemented
in
distr, as well as Convolution, Truncation and Huberization of distributions.
Furthermore,
distr
provides distribution of the maximum or minimum
of two distributions. See Objectorientation above.

User Interface
:
AtelieR
package provides a GTK GUI for teaching basic
concepts in statistical inference, implementing all the R base distributions
as well as the generalized Student, the inverse Chisquare, the inverse gamma
and the lambdaprime distributions.

Transversal functions
:

Histogram, tail plots, distance estimation
:
DistributionUtils
provides loghistogram,
tail plots, functions for testing distributions using inversion tests and the Massart inequality.
GMD
is a package for nonparametric distance measurement
between two discrete frequency distributions.

Paremeter estimation
:
lmomco
and
Lmoments
focus on
univariate/multivariate (L)moments estimation.
VGAM
provides a lot of parameter
estimation for usual and "exotic" distributions.
gaussDiff
provides a collection difference measures for
multivariate Gaussian probability density functions
Package
MASS
implements the flexible
fitdistr
function for parameter estimations.
fitdistrplus
greatly enlarges and enhances
the tools to fit probability distributions.
flexsurv
and
msm
provides a quantile
function for a generic distribution based on numerical
computation based on a dichotomic search.