name	topic	maintainer	email	version	source
Distributions	Probability Distributions	Christophe Dutang, Patrice Kiener, Bruce J. Swihart	dutangc@gmail.com	2024-10-25	https://github.com/cran-task-views/Distributions/

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.

The maintainers gratefully acknowledge 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 send an e-mail to the maintainer or submit an issue or pull request in the GitHub repository linked above.

[Base functionality]{#base}

Base R provides probability distribution functions pfoo() density functions dfoo(), quantile functions qfoo(), and random number generation rfoo() where foo indicates the type of distribution: beta (foo = beta), binomial binom, Cauchy cauchy, chi-squared 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.
Base R provides various one-sample or two-sample tests for univariate distributions, e.g., ks.test, shapiro.test, ansari.test, chisq.test, poisson.test. r pkg("Ecume") provides non-parametric two-sample (or k-sample) distribution comparisons in the univariate or multivariate case allowing observation weights and thresholds.
Probability generating function: no longer implemented.

Some packages may optionally provide the symbolic derivatives with respect to the parameters for the probability functions. For instance, the first and second derivatives of the log-density can be of some help in estimation and inference tasks, and the derivatives of the quantile function can help when inferring on a given quantile. For that purpose, the following base R functions can be used stats::D() for derivatives w.r.t. a single parameter, or stats::deriv() for (partial) derivatives w.r.t. multiple parameters. The r pkg("Deriv") package provides a much more flexible symbolic differentiation interface. One can also use Stan Math library through r pkg("StanHeaders") package, see e.g. this blog. The r pkg("nieve") package provides symbolic differentiation for two probability distribution (Generalized Pareto and Generalized Extreme Value) in order to compute the log-likelihood for example.

Discrete distributions

Discrete univariate distributions

Beta-binomial distribution: provided in r pkg("VGAM", priority = "core"), r pkg("extraDistr"), r pkg("rmutil"), r pkg("emdbook"). ZI/ZM beta binomial distributions are implemented in r pkg("gamlss.dist", priority = "core"). For a continuous analog with d, p, q, r functions, see r pkg("cbbinom").
Beta-geometric distribution: provided in r pkg("VGAM").
Binomial (including Bernoulli) distribution: provided in stats. Zero-modified, zero-inflated, truncated versions are provided in r pkg("gamlss.dist"), r pkg("extraDistr"), r pkg("actuar", priority = "core") and in r pkg("VGAM"). r pkg("LaplacesDemon") provides dedicated functions for the Bernoulli distribution. r pkg("rmutil") provides the double binomial and the multiplicative binomial distributions.

Distribution name Packages Functions Distribution suffix binomial stats d, p, q, r binom zero-infl. binomial extraDistr d, p, q, r zib zero-infl. binomial VGAM d, p, q, r zibinom zero-infl. binomial gamlss.dist d, p, q, r ZIBI zero mod. binomial VGAM d, p, q, r zabinom zero mod. binomial actuar d, p, q, r zmbinom zero mod. binomial gamlss.dist d, p, q, r ZABI zero trunc. binomial actuar d, p, q, r ztbinom trunc. binomial extraDistr d, p, q, r tbinom

: Summary for Binomial-related distributions
Bell Touchard distribution: standard and zero-inflated provided in r pkg("countDM").
Benford distribution: provided in r pkg("VGAM") and r pkg("BenfordTests").
Bernoulli distribution: provided in r pkg("extraDistr").
Borel-Tanner distribution: provided in r pkg("VGAM").
Delaporte distribution: provided in r pkg("gamlss.dist") and r pkg("Delaporte").
Dirac distribution: provided in r pkg("distr", priority = "core").
Discrete Burr-Hatke distribution: r pkg("DiscreteDists") provides d, p, q, r functions.
Discrete categorical distribution: provided in r pkg("LaplacesDemon").
Discrete Cauchy (Cauchy-Cacoullos) distribution: provided in r pkg("CCd").
Discrete exponential distribution: provided in r pkg("poweRlaw"). A generalized version of the second type is in r pkg("DiscreteDists").
Discrete gamma distribution: provided in r pkg("extraDistr").
Discrete inverted Kumaraswamy distribution: r pkg("DiscreteDists") provides d, p, q, r functions.
Discrete inverse Weibull distribution: r pkg("DiscreteInverseWeibull") provides d, p, q, r functions for the inverse Weibull as well as hazard rate function and moments.
Discrete Laplace distribution: The discrete Laplace distribution is provided in r pkg("extraDistr") (d, p, r). The skew discrete Laplace distribution has two parametrization (DSL and ADSL), both provided in r pkg("DiscreteLaplace") and DSL in r pkg("disclap"). r pkg("LaplacesDemon") also provides the DSL parametrization only.
Discrete Lindley distribution: r pkg("DiscreteDists") provides d, p, q, r functions.
Discrete lognormal distribution: provided in r pkg("poweRlaw").
Discrete Marshall–Olkin Length Biased Exponential distribution: r pkg("DiscreteDists") provides d, p, q, r functions.
Discrete normal distribution: provided in r pkg("extraDistr").
Discrete power law distribution: provided in r pkg("poweRlaw").
Discrete uniform distribution: can be easily obtained with the functions sum,cumsum,sample and is provided in r pkg("extraDistr").
Discrete Weibull distribution: provided in r pkg("DiscreteWeibull"): d, p, q, r, m for disc. Weib. type 1, d, p, q, r, m, h for disc. Weib. type 3. r pkg("extraDistr") provides d, p, q, r for Type 1.
Felix distribution: provided in r pkg("VGAM").
gamma count distribution: provided in r pkg("rmutil").
Geometric distribution: provided in stats . Zero-modified, zero-inflated, truncated versions are provided in r pkg("gamlss.dist"), r pkg("actuar") and in r pkg("VGAM"). The time-varying geometric is provided in r pkg("tvgeom").
Geometric (compound) Poisson distribution (also known Polya-Aeppli distribution): provided in r pkg("polyaAeppli"). Uniform-geometric distribution provided in r pkg("new.dist").
Generalized/fractional binomial distribution: r pkg("GenBinomApps") provides the generalized binomial distribution. r pkg("frbinom") provides the fractional binomial distribution where trials are from a generlized Bernoulli process.
Generalized Geometric distribution: provided in r pkg("DiscreteDists").
Generalized Hermite distribution: provided in r pkg("hermite").
Good distribution: provided in r pkg("good").
Hyper-Poisson distribution: provided in r pkg("DiscreteDists").
Hypergeometric distribution: provided in stats . Non-central hypergeometric distribution is provided in r pkg("MCMCpack") (d,r). Extended hypergeometric distribution can be found in r pkg("BiasedUrn") package, which provides not only p, d, q, r functions but also mean, variance, mode functions. Generalized hypergeometric distribution is implemented in r pkg("SuppDists"). Negative hypergeometric distribution is provided in r pkg("tolerance"), r pkg("extraDistr").
Lagrangian Poisson distribution: r pkg("RMKdiscrete") provides d, p, q, r functions for the univariate and the bivariate Lagrangian Poisson distribution.
Lindley's power series distribution: provided in r pkg("LindleyPowerSeries") and in r pkg("new.dist").
Logarithmic distribution: This can be found in r pkg("extraDistr"), r pkg("VGAM"), r pkg("actuar"), and r pkg("gamlss.dist"). Zero-modified and zero-truncated versions is provided in r pkg("actuar"). A fast random generator is available for the logarithmic distribution is implemented in r pkg("Runuran") as well as the 'density' function.
Poisson distribution: provided in stats and in r pkg("poweRlaw"). Zero-modified, zero-inflated, truncated versions are provided in r pkg("extraDistr"), r pkg("gamlss.dist"), r pkg("actuar") and in r pkg("VGAM"). r pkg("extraDistr") provides the truncated Poisson distribution. r pkg("LaplacesDemon") provides the generalized Poisson distribution. r pkg("rmutil") provides the double Poisson, the multiplicative Poisson and the Power variance function Poisson distributions. r pkg("poibin") and r pkg("PoissonBinomial") provide the Poisson binomial distribution. See the mixture section such as the Poisson-lognormal mixture.
Poisson-Lindley distribution: provided in r pkg("tolerance").
Power law distribution: provided in r pkg("poweRlaw").
Mana Clash distribution: provided in r pkg("RMKdiscrete").
Negative binomial distribution: provided in stats . Zero-modified, zero-inflated, truncated versions are provided in r pkg("gamlss.dist"), r pkg("extraDistr"), r pkg("emdbook"), r pkg("actuar") and in r pkg("VGAM"). New parametrization of the negative binomial distribution is available in r pkg("RMKdiscrete"). r pkg("nbconv") provides p, q, r functions for convolutions of negative binomial distributions.
Sichel distribution: provided in r pkg("gamlss.dist").
Skellam distribution: provided in r pkg("extraDistr"), r pkg("VGAM") and r pkg("skellam").
Waring distribution: r pkg("degreenet") provides a random generator, r pkg("cpd") provides d, p, q, r functions for extended biparametric Waring.
Yule-Simon distribution: provided in r pkg("VGAM") and sampling in r pkg("degreenet").
Zeta and Haight's Zeta distribution: provided in r pkg("VGAM"), r pkg("tolerance").
Zipf distribution and extensions: d, p, q, r functions of the Zipf and the Zipf-Mandelbrot distributions are provided in r pkg("tolerance"), r pkg("VGAM"). Package r pkg("zipfR") provides tools for distribution of word frequency, such as the Zipf distribution. r pkg("zipfextR") provides three extensions of the Zipf distribution: the Marshall-Olkin Extended Zipf, the Zipf-Poisson Extreme and the Zipf-Poisson Stopped Sum distributions.

Discrete multivariate distributions

Hyper Dirichlet distribution: provided in r pkg("hyper2") package.
Multinomial distribution: stats, r pkg("mc2d"), r pkg("extraDistr") packages provide d, r functions. r is provided in r pkg("MultiRNG") and r pkg("compositions"). p function is provided by r pkg("pmultinom").
Multinomial Dirichlet distribution: functions d, r are provided in r pkg("MCMCpack"), r pkg("mc2d"), r pkg("dirmult"), r pkg("extraDistr") and r pkg("bayesm"). r is provided in r pkg("MultiRNG").
Multivariate Ewens distribution: not yet implemented?
Multivariate geometric: d, r functions provided in r pkg("bivgeom") for the bivariate geometric distribution. r pkg("BivGeo") provides the Basu-Dhar bivariate geometric distribution.
Multivariate hypergeometric distribution: provided in r pkg("extraDistr"). The conditional hypergeometric distribution is provided (d, p, q, r) in r pkg("chyper").
Multivariate logarithmic distribution: the bivariate logarithmic distribution is provided in r pkg("trawl").
Multiplicative multinomial distribution: The multiplicative multinomial distribution is implemented in r pkg("MM").
Multivariate negative distribution: A bivariate distribution with negative-binomial marginals is available in r pkg("RMKdiscrete") and r pkg("trawl"). r pkg("MNB") provides a generator and diagnostic tool for multivariate negative binomial distribution.
r pkg("bzinb") provides a random generator for the bivariate negative binomial (classic and zero-inflated) distribution.
Multivariate Poisson distribution: r pkg("compositions") provides a random generator. r pkg("bzinb") provides a random generator for the bivariate Poisson (classic and zero-inflated) distribution.
Multivariate Poisson-lognormal: the bivariate Poisson-lognormal distribution is provided in r pkg("poilog").
Multivariate Dirichlet (also known as Polya) distribution: functions d, r of the Dirichlet distribution are provided in r pkg("extraDistr"), r pkg("LaplacesDemon"), r pkg("DirichletReg") and r pkg("Compositional"). The flexible Dirichlet distribution is given in r pkg("FlexDir").
Truncated Stick-Breaking distribution: provided in r pkg("LaplacesDemon").

Continuous distributions

Continuous univariate distributions

Arcsine distribution: implemented in package r pkg("distr").
Argus distribution: implemented in package r pkg("argus").
Beta distribution and its extensions: Base R provides the d, p, q, r functions for this distribution (see above). r pkg("extraDistr") provides the beta distribution parametrized by the mean and the precision. r pkg("actuar") provides moments and limited expected values. r pkg("sadists") implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for doubly non central beta distribution for computing d, p, q, r functions. r pkg("extraDistr") provides the four-parameter beta with lower and upper bounds. The generalized beta of the first kind (GB1) (exponentiation of beta 1) is provided in r pkg("gamlss.dist"), r pkg("mbbefd"), r pkg("actuar"). r pkg("betafunctions") provides the four-parameter beta (that is with location and scale parameters), the beta parametrized by the mean and the variance as well as the beta compound beta distribution. The beta prime (or beta of the second kind), which is the distribution of X/(1-X) when X follows a beta distribution of the first kind, is provided in r pkg("VGAM"), r pkg("extraDistr"), r pkg("LaplacesDemon") and r pkg("mc2d"). The zero and one inflated beta distribution can be found in r pkg("gamlss.dist"). The generalized beta of the second kind (GB2) is provided in r pkg("gamlss.dist"), r pkg("GB2"). Several special cases of the generalized beta distribution are also implemented in r pkg("VGAM"), r pkg("mc2d"): Lomax, inverse Lomax, Dagum, Singh-Maddala, Pert distributions. r pkg("actuar") provides the Feller-Pareto distribution with special cases Burr, log-logistic, paralogistic, generalized Pareto, Pareto, see also the Pareto subsection.

Distribution name Packages Functions Distribution suffix Beta (1st kind) stats d, p, q, r beta Beta actuar m, mgf, lev beta Beta betafunctions d, p, q, r Beta.4P Doubly non central beta sadists d, p, q, r nbeta 4-param beta extraDistr d, p, q, r nsbeta zero-infl beta gamlss.dist d, p, q, r BEZI one-infl beta gamlss.dist d, p, q, r BEOI one-infl beta mbbefd d, p, q, r, m, ec oibeta GB1 gamlss.dist d, p, q, r GB1 GB1 mbbefd d, p, q, r, m, ec gbeta GB1 actuar d, p, q, r, m, lev genbeta one-infl GB1 mbbefd d, p, q, r, m, ec oigbeta

: Summary for Beta-related distributions

Distribution name Packages Functions Distribution suffix Beta (2nd kind) VGAM d, p, q, r beta Beta (2nd kind) extraDistr d, p, q, r invbeta Beta (2nd kind) LaplacesDemon d, r betapr GB2 VGAM d, p, q, r genbetaII GB2 gamlss.dist d, p, q, r GB2 GB2 GB2 d, p, q, r gb2 Trans beta 2 actuar d, p, q, r, m, lev trbeta

: Summary for Beta-2-related distributions
Bell-G distribution: r pkg("BGFD") provides d, p, q, r functions for Bell exponential, Bell extended exponential, Bell Weibull, Bell extended Weibull, Bell-Fisk, Bell-Lomax, Bell Burr-XII, Bell Burr-X, complementary Bell exponential, complementary Bell extended exponential, complementary Bell Weibull, complementary Bell extended Weibull, complementary Bell-Fisk, complementary Bell-Lomax, complementary Bell Burr-XII and complementary Bell Burr-X distribution.
The package also provides hazard function and an estimation procedure.
Benini distribution: provided in r pkg("VGAM").
Bezier-Montenegro-Torres distribution: provided in r pkg("BMT").
Bhattacharjee (normal+uniform) distribution: provided in package r pkg("extraDistr").
Birnbaum-Saunders distribution: provided in r pkg("bsgof"), r pkg("extraDistr"), r pkg("VGAM").
Bridge distribution: provided in r pkg("bridgedist"), as detailed in Wang and Louis (2003). The distribution of random intercept that allows a marginalized random intercept logistic regression to also be logistic regression.
Box Cox distribution: r pkg("gamlss.dist") provides the Box-Cox normal, the Box-Cox power exponential and the Box-Cox t distributions. r pkg("rmutil") provides the Box-Cox normal.
Burr distribution: see Pareto.
Cardioid distribution: provided in r pkg("VGAM") (d,p,q,r) and r pkg("CircStats"), r pkg("circular") (d,r).
Carthwrite's Power-of-Cosine distribution: provided in r pkg("circular") (d,r).
Cauchy distribution: Base R provides the d, p, q, r functions for this distribution (see above). Other implementations are available in r pkg("lmomco", priority = "core") and r pkg("sgt"). The skew Cauchy distribution is provided in r pkg("sn"). r pkg("LaplacesDemon") provides d, p, q, r functions for the Half-Cauchy distribution. The wrapped Cauchy distribution is provided in r pkg("CircStats").
Chen distribution: a special case of the Extended Chen-Poisson Lifetime Distribution as found in r pkg("ecpdist").
Chernoff distribution: r pkg("ChernoffDist") provides d, p, q functions of the distribution of the maximizer of the two-sided Brownian motion minus quadratic drift, known as Chernoff's distribution.
Chi(-squared or not) distribution: Base R provides the d, p, q, r functions for the chi-squared distribution, both central and non-central (see above). Moments, limited expected values and the moment generating function are provided in r pkg("actuar"). r pkg("extraDistr") provides d, p, q, r functions for inverse chi-squared distribution (standard and scaled). Only d,r functions are available for the inverse chi-squared distribution in package r pkg("LaplacesDemon"). A fast random generator is available for the Chi distribution is implemented in r pkg("Runuran") as well as the density function. The non-central Chi distribution is not yet implemented. The chi-bar-squared distribution is implemented in r pkg("emdbook"). r pkg("sadists") implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for sums of non central chi-squared raised to powers distribution and sums of log of non central chi-squared for computing d, p, q, r functions.

Distribution name Packages Functions Distribution suffix Chi-squared stats d, p, q, r chisq Chi-squared actuar m, mgf, lev chisq Chi-squared Runuran d, r chisq Chi-bar-squared emdbook d, p, q, r chibarsq Chi Runuran d, r chi Inverse Chi-squared extraDistr d, p, q, r invchisq Scaled Inverse Chi-squared extraDistr d, p, q, r invchisq Sum of power Chi-squared sadists d, p, q, r sumchisqpow Sum of log Chi-squared sadists d, p, q, r sumlogchisq

: Summary for Chi-related distributions
Circular distribution: uniform circular provided in r pkg("circular") (d,r); Generalized von Mises circular provided in r pkg("circular") (d).
Consul distribution: see r pkg("rmutil").
Continuous binomial distribution: r pkg("cbinom") provides the d/p/q/r functions for a continuous analog to the standard discrete binomial with continuous size parameter and continuous support with x in [0, size + 1].
Dagum distribution: see beta. the power log Dagum provided in r pkg("new.dist").
Davies distribution: The Davies distribution is provided in r pkg("Davies") package.
(non-central) Dunnett's test distribution: no longer provided.
Eta-mu distribution: provided in r pkg("lmomco"). r pkg("sadists") implements Gram Charlier, Edgeworth and Cornish-Fisher 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). r pkg("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. The shifted (or two-parameter exponential) and the truncated exponential distributions are implemented in r pkg("lmomco") and r pkg("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 1-4 distributions are implemented in r pkg("gamlss.dist") and r pkg("lmomco"). The power exponential distribution is also provided in r pkg("normalp"), r pkg("rmutil"), r pkg("LaplacesDemon"). The skew power exponential is provided r pkg("mixSPE"). A fast random generator is available for the power Exponential distribution is implemented in r pkg("Runuran") as well as the density function. r pkg("AEP") implements the Asymmetric Exponential Power Distribution.

Distribution name Packages Functions Distribution suffix Exponential stats d, p, q, r exp Exponential actuar m, mgf, lev 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 Truncated exponential ReIns d, p, q, r texp Power exponential normalp d, p, q, r normp Power exponential Runuran d, r exp Power exponential rmutil d, r powexp Power exponential LaplacesDemon d, p, q, r pe Skew power exp. lmomco d, p, q, r, lm, tlmr aep4 Power and skew power exp. mixSPE r pe, spe Power and skew power exp. gamlss.dist d, p, q, r PE, SEP

: Summary for exponential-related distributions
Externally studentized midrange distribution: Package r pkg("SMR") computes the studentized midrange distribution (d, p, q, r).
Fisher-Snedecor (or F) distribution: Base R provides the d, p, q, r functions for the F distribution, possibly with a non-central parameter. r pkg("sadists") implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for doubly non central Fisher distribution (and product of multiple doubly non central Fisher distribution) for computing d, p, q, r functions. r pkg("flexsurv") provides d, p, q, r functions as well as hazard (h) and integrated hazard rate (i) functions for the generalized F distribution. r pkg("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 r pkg("VGAM"), r pkg("RTDE"), r pkg("ReIns"), r pkg("extraDistr"), r pkg("distributionsrd") and r pkg("evd"). A fast random generator is available for the Frechet distribution is implemented in r pkg("Runuran") as well as the density function. The truncated Frechet distribution is provided in r pkg("ReIns").
Friedman's Chi distribution: provided in r pkg("SuppDists").
Gamma distribution and its extensions: Base R provides the d, p, q, r functions for this distribution (see above). r pkg("EnvStats") provides d, p, q, r functions of the gamma parametrized by the mean and the coefficient of variation. r pkg("actuar") provides d, p, q, r functions of the inverse, the inverse transformed and the log gamma distributions while r pkg("ghyp") provides those functions for the variance gamma distribution. r pkg("extraDistr") and r pkg("LaplacesDemon") provide the inverse gamma distribution. r pkg("CaDENCE") provides the zero-inflated gamma distribution. r pkg("VarianceGamma") provides d, p, q, r functions for the variance gamma distribution as well as moments (skewness, kurtosis, ...). r pkg("VGAM"), r pkg("ggamma") provide d, p, q, r functions of the log gamma and the generalized gamma distribution. The generalized gamma distribution can also be found in r pkg("gamlss.dist"). See Pearson III for a three-parameter gamma distribution with a location parameter. r pkg("flexsurv") provides d, p, q, r functions as well as hazard (h) and integrated hazard rate (i) functions for the generalized gamma distribution. r pkg("coga") provides d, p, r functions for a sum of independent but not identically distributed gamma distributions. r pkg("MCMCpack") provides d, r functions of the Inverse Gamma. r pkg("rmutil") provides the generalized Gamma. r pkg("distTails") provides the full-tail gamma distribution r pkg("sglg") provides the generalized log-Gamma along with various functions to fit semi-parametric regression models. r pkg("ollggamma") provides d, p, q, r for the Odd Log-Logistic Generalized Gamma.

Distribution name Packages Functions Distribution suffix Gamma stats d, p, q, r gamma Gamma actuar m, mgf, lev gamma Gamma EnvStats d, p, q, r gammaAlt zero-inflated Gamma CaDENCE d, p, q, r bgamma Inverse gamma actuar d, p, q, r, m, lev, mgf invgamma Inverse gamma extraDistr d, p, q, r invgamma Inverse gamma LaplacesDemon d, r invgamma Inverse gamma MCMCpack d, r invgamma Log-gamma actuar d, p, q, r, m, lev lgamma Log-gamma 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 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 Generalized gamma rmutil d, p, q, r ggamma Generalized gamma ggamma d, p, q, r ggamma convolution of gamma coga d, p, r coga Full-taill gamma distTails d, p, r dFTG Generalized log-gamma sglg d, p, q, r glg

: Summary for gamma-related distributions
Pólya–Gamma distribution: r function random sampling routines for the distribution are provided by r pkg("BayesLogit"), r pkg("pg"), and r pkg("pgdraw").
Gaussian (or normal) distribution and its extensions: Base R provides the d, p, q, r functions for this distribution (see above). r pkg("actuar") provides the moment generating function and moments. The r pkg("truncnorm") package provides d, p, q, r functions for the truncated gaussian distribution as well as functions for the first two moments. r pkg("EnvStats") provides d, p, q, r functions for the truncated normal distribution and the zero-modified distribution. r pkg("extraDistr") provides the truncated normal. r pkg("LaplacesDemon") provides d, p, q, r functions for the Half-normal distribution. The wrapped normal distribution is provided in r pkg("CircStats"). r pkg("lmomco") implements the generalized normal distribution. The Exponentially modified Gaussian is available in r pkg("emg"), r pkg("gamlss.dist"), r pkg("tsdistributions"), r pkg("sn") implements the skew normal distribution. r pkg("greybox") implements the folded normal distribution. r pkg("VGAM") implements the folded and the skewed normal distribution, and r pkg("csn") provides d, r functions for the closed skew normal distribution. r pkg("NormalLaplace") provides d, p, q, r functions for the sum of a normal and a Laplace random variables, while r pkg("LaplacesDemon") provides d, r functions of the sum of a normal and a Laplace random variables. r pkg("PSDistr") provides d, p, q, r functions of transformations of the normal distribution, such as expnormal and sinh-normal distributions.

Distribution name Packages Functions Distribution suffix Normal stats d, p, q, r norm Normal actuar m, mgf norm Truncated normal truncnorm d, p, q, r, m truncnorm Truncated normal EnvStats d, p, q, r normTrunc Truncated normal extraDistr d, p, q, r tnorm Truncated normal crch d, p, q, r cnorm Generalized normal lmomco d, p, q, r gno Zero modified Gaussian EnvStats d, p, q, r zmnorm Exponentially modified Gaussian emg d, p, q, r emg Exponentially modified Gaussian gamlss.dist d, p, q, r exGAUSS Folded and skew normal gamlss.dist d, p, q, r SN1, SN2 Folded normal greybox d, p, q, r fnorm Closed skew normal csn d, p, q, r csn Skew normal sn d, p, q, r sn Skew normal snorm d, p, q, r tsdistributions

: Summary for Gaussian-related distributions
General error distribution (also known as exponential power distribution): see exponential item.
Generalized extreme value distribution: d, p, q provided in r pkg("lmomco"); d, p, q, r, provided in r pkg("VGAM"), r pkg("evd"), r pkg("evir"), r pkg("FAdist"), r pkg("extraDistr"), r pkg("EnvStats"), r pkg("TLMoments"), r pkg("rmutil"), r pkg("QRM"), r pkg("ROOPSD") and r pkg("fExtremes"). r pkg("revdbayes") provide d, p, q, r functions of the GEV distribution in a Bayesian setting. r pkg("bgev") provide d, p, q, r functions of the bimodal GEV distribution
Gompertz distribution: provided in r pkg("flexsurv"), r pkg("extraDistr"). r pkg("flexsurv") also provides hazard (h) and integrated hazard rate (i) functions. The shifted Gompertz distribution is implemented in r pkg("extraDistr"). The unit-Gompertz is provided in r pkg("ugomquantreg").
Govindarajulu distribution: provided in r pkg("lmomco").
Gumbel distribution: provided in packages r pkg("lmomco"), r pkg("VGAM"), r pkg("gamlss.dist"), r pkg("FAdist"), r pkg("extraDistr"), r pkg("QRM"), r pkg("TLMoments"), r pkg("dgumbel"), r pkg("EnvStats") and r pkg("evd"). r pkg("actuar") provides the raw moments and the moment generating function (mgf) in addition to the d, p, q, r functions. A fast random generator is available for the Gumbel distribution is implemented in r pkg("Runuran") as well as the density function. The reverse Gumbel distribution is implemented in r pkg("lmomco") and r pkg("gamlss.dist"). r pkg("bgumbel") provides the bimodel Gumbel distribution.
Hjorth distribution: provided in r pkg("rmutil").
Huber distribution: Huber's least favourable distribution provided in package r pkg("smoothmest") (d, r), and in r pkg("VGAM"), r pkg("marg"), r pkg("extraDistr") (d, p, q, r).
(generalized) G-and-K, G-and-H distributions: r pkg("gk") provides d, p, q, r functions for the g-and-k and generalized g-and-h distributions which are nonlinear transforms of the Gaussian variables.
(generalized) Hyperbolic distribution: r pkg("fBasics"), r pkg("ghyp"), r pkg("tsdistributions") , r pkg("GeneralizedHyperbolic") and r pkg("HyperbolicDist") packages provide d, p, q, r functions for the generalized hyperbolic distribution. r pkg("QRM") provides d, r functions for the generalized hyperbolic distribution. r pkg("SkewHyperbolic") provides the skewed Hyperbolic Student t-Distribution. r pkg("fBasics") also implements the standardized generalized Hyperbolic distribution. A fast random generator is available for the hyperbolic distribution is implemented in r pkg("Runuran") as well as the density function.
Hyperbolic sine distribution and extension: r pkg("gamlss.dist") provides the sinh and the asinh distributions. Generalized Power Hyperbolic sine distributions are provided in r pkg("FatTailsR").
Inverse Gaussian (also known Wald) distribution: d, p, q, and r functions of the inverse Gaussian are provided in r pkg("statmod"), r pkg("extraDistr"), r pkg("SuppDists"), r pkg("rmutil"). r pkg("LaplacesDemon") provides d, r functions for the inverse Gaussian distribution. r pkg("actuar") provides d, p, q, r, m, lev, mgf functions for the Inverse Gaussian distribution. r pkg("SuppDists") also provides a function that returns moments, skewness, kurtosis. r pkg("fBasics") the normal inverse Gaussian and standardized normal inverse Gaussian distributions. r pkg("tsdistributions") provides the normal inverse Gaussian distribution. The generalized inverse gaussian (GIG) distribution can be found in r pkg("gamlss.dist"), r pkg("ginormal"), r pkg("HyperbolicDist"), r pkg("QRM"), r pkg("rmutil"). The truncated GIG is also available in r pkg("ginormal"). A random generator is available for the (generalized) Inverse Gaussian distribution is implemented in r pkg("Runuran") as well as the density function. r pkg("GIGrvg") generates random variables from the generalized inverse Gaussian distribution. Unit inverse Gaussian provided in r pkg("new.dist").
Johnson distribution: provided in r pkg("SuppDists"), r pkg("ForestFit"), r pkg("tsdistributions") provides d, p of Johnson SB distribution.
Jones and Pewsey distribution: provided in r pkg("circular") (d).
K-prime distribution: r pkg("sadists") implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for K-prime distribution for computing d, p, q, r functions.
Kappa distribution: A 4-parameter Kappa distribution is provided in r pkg("lmomco") and r pkg("FAdist").
Kappa-mu distribution: provided in r pkg("lmomco").
Kato-Jones distribution: provided in r pkg("circular") (d, r).
Kendall's tau distribution: provided in r pkg("SuppDists").
Kiener distribution: a family of distributions generalizing hyperbolic sine distributions (see hyperbolic sine section), d, p, q, r, m provided in r pkg("FatTailsR").
Kruskal Wallis distribution: provided in r pkg("SuppDists").
Kumaraswamy distribution: provided in packages r pkg("VGAM"), r pkg("extraDistr"), r pkg("lmomco"), r pkg("new.dist"). r pkg("elfDistr") provides the Kumaraswamy Complementary Weibull Geometric Probability Distribution.
(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 (Ramberg-Schmeiser or tail-index param), FMKL (Freimer-Mudholkar-Kollia-Lin), FM5 (Five-parameter version of FKML by Gilchrist), GPD (gen. Pareto dist.) and AS (Asymmetry-steepness). The following packages implement such distributions (with d, p, q, r functions): r pkg("gld") (RS, FKML, FM5, GPD), r pkg("Davies") (RS), r pkg("gb") (RS), r pkg("lmomco") (FMKL), r pkg("extraDistr") (original Tukey).
Tukey's G/H distribution: provided in r pkg("TukeyGH77"), and Tukey's H distribution is provided as a special case of Lambert W x F distribution.
Lambda-prime distribution: r pkg("sadists") implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for K-prime distribution for computing d, p, q, r functions.
Lambert W x F distribution: r pkg("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 asymmetric Laplace distribution: provided in r pkg("distr"), r pkg("lmomco"), r pkg("LaplacesDemon"), r pkg("L1pack"), r pkg("VGAM"), r pkg("sgt"), r pkg("extraDistr"), r pkg("greybox"), r pkg("rmutil") and r pkg("HyperbolicDist") packages. r pkg("LaplacesDemon") provides the Laplace distribution parametrized by the precision parameter as well as the skew Laplace distribution. Asymetric Laplace distribution is implemented in r pkg("ald"), r pkg("greybox"). A fast random generator is available for the Laplace distribution is implemented in r pkg("Runuran") as well as the density function. r pkg("smoothmest") implements the density and the random generator. The skew Laplace distribution is available in r pkg("sgt"). r pkg("LaplacesDemon") provides the log-Laplace distribution.
LASSO distribution: provided in r pkg("LaplacesDemon").
Lévy distribution: provided in r pkg("rmutil").
Lindley distribution: provided in r pkg("VGAM"), r pkg("gambin"), r pkg("new.dist").
Linear failure rate distribution: no longer implemented.
Loglog distribution: no longer implemented.
Lomax distribution: see beta.
Logistic distribution and its extensions: Base R provides the d, p, q, r functions for this distribution (see above). r pkg("actuar") and r pkg("VGAM") provide d, p, q, r functions for the log-logistic (also called Fisk), the paralogistic and the inverse paralogistic distributions. r pkg("FAdist") the log-logistic distribution with two and three parameters. r pkg("llogistic") provides the log-logistic parametrized by the median. r pkg("trdist") provides the log-logistic distribution. The generalized logistic distribution (Type I, also known as skew-logistic distribution) is provided in r pkg("lmomco"), r pkg("sld"), r pkg("rmutil"), r pkg("SCI") and r pkg("glogis"). r pkg("GTDL") implements generalized Time-Dependent Logistic distribution.

Distribution name Packages Functions Distribution suffix Logistic stats d, p, q, r logis Logistic actuar m, mgf logis 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 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 Truncated logistic crch d, p, q, r tlogis 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 Generalized logistic rmutil d, p, q, r glogis

: Summary for Logistic-related distributions
Logit-normal distribution: provided in r pkg("logitnorm").
Log-normal distribution and its extensions: The log normal distribution is implemented in Base R (see above) and r pkg("poweRlaw"). The log normal distribution parametrized by its mean and its coefficient of variation is also provided in r pkg("EnvStats"). r pkg("LaplacesDemon") provides the lognormal parametrized by the precision parameter. The truncated lognormal distribution is provided in r pkg("EnvStats") with two possible parametrizations as well as in r pkg("ReIns"). The 3-parameter lognormal distribution is available in r pkg("lmomco"), r pkg("greybox"), r pkg("TLMoments"), r pkg("EnvStats") and r pkg("FAdist"). The package r pkg("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. r pkg("EnvStats") provides d, p, q, r functions for the zero-modified lognormal distribution with two possible parametrizations. r pkg("distributionsrd") provides the double Pareto-lognormal distribution, the left Pareto-lognormal distribution, the truncated lognormal distribution.
Makeham distribution: provided in r pkg("VGAM").
Maxwell distribution: provided in r pkg("VGAM"), r pkg("new.dist").
Minimax distribution: provided in r pkg("minimax").
Mittag-Leffler distribution: d, p, q, r functions provided in r pkg("MittagLeffleR").
Muth distribution: provided in r pkg("new.dist").
Nakagami distribution: provided in r pkg("VGAM").
Neutrosophic: provided in r pkg("ntsDists").
Pareto distribution: d, p, q, r functions are implemented in r pkg("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, r pkg("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, all special cases of the transformed beta II distribution. A fast random generator for the Burr and the Pareto II distribution is implemented in r pkg("Runuran") as well as the density. r pkg("EnvStats") and r pkg("LaplacesDemon") provides d, p, q, r functions for Pareto I distribution. r pkg("extremefit") provides the Burr, the Pareto II, mixture of Pareto I distributions and a composite distribution of two Pareto I distributions. r pkg("lmomco"), r pkg("evd"), r pkg("fExtremes"), r pkg("extraDistr"), r pkg("QRM"), r pkg("Renext"), r pkg("revdbayes"), r pkg("FAdist"), r pkg("LaplacesDemon"), r pkg("TLMoments") r pkg("qrmtools") and r pkg("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. r pkg("ParetoPosStable") implements the Pareto positive stable distribution. The extended Pareto distribution is implemented in r pkg("RTDE") and the shifted truncated (to unit interval) Pareto is implemented in r pkg("mbbefd"). r pkg("ReIns") provides Burr, extended Pareto, generalized Pareto, Pareto 1 distributions and their truncated version. r pkg("CaDENCE") provides the Pareto 2 and the zero-inflated Pareto 2 distribution. r pkg("Pareto") provides the Pareto 1, piecewise Pareto and the generalized Pareto (from actuarial theory). The gamma-Lomax distribution is provided in r pkg("new.dist").

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 I EnvStats d, p, q, r pareto Pareto I extraDistr d, p, q, r pareto Pareto I ReIns d, p, q, r pareto Pareto I LaplacesDemon d, p, q, r pareto Pareto I distributionsrd d, p, q, r pareto Pareto I Pareto d, p, q, r Pareto Trunc. Pareto I ReIns d, p, q, r tpareto 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 extraDistr d, p, q, h lomax Pareto II extremefit d, p, q, h pareto Pareto II Renext d, p, q, r lomax Pareto II rmutil d, p, q, r pareto Pareto II CaDENCE d, p, q, r pareto2 zero-inflated Pareto II CaDENCE d, p, q, r bpareto2 Pareto III VGAM d, p, q, r paretoIII Pareto III actuar d, p, q, r pareto3 Pareto IV VGAM d, p, q, r paretoIV Pareto IV actuar d, p, q, r pareto4 Inverse Pareto actuar d, p, q, r, m, lev invpareto Inverse Pareto distributionsrd d, p, q, r, m, lev invpareto Extended Pareto RTDE d, p, q, r EPD Extended Pareto ReIns d, p, q, r epd Shift. trunc. Pareto mbbefd d, p, q, r, m, ec stpareto Gen. Pareto (actuarial) actuar d, p, q, r, m, lev genpareto Gen. Pareto (actuarial) Pareto d, p, q, r 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 Gen. Pareto (EVT) extraDistr d, p, q, r gpd Gen. Pareto (EVT) QRM d, p, q, r GPD Gen. Pareto (EVT) ReIns d, p, q, r gpd Gen. Pareto (EVT) LaplacesDemon d, r gpd Gen. Pareto (EVT) TLMoments d, p, q, r gpd Trunc. Gen. Pareto (EVT) ReIns d, p, q, r tgpd Gen. Pareto (EVT) revdbayes d, p, q, r gp Gen. Pareto (EVT) Renext d, p, q, r GPD Gen. Pareto (EVT) qrmtools d, p, q, r GPD Gen. Pareto (EVT) ROOPSD d, p, q, r gpd Feller-Pareto actuar d, p, q, r, m, lev fpareto Burr actuar d, p, q, r, m, lev burr Burr extremefit d, p, q, r burr Burr ReIns d, p, q, r burr Burr rmutil d, p, q, r burr Trunc. Burr ReIns d, p, q, r tburr Inverse Burr actuar d, p, q, r, m, lev invburr

: Summary for Pareto-related distributions
Pearson's distribution: Pearson type III available in r pkg("lmomco") and r pkg("FAdist"). A log-Pearson type III distribution is also available in r pkg("FAdist"). r pkg("PearsonDS", priority = "core") 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. r pkg("cpd") provides d, p, q, r for complex bi/triparametric Pearson distributions.
Pearson's Rho distribution: provided in r pkg("SuppDists").
Perks distribution: provided in r pkg("VGAM").
Planck's distribution: a random generator is available in r pkg("Runuran").
Phase-type distribution: provided in r pkg("actuar"), r pkg("mapfit"), r pkg("matrixdist"), r pkg("PhaseTypeR").
Power distribution: r r pkg("poweRlaw")implement the exponential power distribution. Two-sided power distribution provided inr pkg("rmutil")`.
Proportion distribution: this is the distribution for the difference between two independent beta distributions. d, p, q, r functions in r pkg("tolerance").
Omega distribution: provided in r pkg("new.dist").
Quadratic forms and their ratios: r pkg("CompQuadForm") provides several exact and approximate methods to evaluate the distribution function of quadratic forms in normal variables. r pkg("Qapprox") provides fast approximations for the distribution function in nonnegative definite cases. r pkg("QF") provides d, p, q, r for nonnegative definite quadratic forms in normal variables and their ratios where the numerator and denominator are independent, as well as p for ratios of central quadratic forms in the same normal variables. r pkg("qfratio") provides d, p, q, r for the distribution of ratios of potentially noncentral quadratic forms in the same normal variables, as well as moment.
Rayleigh distribution: provided in packages r pkg("VGAM"), r pkg("extraDistr") and r pkg("lmomco"). The slashed generalized Rayleigh distribution provided in r pkg("new.dist"). The two-parameter Rayleigh provided in r pkg("new.dist").
Ram Awadh: provided in r pkg("new.dist").
Response time distribution: r pkg("rtdists") provides d, p, q, r functions for the (Ratcliff) diffusion distribution and for the linear ballistic accumulator (LBA) with different underlying drift-distributions (Normal, Gamma, Frechet, and log-normal).
Rice distribution: provided in r pkg("VGAM") and r pkg("lmomco").
Simplex distribution: provided in r pkg("rmutil").
Singh-Maddala distribution: see beta.
Slash distribution: provided in r pkg("lmomco"), r pkg("extraDistr") and r pkg("VGAM").
Spearman's Rho distribution: provided in r pkg("SuppDists").
Stable distribution: d, p, q, r functions are available in r pkg("fBasics") and r pkg("stabledist"), the functions use the approach of J.P. Nolan for general stable distributions. r pkg("stable") (d, p, q, r, h) is also used for general stable and uses a modified Buck parametrization. r pkg("MixedTS") provides mixed tempered stable distribution (d, p, q, r). r pkg("FMStable") provides (d, p, q) the extremal or maximally skew stable and the finite moment log stable distributions. r pkg("SymTS") provides d, p, q, r functions for symmetric stable, symmetric classical tempered stable, and symmetric power tempered stable distributions. r pkg("TempStable") provides d, p, q, r functions for tempered stable distributions. r pkg("libstable4u") provides d, p, q, r functions for skew stable distributions. r pkg("dstabledist") provides d, p, r functions for skew stable distributions. r pkg("StableEstim") provides fitting functions, characteristic functions, and simulation capabilities for 4-parameter 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). r pkg("extraDistr") and r pkg("LaplacesDemon") provides the Student distribution with location and scale parameters. r pkg("LaplacesDemon") provides d, p, q, r functions for the Half-Student distribution. r pkg("sadists") implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for doubly non central Student distribution for computing d, p, q, r functions. The skewed Student distribution is provided in r pkg("skewt"), r pkg("sn"), r pkg("tsdistributions") and r pkg("gamlss.dist") packages. The generalized skew distribution is provided in r pkg("sgt"). d, p, q, r functions for the generalized t-distribution can be found in r pkg("gamlss.dist"). r pkg("fBasics") provides d, p, q, r functions for the skew and the generalized hyperbolic t-distribution. The L-moments of the Student t (3-parameter) are provided in r pkg("lmomco"). r pkg("crch") provides d, p, q, r functions for the truncated student distribution.

Distribution name Packages Functions Distribution suffix Student stats d, p, q, r t Student with loc. and scal. extraDistr d, p, q, r lst Student with loc. and scal. LaplacesDemon d, p, q, r st Doubly non central St. sadists d, p, q, r dnt Skew Student skewt d, p, q, r skt Skew Student sn d, p, q, r st Skew St. Type 1-5 gamlss.dist d, p, q, r ST1, ST2, ST3, ST4, ST5 Gen. Student gamlss.dist d, p, q, r GT Gen. Hyp. Student fBasics d, p, q, r ght Skew Gen. Student sgt d, p, q, r sgt

: Summary for Student-related distributions
Topp-Leone Cauchy Rayleigh (TLCAR) distribution: provided in r pkg("TLCAR") (d, p, q, r).
Triangle/trapezoidal distribution: packages r pkg("triangle"), r pkg("extraDistr"), r pkg("mc2d"), r pkg("EnvStats") and r pkg("VGAM") provide d, p, q, r functions for the triangle or triangular distribution, while the package r pkg("trapezoid") provides d, p, q, r functions for the Generalized Trapezoidal Distribution. r pkg("CircStats"), r pkg("circular") provide d, r functions for triangular distribution. A fast random generator is available for the triangle distribution is implemented in r pkg("Runuran") as well as the density function.
Tsallis or q-Exponential distribution: r pkg("tsallisqexp") provides d, p, q, r functions for two parametrizations of the Tsallis distribution and also implements a left-censored version.
Tweedie distribution: the Tweedie distribution is implemented in package r pkg("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. r pkg("KScorrect") provides d, p, q, r functions for the log-uniform distribution.
Upsilon distribution: r pkg("sadists") implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for Upsilon distribution for computing d, p, q, r functions.
Vasicek distribution: r pkg("vasicek") implements d, p, r functions. r pkg("vasicekreg") implements d, p, q, r functions.
von Mises distribution: The r pkg("CircStats") package provides d, p, r functions; the r pkg("circular") package provides d, p, q, r functions. r pkg("rvMF") package provides a fast random generator for von Mises Fisher distribution.
Wakeby distribution: A 5-parameter Wakeby is provided in r pkg("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 r pkg("actuar") package and also the moments and the limited expected value for both the raw and the inverse Weibull distribution. r pkg("FAdist") implements the three-parameter Weibull distribution. Furthermore, r pkg("lmomco") implements the Weibull distribution while r pkg("evd") implements the reverse Weibull distribution. The reverse generalized extreme value distribution are provided in r pkg("gamlss.dist") (d, p, q, r) and the shifted left truncated Weibull distribution is provided in r pkg("Renext"). The right truncated Weibull is provided in r pkg("ReIns"). The generalized Weibull is provided in r pkg("rmutil"). The tail Weibull is provided in r pkg("distTails"). r pkg("CaDENCE") provides the zero-inflated Weibull distribution. The bimodal Weibull distribution is provided in r pkg("new.dist").
First-passage time of a Wiener process: r pkg("WienR") provides d, p functions of the first-passage time of a diffusion model.

Continuous multivariate distributions

Bivariate Pareto: r pkg("Bivariate.Pareto") provides a random generator for the bivariate Pareto distribution.
Multivariate beta distribution: r pkg("NonNorMvtDist") provides d, p, q, r, s functions for inverted beta distribution.
Multivariate Burr distribution: r pkg("NonNorMvtDist") provides d, p, q, r, s functions.
Multivariate Cauchy distribution: r pkg("sn") provide d, p, r functions for the multivariate skew Cauchy distribution, while r pkg("LaplacesDemon") provides d, r functions for the multivariate Cauchy distribution parametrized either by sigma, by the Cholesky decomposition of sigma, by the precision matrix omega or by the Cholesky decomposition of omega. r pkg("mcauchyd") provides d, p, r functions of the multivariate Cauchy distribution..
Cook-Johnson's Multivariate Uniform Distribution: r pkg("NonNorMvtDist") provides d, p, q, r, s functions.
Multivariate Dirichlet distribution: r pkg("Compositional"), r pkg("LaplacesDemon"), r pkg("MCMCpack") packages provide d, r functions as well as a fitting function for r pkg("Compositional"). r pkg("compositions"), r pkg("bayesm") provide r function. r pkg("SGB") provides a generalization of the Dirichlet distribution called Simplicial Generalized Beta distribution.
Multivariate exponential distribution: while r pkg("LaplacesDemon") provides d, r functions for the multivariate power exponential distribution parametrized either by sigma, or by the Cholesky decomposition of sigma.
Multivariate F distribution: r pkg("NonNorMvtDist") provides d, p, q, r, s functions.
Multivariate Gaussian (or normal) distribution: The multivariate Gaussian distribution is provided in the packages r pkg("mvtnorm", priority = "core") (d, p, r), r pkg("mnormt", priority = "core") (d, p, r), r pkg("mnorm") (d, p, r), r pkg("mniw") (d, r), r pkg("Compositional") (r), r pkg("compositions") (r). r pkg("pbv") provides d, p functions for bivariate normal distributions. r pkg("symmoments") computes central and non-central moments of the multivariate Gaussian distribution. r pkg("LaplacesDemon") provides d, r functions for the multivariate normal distribution parametrized either by sigma, by the Cholesky decomposition of sigma, by the precision matrix omega or by the Cholesky decomposition of omega. Futhermore, the multivariate truncated normal is implemented in r pkg("TruncatedNormal") for d, p, r functions; r pkg("tmvtnorm") for p, q, r, m(oments) functions; r pkg("tmvmixnorm") for a fast RNG; r pkg("nntmvn") for RNG using SNN method. r pkg("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. r pkg("cmvnorm") implements the complex multivariate normal distribution (d, r). Furthermore, r pkg("condMVNorm") implements d, p, r functions for the conditional multivariate normal distribution. r pkg("condTruncMVN") implements d, p, r functions of the conditional truncated multivariate normal distribution. Finally, r pkg("sn") besides providing facilities for their distribution functions, r pkg("sn") allows the creation of S4 objects which encapsulate these distributions and provide facilities for plotting, summary, marginalization, conditioning, affine transformations of these S4 objects. r pkg("Compositional") provides random generator for the multivariate normal distribution on the simplex and multivariate skew normal distribution on the simplex. A random generator of the multivariate normal is provided in r pkg("MultiRNG"). r pkg("mggd") provides d, r function of the multivariate generalized Gaussian distribution.
Multivariate generalized hyperbolic distribution: r pkg("QRM") provides d, r functions of the standard and the symmetric multivariate generalized hyperbolic distribution. r pkg("ghyp") provides d, p, r functions of the standard multivariate generalized hyperbolic distribution.
Multivariate generalized extreme value distribution: Both bivariate and multivariate Extreme Value distributions as well as order/maxima/minima distributions are implemented in r pkg("evd") (d, p, r).
Multivariate inverse Gaussian distribution: r pkg("mig") provides (d, p, r) functionality as well as a fitting function.
Multivariate Laplace distribution: r pkg("LaplacesDemon") provides d, r functions for the multivariate Laplace distribution parametrized either by sigma, or by the Cholesky decomposition of sigma. r is provided in r pkg("MultiRNG"). r pkg("L1pack") provides d, r functions of the multivariate Laplace distribution.
Multivariate logistic distribution: r pkg("VGAM") package implements the bivariate logistic distribution, while r pkg("NonNorMvtDist") implements the multivariate logistic distribution.
Multivariate lognormal distribution: r pkg("compositions") provides r function.
Multivariate Pareto distribution: r pkg("evd") provides the density for the multivariate generalized Pareto type I. r pkg("NonNorMvtDist") provides d, p, q, r, s functions for multivariate Lomax (type II) distributions and its generalized version. r pkg("NonNorMvtDist") provides d, p, q, r, s functions for Mardia's Multivariate Pareto Type I Distribution
Multivariate Stable distribution: For elliptically contoured (subgaussian stable), r pkg("alphastable") provides d, r functions as well as a fitting function, r pkg("mvgb") provides p function. The multivariate subgaussian stable distribution (d, p, r) is available in r pkg("mvpd").
Multivariate Student distribution: The multivariate Student distribution is provided in the packages r pkg("mvtnorm") (d, r), r pkg("mnormt") (d, p, r), r pkg("Compositional") (r), r pkg("tmvmixnorm") (r), r pkg("QRM") (d, r), r pkg("bayesm") (r), r pkg("MVT") (r). r pkg("TruncatedNormal") for d, p, r functions; r pkg("tmvtnorm") for d, p, q, r functions. r pkg("sn") provides d, p, r functions for the multivariate skew t distribution. r pkg("LaplacesDemon") provides d, r functions for the multivariate Student distribution parametrized either by sigma, by the Cholesky decomposition of sigma, by the precision matrix omega or by the Cholesky decomposition of omega. Random generator r is provided in r pkg("MultiRNG"). A special case of a bivariate noncentral t-distribution called Owen distribution is provided in r pkg("OwenQ"). Distance between multivariate t distributions are provided in r pkg("mstudentd").
Multivariate Uniform distribution: r is provided in r pkg("MultiRNG"). r pkg("compositions") provides a random generator on the simplex.

[Other distributions]{#other}

[Mixed-type distributions]{#mixedType}

Maxwell-Boltzmann-Bose-Einstein-Fermi-Dirac (MBBEFD) distribution : provided in r pkg("mbbefd") and r pkg("MBBEFDLite").
Mixed ordinal and normal distribution: provided in r pkg("OrdNor").
One-inflated distributions: a generic distribution as well as special cases (OI-beta, OI-uniform, OI-GB1, OI-Pareto) are provided in r pkg("mbbefd"). The zero and one inflated beta distribution can be found in r pkg("gamlss.dist").
Zero-modified distributions: r pkg("EnvStats") provides the zero-modified normal distribution and the zero-modified lognormal distribution.

[Mixture of probability laws]{#mixture}

Bernoulli-dist mixture: d, p, q, r functions for Bernoulli-exponential, Bernoulli-Gamma, Bernoulli-lognormal, Bernoulli-Weibull distributions are provided in r pkg("qmap").
Cauchy-polynomial quantile mixture: d, p, q, r functions are provided in r pkg("Lmoments").
Chi-square mixture: d, p, q, r functions are provided in r pkg("emdbook").
Gaussian mixture: Functions d, r are provided in r pkg("mixtools"), r pkg("bmixture") package when dealing with finite mixture models. r pkg("nor1mix"), r pkg("extraDistr"), r pkg("mclust"), r pkg("LaplacesDemon"), r pkg("KScorrect") provides d, p, r functions for Gaussian mixture. r pkg("EnvStats") provides d, p, q, r functions for mixture of two normal distributions. r pkg("bayesm") provides d function for the mixture of multivariate normals.
Gamma Poisson: provided in r pkg("extraDistr").
Gamma mixture: Ga r pkg("GSM") package provides d, p, r, r pkg("bmixture") provides d, r, r pkg("evmix") provides d, p, q, r.
Generic mixtures: there is an implementation via S4-class UnivarMixingDistribution in package r pkg("distr"). r pkg("gendist") provides d, p, q, r functions for two-distribution mixture models working with any distribution defined by its d, p, q, r functions. r pkg("fmx") provides d, p, q, r functions for finite parametrized distributions.
Horseshoe distribution: provided in r pkg("LaplacesDemon").
Laplace mixture distribution: provided in r pkg("LaplacesDemon").
Left-truncated mixtures of Gamma, Weibull, and Lognormal distributions: provided in r pkg("ltmix").
Log normal mixture: d, p, q, r functions are provided in r pkg("EnvStats") with two possible parametrizations.
Normal-polynomial quantile mixture: d, p, q, r functions are provided in r pkg("Lmoments").
Pareto distribution: r pkg("extremefit") implements the mixture of two Pareto I distributions.
Poisson beta distribution: provided in r pkg("scModels").
Poisson Binomial distribution: r pkg("poibin") implements the Poisson Binomial distribution.
Poisson lognormal distribution: r pkg("poilog") implements the Poisson lognormal distribution.
Poisson mixture: provided in r pkg("extraDistr").
Poisson-Tweedie exponential family models: provided in r pkg("poistweedie").
Student mixture: The r pkg("AdMit") package provides d, r functions for Student mixtures in the context of Adaptive Mixture of Student-t distributions. r pkg("bmixture") package also provide d, r functions for mixture of Student-t distributions.
von Mises Fisher (or Langevin) mixture: The r pkg("movMF") and r pkg("CircStats") packages provide d, r functions for finite von Mises Fisher mixtures.

[Random matrices]{#matrix}

Huang-Wan distribution: provided in r pkg("LaplacesDemon").
Inverse matrix gamma distribution: provided in r pkg("LaplacesDemon").
Inverse Wishart distribution: r pkg("LaplacesDemon") provides inverse Wishart distribution parametrized either by Sigma or by its Cholesky decomposition. r pkg("LaplacesDemon") provides the scaled inverse Wishart distribution. r pkg("MCMCpack") and r pkg("mniw") provides the inverse Wishart distribution. r pkg("wishmom") allows to computes the theoretical moments of the inverse beta-Wishart distribution.
Marcenko-Pastur distribution: provided in r pkg("RMTstat"), r pkg("MCMCpack") and r pkg("bayesm").
Matrix gamma distribution: provided in r pkg("LaplacesDemon").
Matrix normal distribution: r pkg("MBSP") (r) provides a random generator using a Cholesky decomposition; r pkg("matrixsampling") (r) provides a random generator using a spectral decomposition; r pkg("LaplacesDemon") and r pkg("mniw") (d, r); r pkg("matrixNormal") (d, p, r) collects these forms in one place and allows users to be flexible in simulating random variates (Cholesky, spectral, SVD).
Matrix student distribution: provided in r pkg("mniw").
Normal Inverse Wishart distribution: provided in r pkg("LaplacesDemon"), r pkg("mniw").
Normal Wishart distribution: provided in r pkg("LaplacesDemon").
Tracy-Widom distribution: provided in r pkg("RMTstat"), r pkg("MCMCpack") and r pkg("bayesm"): supported beta values are 1 (Gaussian Orthogonal Ensemble), 2 (Gaussian Unitary Ensemble), and 4 (Gaussian Symplectic Ensemble).
Sparse matrix: r pkg("spam") provides functionalities to draw random numbers from a user-supplied RNG (e.g. rexp) or from a multivariate normal distribution for large sparse matrices: typically for sparse covariance matrices.
Spiked Wishart Maximum Eigenvalue Distribution: provided in r pkg("RMTstat"), r pkg("MCMCpack") and r pkg("bayesm").
Wishart distributions: Base R provides the r function for the Wishart distribution. r pkg("MCMCpack"), r pkg("RMTstat"), r pkg("bayesm"), r pkg("mniw") provides d, r functions, r pkg("bayesm") provides r function. r pkg("LaplacesDemon") provides Wishart distribution parametrized either by Sigma or by its Cholesky decomposition. r pkg("wishmom") allows to computes the theoretical moments of the beta-Wishart distribution.
White Wishart Maximum Eigenvalue Distribution: provided in r pkg("RMTstat"), r pkg("MCMCpack") and r pkg("bayesm").
Yang-Berger distribution: provided in r pkg("LaplacesDemon").
Zellner distribution: provided in r pkg("LaplacesDemon").

Copulas

Unified approaches: The packages r pkg("fCopulae", priority = "core"), r pkg("copula", priority = "core"), and r pkg("copBasic") provide a lot of general functionality for copulas. Although lacking support for many existing copulas themselves, r pkg("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: r pkg("gumbel") is a standalone package for the Gumbel copula r pkg("fCopulae") implements the 22 Archimedean copulas of Nelsen (1998, Introduction to Copulas , Springer-Verlag) including Gumbel, Frank, Clayton, and Ali-Mikhail-Haq. r pkg("VGAM") provides Ali-Mikhail-Haq, Clayton, Frank, Frechet copulas. r pkg("copula") provides Ali-Mikhail-Haq, Clayton, Frank, Gumbel and Joe copulas. The Frank bivariate distribution is available in r pkg("RTDE"). r pkg("VineCopula") provides Clayton, Gumbel, Frank, Joe, BB1, BB6, BB7 and BB8 copulas. Nested Archimedean copulas are available in the r pkg("HAC") package. r pkg("copBasic") provides functions for Ali-Mikhail-Haq, Clayton, Frechet copulas. r pkg("QRM") provides pdf and random generator for Clayton, Gumbel, Frank, BB9 copula. r pkg("Bivariate.Pareto") provides a random generator for the Frank copula with Pareto margins. r pkg("HAC") provides hierarchical archimedean copulas. r pkg("lcopula") provides the Liouville copula. r pkg("CopulaGAMM") provides the bivariate version of Frank, FGM, Galambos, Gumbel, Huesler-Reiss, Joe, MTCJ, Plackett copulas.
Blomqvist copula: provided in r pkg("copBasic").
Composition of copula: r pkg("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 r pkg("copBasic"), r pkg("copent"), r pkg("HAC"). r pkg("GenOrd") provides sampling function for multivariate discrete random vectors with a specified correlation matrix.
Elliptical copulas: Gaussian, Student and Cauchy copulas are implemented in r pkg("fCopulae") for the bivariate cases. r pkg("copula"), r pkg("VGAM"), r pkg("VineCopula") provide the Gaussian and the Student copulas. r pkg("QRM") provides pdf and random generator for Gaussian, Student copulas. r pkg("relliptical") provides a random generator for multivariate truncated Normal, Student-t, Power Exponential, Pearson VII, Slash and Contaminated Normal distributions. r pkg("CopulaGAMM") provides the bivariate Gaussian and student copula.
Extreme value copulas: r pkg("fCopulae") provides the following copulas Gumbel, Galambos, Husler-Reiss, Tawn, or BB5. r pkg("copula") implements Gumbel, Galambos and Husler-Reiss.
Eyraud-Farlie-Gumbel-Morgenstern copula: provided in r pkg("VGAM"), r pkg("RTDE"), and r pkg("copula").
Integrated gamma copula: provided in r pkg("igcop").
Mardia copula: Not yet implemented?
Nested copulas: arbitrary nested versions of copulas can be implemented in r pkg("copula").
Plackett: provided in r pkg("VGAM"), r pkg("copBasic") and r pkg("copula").
Vine copulas: Package r pkg("vines") provides functions for C- and D-vine copulas and r pkg("VineCopula") for general R-vine copulas.

[Compound, composite, discretized, exponentiated and transformation of distributions]{#transform}

Absolute value or half distribution: Half-Cauchy, half normal and half-student are implemented both in r pkg("extraDistr") and in r pkg("LaplacesDemon").
Composite distribution also known as spliced distribution: Split-normal (also known as the two-piece normal distribution) not yet implemented. Split-student provided in package r pkg("dng"). r pkg("evmix") provides d, p, q, r of the following composite distributions: gamma-GPD, lognormal GPD, normal-GPD, Weibull-GPD as well as bulk models such as GPD-normal-GPD distribution. r pkg("gendist") provides d, p, q, r functions for composite models working with any distribution defined by its d, p, q, r functions.
Compound distribution: r pkg("kdist") provides d, p, q, r functions of the K distribution.
Discretized distribution: r pkg("distcrete") allows discretised versions of continuous distribution by mapping continuous values to an underlying discrete grid, based on a (uniform) frequency of discretisation, a valid discretisation point, and an integration range.
Quantile-based asymmetric (QBA) family of distributions: no longer implemented.
Transformed distribution: r pkg("Newdistns") provides G-transformed distributions for a selected number of distributions 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, truncated-exponential skew-symmetric G distribution, modified beta G distribution, and exponentiated exponential Poisson G distribution. r pkg("MPS") provides also G-transformed distributions, such as beta exponential G distribution, beta G distribution, exponentiated exponential Poisson G distribution, exponentiated G distribution, exponentiated generalized G distribution, exponentiated Kumaraswamy G distribution, gamma uniform G distribution, gamma uniform type I/II G distribution, generalized beta G distribution, geometric exponential Poisson G distribution, gamma-X family of modified beta exponential G distribution, exponentiated exponential Poisson G distribution, gamma-X generated of log-logistic-X familiy of G distribution, Kumaraswamy G distribution, log gamma G type I/II distribution, modified beta G distribution, Marshall-Olkin Kumaraswamy G distribution, odd log-logistic G distribution, truncated-exponential skew-symmetric G distribution, T-X{log-logistic}G distribution, Weibull G distribution. r pkg("gendist") provides d, p, q, r functions for composite models, folded models, skewed symmetric models and arctan models working with any distribution defined by its d, p, q, r functions. r pkg("ComRiskModel") provides also G-transformed such as binomial-G, complementary negative binomial-G and complementary geometric-G families of distributions taking baseline models such as exponential, extended exponential, Weibull, extended Weibull, Fisk, Lomax, Burr-XII and Burr-X. r pkg("geppe") provides exponential-Poisson (EP), the generalised EP (GEP) and the Poisson-exponential (PE) distributions.
Truncated distribution: A generic code snippet is available in the JSS . This code is now available in two packages: r pkg("truncdist"), r pkg("trdist") are dedicated packages providing d, p, q, r, m(oments) functions of a univariate truncated distribution for a base distribution and a user-supplied distribution; r pkg("LaplacesDemon") provides a generic function in a Bayesian environment. r pkg("TruncExpFam") provides d, r functions for truncated distributions of the exponential family, e.g. truncated gamma or truncated Poisson as well as fitting procedures. It also provides functions to retrieve the original distribution parameters from a truncated sample by maximum-likelihood estimation.

[Moments, skewness, kurtosis and etc]{#moments}

Empirical mean, standard deviation and variance: base R provides mean(), sd(), var() functions to compute the mean, standard deviation and variance, respectively.
Empirical skewness: available in r pkg("agricolae"), r pkg("e1071"), r pkg("GLDEX"), r pkg("HyperbolicDist"), r pkg("modeest"), r pkg("moments"), r pkg("s20x"), r pkg("fromo"), r pkg("DistributionUtils"), r pkg("EnvStats"), r pkg("parameters") packages.
Empirical kurtosis: available in r pkg("agricolae"), r pkg("DistributionUtils"), r pkg("e1071"), r pkg("EnvStats"), r pkg("GLDEX"), r pkg("HyperbolicDist"), r pkg("fromo"), r pkg("moments"), r pkg("parameters") packages. The raw or centered moments are provided in r pkg("e1071"), r pkg("moments").
Empirical L-moments: L-moments are available in r pkg("lmom"), r pkg("lmomco"), r pkg("Lmoments"), r pkg("GLDEX"), r pkg("EnvStats"), trimmed L-moments are available in r pkg("lmomco"), r pkg("TLMoments") and r pkg("Lmoments"), right-censored L-moments are available in r pkg("lmomco"), and cumulants in r pkg("GLDEX"). r pkg("TLMoments") provides a function to convert them to some distribution parameters.
Empirical probability weighted moments: Probability weighted moments are available in r pkg("EnvStats") and r pkg("fromo").
Empirical cumulants: r pkg("fromo") provides centered and standardized cumulants.
Mode estimation: Package r pkg("modeest") provides mode computation of known distributions and mode estimation on datasets in the unimodal case. Package r pkg("ModEstM") provides mode estimation in unimodal and multimodal cases. Package r pkg("multimode") provides for testing and exploring the number of modes on data using non-parametric procedures.
Order statistics: Distribution function of the jth order statistic can be obtained with base R functions. r pkg("orders") allows to generate samples of k-th order statistics and others quantities of interest for the following distributions: Burr, Feller-Pareto, Generalized Pareto, The Inverse Paralogistic, Marshall-Olkin G, exponentiated G, beta G, gamma G, Kumaraswamy G, generalized beta G, beta extended G, gamma G, gamma uniform G, beta exponential G, Weibull G, log gamma G I/II, exponentiated generalized G, exponentiated Kumaraswamy G, geometric exponential Poisson G, truncated-exponential skew-symmetric G, modified beta G, exponentiated exponential Poisson G, Poisson-inverse gaussian, Skew normal type 1, Skew student t, Sinh-Arcsinh, Sichel, Zero inflated Poisson.
Empirical characteristic function: r pkg("empichar") evaluates the empirical characteristic function of univariate and multivariate samples.
Dispersion index: Package r pkg("GWI") provides univariate dispersion index against a particular distribution.
Theoretical moments:
- common distributions: The r pkg("actuar") package implements raw moments, limited expected values and moment generating function for base R distributions. r pkg("lmomco") provides L-moments (L), trimmed L-moments (TL), and right-censored [RC] for the following distributions: Asymmetric Exponential Power (L), Cauchy (TL), Eta-Mu (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), Kappa-Mu (L), Kumaraswamy (L), Laplace (L), Normal (L), 3-parameter log-Normal (L), Pearson Type III (L), Rayleigh (L), Reverse Gumbel (L[RC]), Rice/Rician (L), Slash (TL), 3-parameter Student T (L), Truncated Exponential (L), Wakeby (L), and Weibull (L). Multivariate L-moments (L-comoments). r pkg("Distributacalcul") provides first few moments for most common distributions.
- hyperbolic distributions: r pkg("HyperbolicDist") provides the mean, variance, skewness, kurtosis, mode, raw and centered moments for the hyperbolic, the generalized hyperbolic and the generalized inverse Gaussian distributions.
- Lambda distribution: r pkg("GLDEX") also provides the mean, variance, skewness, kurtosis of generalized Lambda distribution.
- multivariate distributions: r pkg("MomTrunc") provides mean vector, covariance matrices and raw moments for truncated or folded of the following multivariate distributions: normal, skew normal, extended skew normal and student.

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 Mersenne-Twister algorithm. Other generators include Wichmann-Hill, Marsaglia-Multicarry, Super-Duper, Knuth-TAOCP, Knuth-TAOCP-2002, as well as user-supplied RNGs. For normal random numbers, the following algorithms are available: Kinderman-Ramage, Ahrens-Dieter, Box-Muller, Inversion (default). In addition to the tools above, r pkg("setRNG") provides an easy way to set, retain information about the setting, and reset the RNG.
Pseudo-randomness: r pkg("RDieHarder") offers several dozen new RNGs from the GNU GSL. r pkg("randtoolbox") provides more recent RNGs such as SF Mersenne-Twister and WELL, which are generators of Mersenne Twister type, but with improved quality parameters. r pkg("SuppDists") implements two RNGs of G. Marsaglia. r pkg("dqrng") provides PCG family by O'Neill (2014) as well as Xoroshiro128+ and Xoshiro256+ by Blackman and Vigna (2018).
- For non-uniform generation, the r pkg("Runuran") package interfaces to the UNU.RAN library for universal non-uniform generation as well as customised distributions based on polynomial interpolation of the inverse cumulative distribution function. r pkg("rust") performs non-uniform random variate generation from unimodal (low-dimensional) multivariate continuous distributions, using the generalized ratio-of-uniforms method. r pkg("UnivRNG") provides 17 non-uniform generators either using an acceptance/rejection algorithm or the inverse CDF method. r pkg("MultiRNG") provides 11 multivariate generators, see each distribution. r pkg("Tinflex") provides a non-uniform random number generator for quite arbitrary distributions with piecewise twice differentiable densities.
- r pkg("kernelboot") provides functions for random generation from univariate and multivariate kernel densities (in particular multivariate Gaussian kernels).
Quasi-randomness: The r pkg("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). r pkg("lhs") and r pkg("mc2d") packages implement the latin hypercube sampling, an hybrid quasi/pseudo random method. r pkg("sfsmisc") also provides the Halton sequence. r pkg("qrng") provides Korobov, generalize Halton and Sobol quasi-random sequences. r pkg("spacefillr") provides Halton and Sobol sequences.
True randomness: The r pkg("random") package provides several functions that access the true random number service at random.org .
RNG tests: r pkg("RDieHarder") offers numerous tests of RNGs based on a reimplementation and extension of Marsaglia's DieHarder battery. r pkg("randtoolbox") provides basic RNG tests.
Parallel computing: Support for several independent streams:
- r pkg("rstream") focuses on multiple independent streams of random numbers from different sources (in an object oriented approach).
- r pkg("dqrng") provides RNG for parallel computation either in R or in C++.
- r pkg("rlecuyer") provides an interface to the C implementation of the random number generator with multiple independent streams.
- See the r view("HighPerformanceComputing") task view for more details.
Multivariate random vectors: for parametric multivariate distributions, we refer to Multivariate Continuous and Multivariate Discrete. For non-parametric distributions, r pkg("SimJoint") offers various to simulate multivariate distributions with non-parametric marginals given a Pearson or Spearman correlation matrix.
Unit sphere and other: r pkg("simdd") provides a generator for the Fisher Bingham distribution on the unit sphere, the matrix Bingham distribution on a Grassmann manifold, the matrix Fisher distribution on SO(3), and the bivariate von Mises sin model on the torus. r pkg("uniformly") provides sampling on various geometric shapes, such as spheres, ellipsoids, simplices. r pkg("watson") allows simulating mixtures of Watson distributions.
Tidyverse: r pkg("TidyDensity") maps the RNG of stats (and actuar) distributions to a tidy tibble which allows to work with the rest of the tidyverse.

Miscellaneous

Computation/benchmark:
- Approximation of d, p, q, r functions: r pkg("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 Cornish-Fisher approximations. r pkg("sadists") is a showcase for PDQutils, providing density, cumulative distribution, quantile, and random generation for the doubly non-central t, doubly non-central F, K-prime, Lambda-prime, Upsilon, and sum of (non-central) chi-squares to powers distributions. Various approximations and alternative computations for d, p, q functions of probability distributions in R are given r pkg("DPQ").
- r pkg("benchden") implements the 28 distributions introduced as kernel benchmarks for nonparametric density estimation by Berlinet and Devroye (1994): includes d, p, q, r functions as well as additional information on features of the densities.
- For non-uniform generation, see the r pkg("Runuran") above.
Non parametric models:
- Binned Empirical distributions: no longer provided.
- 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 with or without replacement (see sample()). r pkg("distributionsrd") provides d, p, q, r user-friendly functions for the empirical distributions as well as moments. r pkg("distfromq") provides d, p, q, r user-friendly functions for the empirical distributions and options for estimating the tails. r pkg("mded") provides a function for measuring the difference between two independent or non-independent empirical distributions and returning a significance level of the difference.
  r pkg("MEPDF") provides functions to compute and visualize empirical density functions for multivariate data.
- Non Parametric distributions : r pkg("spd") provides the Semi Parametric Piecewise Distribution, while r pkg("fBasics") implements spline smoothed distributions.
Hierarchical models: Distributions whose some parameters are no longer constant but random according to a particular distribution. r pkg("VGAM") provides a lot of hierarchical models: beta/binomial, beta/geometric and beta/normal distributions. r pkg("bayesm") implements: binary logit, linear, multivariate logit and negative binomial models. Furthermore r pkg("LearnBayes") and r pkg("MCMCpack") provides poisson/gamma, beta/binomial, normal/normal and multinomial/Dirichlet models.
Unified interface to handle distributions:
- S3 Object-orientation: r pkg("distributions3") provides tools to create and to manipulate probability distributions using S3, that is r pkg("distributions3"), generics random(), pdf(), cdf() and quantile() provide replacements for base R's r/d/p/q style functions. r pkg("distributional") also provides tools to create and to manipulate probability distributions using S3, with cdf(), density(), hdr(), mean(), median(), quantile(),...
- S4 Object-orientation: General discrete and continuous distributions are implemented in package r pkg("distr") respectively via S4-class DiscreteDistribution and AbscontDistribution providing the classic d, p, q and r functions. r pkg("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 r pkg("distrMod") provides functions for the computation of minimum criterion estimators (maximum likelihood and minimum distance estimators). See other packages of the distr-family (r pkg("distrSim"), r pkg("distrTEst"), r pkg("distrTeach"), r pkg("distrDoc"), r pkg("distrEllipse")).
- R6 Object-orientation: r pkg("ROOPSD") provides a R6 class interface to classic statistical distribution.
- Transformation: Lebesgue decomposition are implemented in r pkg("distr"), as well as Convolution, Truncation and Huberization of distributions. Furthermore, r pkg("distr") provides distribution of the maximum or minimum of two distributions. See Object-orientation above. r pkg("convdistr") provides functions to convolute probabilistic distributions using RNG for a set of distributions.
Transversal functions:
- Histogram, tail plots, distance estimation: r pkg("DistributionUtils") provides log-histogram, tail plots, functions for testing distributions using inversion tests and the Massart inequality. r pkg("visualize") provides functions to plot the pdf or pmf with highlights on area or when probability is present in user defined locations, as well as the graph is the mean and variance of the distribution. r pkg("visualize") provides lower tail, bounded, upper tail, and two tail calculations. r pkg("visualize") contains convenience functions for constructing and plotting bivariate probability distributions (probability mass functions, probability density functions and cumulative distribution functions). r pkg("vistributions") provides visualization tools for a selected number of distributions.
- Parameter estimation: r pkg("lmomco") and r pkg("Lmoments") focus on univariate/multivariate (L-)moments estimation. r pkg("VGAM") provides a lot of parameter estimation for usual and "exotic" distributions. r pkg("gaussDiff") provides a collection difference measures for multivariate Gaussian probability density functions Package r pkg("MASS") implements the flexible fitdistr function for parameter estimations. r pkg("fitdistrplus") greatly enlarges fitdistr and enhances the tools to fit a user-supplied probability distribution. r pkg("OneStep") is based upon r pkg("fitdistrplus") to provide one-step estimation procedures. r pkg("EnvStats"), r pkg("fitteR"), r pkg("ExtDist"), r pkg("MLE") also provide tools to fit and select a set of probability distributions. r pkg("flexsurv") and r pkg("msm") provides a quantile function for a generic distribution based on numerical computation based on a dichotomic search. r pkg("reservr") provides fitting procedures for censored and truncated dataset on a set of selected distributions.

Bibliography

General books

N. L. Johnson, S. Kotz, N. Balakrishnan (1994). Continuous univariate distributions, Volume 1, Wiley.
N. L. Johnson, S. Kotz, N. Balakrishnan (1995). Continuous univariate distributions, Volume 2, Wiley.
N. L. Johnson, S. Kotz, N. Balakrishnan (1997). Discrete multivariate distributions, Wiley.
N. L. Johnson, A. W. Kemp, S. Kotz (2008). Univariate discrete distributions, Wiley. r doi("10.1002/0471715816")
S. Kotz, N. Balakrishnan, N. L. Johnson (2000). Continuous multivariate distributions Volume 1, Wiley.
G. Wimmer (1999), Thesaurus of univariate discrete probability distributions.

Books dedicated to a distribution family

M. Ahsanullah, B.M. Golam Kibria, M. Shakil (2014). Normal and Student's t Distributions and Their Applications, Springer. r doi("10.2991/978-94-6239-061-4")
B. C. Arnold (2010). Pareto Distributions, Chapman and Hall. r doi("10.1201/b18141")
A. Azzalini (2013). The Skew-Normal and Related Families. r doi("10.1017/CBO9781139248891")
N. Balakrishnan (2014). Handbook of the Logistic Distribution, CRC Press. r doi("10.1201/9781482277098 ")

Books with applications

C. Forbes, M. Evans, N. Hastings, B. Peacock (2011). Statistical Distributions, Wiley. r doi("10.1002/9780470627242")
K. Krishnamoorthy (2015). Handbook of Statistical Distributions with Applications, Chapman and Hall. r doi("10.1201/b19191")
Z. A. Karian, E. J. Dudewicz, K. Shimizu (2010). Handbook of Fitting Statistical Distributions with R, CRC Press. r doi("10.1201/b10159-3")

Links

Clickable diagram of distribution relationships
Compendium of distributions.
Comprehensive list of data types
Diagram of discrete distribution relationships
Diagram of continuous distribution relationships
Journal of Statistical Software: R programs for truncated distributions
List and diagram of distribution relationship.

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Distributions.md

Distributions.md

Table of contents

[Base functionality]{#base}

Discrete distributions

Discrete univariate distributions

Discrete multivariate distributions

Continuous distributions

Continuous univariate distributions

Continuous multivariate distributions

[Other distributions]{#other}

[Mixed-type distributions]{#mixedType}

[Mixture of probability laws]{#mixture}

[Random matrices]{#matrix}

Copulas

[Compound, composite, discretized, exponentiated and transformation of distributions]{#transform}

[Moments, skewness, kurtosis and etc]{#moments}

Random number generators

Miscellaneous

Bibliography

General books

Books dedicated to a distribution family

Books with applications

Links

Files

Distributions.md

Latest commit

History

Distributions.md

File metadata and controls

Table of contents

[Base functionality]{#base}

Discrete distributions

Discrete univariate distributions

Discrete multivariate distributions

Continuous distributions

Continuous univariate distributions

Continuous multivariate distributions

[Other distributions]{#other}

[Mixed-type distributions]{#mixedType}

[Mixture of probability laws]{#mixture}

[Random matrices]{#matrix}

Copulas

[Compound, composite, discretized, exponentiated and transformation of distributions]{#transform}

[Moments, skewness, kurtosis and etc]{#moments}

Random number generators

Miscellaneous

Bibliography

General books

Books dedicated to a distribution family

Books with applications

Links