# Univariate Discrete Distributions

In statistics, a univariate distribution is a probability distribution of only one random variable. This is in contrast to a multivariate distribution, the probability distribution of a random vector (consisting of multiple random variables).

## Univariate Discrete Distributions

One of the simplest examples of a discrete univariate distribution is the discrete uniform distribution, where all elements of a finite set are equally likely. It is the probability model for the outcomes of tossing a fair coin, rolling a fair die, etc. The univariate continuous uniform distribution on an interval [a, b] has the property that all sub-intervals of the same length are equally likely.

Other examples of discrete univariate distributions include the binomial, geometric, negative binomial, and Poisson distributions.[1] At least 750 univariate discrete distributions have been reported in the literature.[2]

Examples of commonly applied continuous univariate distributions[3] include the normal distribution, Student's t distribution, chisquare distribution, F distribution, exponential and gamma distributions.

The Third Edition of the critically acclaimed Univariate Discrete Distributions provides a self-contained, systematic treatment of the theory, derivation, and application of probability distributions for count data. Generalized zeta-function and q-series distributions have been added and are covered in detail. New families of distributions, including Lagrangian-type distributions, are integrated into this thoroughly revised and updated text. Additional applications of univariate discrete distributions are explored to demonstrate the flexibility of this powerful method.

A thorough survey of recent statistical literature draws attention to many new distributions and results for the classical distributions. Approximately 450 new references along with several new sections are introduced to reflect the current literature and knowledge of discrete distributions.

Emphasis continues to be placed on the increasing relevance of Bayesian inference to discrete distribution, especially with regard to the binomial and Poisson distributions. New derivations of discrete distributions via stochastic processes and random walks are introduced without unnecessarily complex discussions of stochastic processes. Throughout the Third Edition, extensive information has been added to reflect the new role of computer-based applications.

Note: params are defined for all univariate distributions, while other parameter retrieval methods are only defined for those distributions for which these parameters make sense. See below for details.

The set of all possible outcomes from an experiment or sampling scheme is called the sample space, Î©. In univariate situations a single real value is associated with every outcome. The function Xthat determines these numerical values is the random variable and...

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.

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. Ecume provides non-parametric two-sample (or k-sample) distribution comparisons in the univariate or multivariate case allowing observation weights and thresholds.

Binomial (including Bernoulli) distribution: provided in stats . Zero-modified, zero-inflated, truncated versions are provided in gamlss.dist, extraDistr, actuar and in VGAM. LaplacesDemon provides dedicated functions for the Bernoulli distribution. rmutil provides the double binomial and the multiplicative binomial distributions.

Discrete Laplace distribution: The discrete Laplace distribution is provided in extraDistr (d, p, r). The skew discrete Laplace distribution has two parametrization (DSL and ADSL), both provided in DiscreteLaplace and DSL in disclap. LaplacesDemon also provides the DSL parametrization only.

Poisson distribution: provided in stats and in poweRlaw. Zero-modified, zero-inflated, truncated versions are provided in extraDistr, gamlss.dist, actuar and in VGAM. extraDistr provides the truncated Poisson distribution. LaplacesDemon provides the generalized Poisson distribution. rmutil provides the double Poisson, the multiplicative Poisson and the Power variance function Poisson distributions. poibin and PoissonBinomial provide the Poisson binomial distribution. See the mixture section such as the Poisson-lognormal mixture.

Negative binomial distribution: provided in stats . Zero-modified, zero-inflated, truncated versions are provided in gamlss.dist, extraDistr, emdbook, actuar and in VGAM. New parametrization of the negative binomial distribution is available in RMKdiscrete.

Zipf distribution and extensions: d, p, q, r functions of the Zipf and the Zipf-Mandelbrot distributions are provided in tolerance, VGAM. Package zipfR provides tools for distribution of word frequency, such as the Zipf distribution. zipfextR provides three extensions of the Zipf distribution: the Marshall-Olkin Extended Zipf, the Zipf-Poisson Extreme and the Zipf-Poisson Stopped Sum distributions.

Beta distribution and its extensions: Base R provides the d, p, q, r functions for this distribution (see above). extraDistr provides the beta distribution parametrized by the mean and the precision. actuar provides moments and limited expected values. sadists implements Gram Charlier, Edgeworth and Cornish-Fisher approximations for doubly non central beta distribution for computing d, p, q, r functions. 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 gamlss.dist, mbbefd, actuar. 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 VGAM, extraDistr, LaplacesDemon and mc2d. The zero and one inflated beta distribution can be found 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, mc2d: Lomax, inverse Lomax, Dagum, Singh-Maddala, Pert distributions. actuar provides the Feller-Pareto distribution as special cases Burr, loglogistic, paralogistic, generalized Pareto, Pareto, see also the Pareto subsection. llogistic provides the log-logistic parametrized by the median.

Continuous binomial distribution: 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].

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. The shifted (or two-parameter 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 1-4 distributions are implemented in gamlss.dist and lmomco. The power exponential distribution is also provided in normalp, rmutil, LaplacesDemon. The skew power exponential is provided mixSPE. A fast random generator is available for the power Exponential distribution is implemented in Runuran as well as the density function.

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

Gamma distribution and its extensions: Base R provides the d, p, q, r functions for this distribution (see above). EnvStats provides d, p, q, r functions of the gamma parametrized by the mean and the coefficient of variation. actuar provides d, p, q, r functions of the inverse, the inverse transformed and the log gamma distributions while ghyp provides those functions for the variance gamma distribution. extraDistr and LaplacesDemon provide the inverse gamma distribution. CaDENCE provides the zero-inflated gamma distribution. VarianceGamma provides d, p, q, r functions for the variance gamma distribution as well as moments (skewness, kurtosis, ...). VGAM, 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 gamlss.dist. See Pearson III for a three-parameter 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. coga provides d, p, r functions for a sum of independent but not identically distributed gamma distributions. MCMCpack provides d, r functions of the Inverse Gamma. rmutil provides the generalized Gamma. distTails provides the full-tail gamma distribution sglg provides the generalized log-Gamma along with various functions to fit semi-parametric regression models.

Inverse Gaussian (also known Wald) distribution: d, p, q, and r functions of the inverse Gaussian are provided in statmod, extraDistr, SuppDists, rmutil. LaplacesDemon provides d, r functions for the inverse Gaussian distribution. actuar provides d, p, q, r, m, lev, mgf functions for the Inverse Gaussian distribution. SuppDists also provides a function that returns moments, skewness, kurtosis. fBasics the normal inverse Gaussian and standardized normal inverse Gaussian distributions. The generalized inverse gaussian distribution can be found in gamlss.dist, QRM, rmutil, 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.