Many

probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

s that are important in theory or applications have been given specific names.

Discrete distributions

With finite support

*The

Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with pro ...

, which takes value 1 with probability ''p'' and value 0 with probability ''q'' = 1 − ''p''. *The Rademacher distribution, which takes value 1 with probability 1/2 and value −1 with probability 1/2. *The

binomial distribution In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...

, which describes the number of successes in a series of independent Yes/No experiments all with the same probability of success. *The beta-binomial distribution, which describes the number of successes in a series of independent Yes/No experiments with heterogeneity in the success probability. *The

degenerate distribution In probability theory, a degenerate distribution on a measure space (E, \mathcal, \mu) is a probability distribution whose support is a null set with respect to \mu. For instance, in the -dimensional space endowed with the Lebesgue measure, an ...

at ''x''₀, where ''X'' is certain to take the value ''x''₀. This does not look random, but it satisfies the definition of

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

. This is useful because it puts deterministic variables and random variables in the same formalism. *The

discrete uniform distribution In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein each of some finite whole number ''n'' of outcome values are equally likely to be observed. Thus every one of the ''n'' out ...

, where all elements of a finite

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

are equally likely. This is the theoretical distribution model for a balanced coin, an unbiased die, a casino roulette, or the first card of a well-shuffled deck. *The

hypergeometric distribution In probability theory and statistics, the hypergeometric distribution is a Probability distribution#Discrete probability distribution, discrete probability distribution that describes the probability of k successes (random draws for which the ...

, which describes the number of successes in the first ''m'' of a series of ''n'' consecutive Yes/No experiments, if the total number of successes is known. This distribution arises when there is no replacement. *The negative hypergeometric distribution, a distribution which describes the number of attempts needed to get the ''n''th success in a series of Yes/No experiments without replacement. *The Poisson binomial distribution, which describes the number of successes in a series of independent Yes/No experiments with different success probabilities. * Fisher's noncentral hypergeometric distribution * Wallenius' noncentral hypergeometric distribution * Benford's law, which describes the frequency of the first digit of many naturally occurring data. *The ideal and robust soliton distributions. *

Zipf's law Zipf's law (; ) is an empirical law stating that when a list of measured values is sorted in decreasing order, the value of the -th entry is often approximately inversely proportional to . The best known instance of Zipf's law applies to the ...

or the Zipf distribution. A discrete power-law distribution, the most famous example of which is the description of the frequency of words in the English language. *The Zipf–Mandelbrot law is a discrete power law distribution which is a generalization of the Zipf distribution. CMP PMF

With infinite support

*The

beta negative binomial distribution In probability theory, a beta negative binomial distribution is the probability distribution of a discrete probability distribution, discrete random variable X equal to the number of failures needed to get r successes in a sequence of indepe ...

*The

Boltzmann distribution In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability tha ...

, a discrete distribution important in

statistical physics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

which describes the probabilities of the various discrete energy levels of a system in

thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in t ...

. It has a continuous analogue. Special cases include: **The Gibbs distribution **The Maxwell–Boltzmann distribution *The

Borel distribution The Borel distribution is a discrete probability distribution, arising in contexts including branching processes and queueing theory. It is named after the French mathematician Émile Borel. If the number of offspring that an organism has is Poi ...

*The discrete phase-type distribution, a generalization of the geometric distribution which describes the first hit time of the absorbing state of a finite terminating

Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ...

. *The extended negative binomial distribution *The generalized log-series distribution *The Gauss–Kuzmin distribution *The

geometric distribution In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: * The probability distribution of the number X of Bernoulli trials needed to get one success, supported on \mathbb = \; * T ...

, a discrete distribution which describes the number of attempts needed to get the first success in a series of independent Bernoulli trials, or alternatively only the number of losses before the first success (i.e. one less). *The Hermite distribution *The logarithmic (series) distribution *The

mixed Poisson distribution A mixed Poisson distribution is a univariate discrete probability distribution in stochastics. It results from assuming that the conditional distribution of a random variable, given the value of the rate parameter, is a Poisson distribution, and ...

*The

negative binomial distribution In probability theory and statistics, the negative binomial distribution, also called a Pascal distribution, is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Berno ...

or Pascal distribution, a generalization of the geometric distribution to the ''n''th success. *The discrete compound Poisson distribution *The

parabolic fractal distribution Parabolic usually refers to something in a shape of a parabola, but may also refer to a parable. Parabolic may refer to: *In mathematics: **In elementary mathematics, especially elementary geometry: **Parabolic coordinates **Parabolic cylindrical ...

*The

Poisson distribution In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...

, which describes a very large number of individually unlikely events that happen in a certain time interval. Related to this distribution are a number of other distributions: the displaced Poisson, the hyper-Poisson, the general Poisson binomial and the Poisson type distributions. **The Conway–Maxwell–Poisson distribution, a two-parameter extension of the

with an adjustable rate of decay. **The

zero-truncated Poisson distribution In probability theory, the zero-truncated Poisson distribution (ZTP distribution) is a certain discrete probability distribution whose support is the set of positive integers. This distribution is also known as the conditional Poisson distributi ...

, for processes in which zero counts are not observed *The Polya–Eggenberger distribution *The Skellam distribution, the distribution of the difference between two independent Poisson-distributed random variables. *The skew elliptical distribution *The Yule–Simon distribution *The zeta distribution has uses in applied statistics and statistical mechanics, and perhaps may be of interest to number theorists. It is the

Zipf distribution Zipf's law (; ) is an empirical law stating that when a list of measured values is sorted in decreasing order, the value of the -th entry is often approximately inversely proportional to . The best known instance of Zipf's law applies to the ...

for an infinite number of elements. *The Hardy distribution, which describes the probabilities of the hole scores for a given golf player.

Absolutely continuous distributions

Supported on a bounded interval

*The

Beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval

, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...

or (0, 1) in terms of two positive Statistical parameter, parameters, denoted by ''alpha'' (''α'') an ...

on ,1 a family of two-parameter distributions with one mode, of which the uniform distribution is a special case, and which is useful in estimating success probabilities. **The four-parameter Beta distribution, a straight-forward generalization of the Beta distribution to arbitrary bounded intervals

, b The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...

/math>. *The arcsine distribution on 'a'',''b'' which is a special case of the Beta distribution if ''α'' = ''β'' = 1/2, ''a'' = 0, and ''b'' = 1. *The

PERT distribution In probability and statistics, the PERT distributions are a family of continuous probability distributions defined by the minimum (''a''), most likely (''b'') and maximum (''c'') values that a variable can take. It is a transformation of the four ...

is a special case of the four-parameter

beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval

or (0, 1) in terms of two positive Statistical parameter, parameters, denoted by ''alpha'' (''α'') an ...

. *The uniform distribution or rectangular distribution on 'a'',''b'' where all points in a finite interval are equally likely, is a special case of the four-parameter Beta distribution. *The Irwin–Hall distribution is the distribution of the sum of ''n'' independent random variables, each of which having the uniform distribution on ,1 *The Bates distribution is the distribution of the mean of ''n'' independent random variables, each of which having the uniform distribution on ,1 *The logit-normal distribution on (0,1). *The

Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...

, although not strictly a probability distribution, is a limiting form of many continuous probability functions. It represents a ''discrete'' probability distribution concentrated at 0 — a

— it is a

Distribution (mathematics) Distributions, also known as Schwartz distributions are a kind of generalized function in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose derivatives do not exist in the classical sense. In par ...

in the generalized function sense; but the notation treats it as if it were a continuous distribution. *The

Kent distribution In directional statistics, the Kent distribution, also known as the 5-parameter Fisher–Bingham distribution (named after John T. Kent, Ronald Fisher, and Christopher Bingham), is a probability distribution on the unit sphere (2-sphere ''S''2 i ...

on the two-dimensional sphere. *The Kumaraswamy distribution is as versatile as the Beta distribution but has simple closed forms for both the cdf and the pdf. *The logit metalog distribution, which is highly shape-flexible, has simple closed forms, and can be parameterized with data using linear least squares. *The Marchenko–Pastur distribution is important in the theory of

random matrices In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability distribution. Random matrix theory (RMT) is the ...

. *The bounded

quantile-parameterized distribution A quantile-parameterized distribution (QPD) is a probability distributions that is directly parameterized by data. They were created to meet the need for easy-to-use continuous probability distributions flexible enough to represent a wide range of u ...

s, which are highly shape-flexible and can be parameterized with data using linear least squares (see Quantile-parameterized distribution#Transformations) *The raised cosine distribution on math>\mu-s,\mu+s*The

reciprocal distribution In probability and statistics, the reciprocal distribution, also known as the log-uniform distribution, is a continuous probability distribution. It is characterised by its probability density function, within the support of the distribution, bei ...

*The

triangular distribution In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit ''a'', upper limit ''b'', and mode ''c'', where ''a'' < ''b'' and ''a'' ≤ ''c'' ≤ ''b''. ...

on 'a'', ''b'' a special case of which is the distribution of the sum of two independent uniformly distributed random variables (the ''convolution'' of two uniform distributions). *The trapezoidal distribution *The

truncated normal distribution In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated no ...

on 'a'', ''b'' *The U-quadratic distribution on 'a'', ''b'' *The

von Mises–Fisher distribution In directional statistics, the von Mises–Fisher distribution (named after Richard von Mises and Ronald Fisher), is a probability distribution on the (p-1)-sphere in \mathbb^. If p=2 the distribution reduces to the von Mises distribution on the c ...

on the ''N''-dimensional sphere has the

von Mises distribution In probability theory and directional statistics, the Richard von Mises, von Mises distribution (also known as the circular normal distribution or the Andrey Nikolayevich Tikhonov, Tikhonov distribution) is a continuous probability distribution ...

as a special case. *The

Bingham distribution In statistics, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric probability distribution on the ''n''-sphere. It is a generalization of the Watson distribution and a special case of the Kent and Fisher–Bin ...

on the ''N''-dimensional sphere. *The

Wigner semicircle distribution The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution defined on the domain minus;''R'', ''R''whose probability density function ''f'' is a scaled semicircle, i.e. a semi-ellipse, centered at ...

is important in the theory of

. *The continuous Bernoulli distribution is a one-parameter

exponential family In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate ...

that provides a probabilistic counterpart to the binary

cross-entropy In information theory, the cross-entropy between two probability distributions p and q, over the same underlying set of events, measures the average number of bits needed to identify an event drawn from the set when the coding scheme used for the ...

loss.

Supported on intervals of length 2 – directional distributions

*The Henyey–Greenstein phase function *The Mie phase function *The

*The wrapped normal distribution *The wrapped exponential distribution *The wrapped Lévy distribution *The wrapped Cauchy distribution *The wrapped Laplace distribution *The wrapped asymmetric Laplace distribution *The

Dirac comb In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic function, periodic Function (mathematics), function with the formula \operatorname_(t) \ := \sum_^ \delta(t - k T) for some given perio ...

of period 2, although not strictly a function, is a limiting form of many directional distributions. It is essentially a wrapped

. It represents a ''discrete'' probability distribution concentrated at 2''n'' — a

— but the notation treats it as if it were a continuous distribution.

Supported on semi-infinite intervals, usually ,∞)

*The Beta prime distribution *The Birnbaum–Saunders distribution">Beta_prime_distribution.html" ;"title=",∞)

*The Beta prime distribution">,∞)

*The Beta prime distribution *The Birnbaum–Saunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times. *The chi distribution **The noncentral chi distribution *The chi-squared distribution, which is the sum of the squares of ''n'' independent Gaussian random variables. It is a special case of the Gamma distribution, and it is used in goodness-of-fit tests in

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

. **The inverse-chi-squared distribution **The

noncentral chi-squared distribution In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power ...

**The scaled inverse chi-squared distribution *The

Dagum distribution The Dagum distribution (or Mielke Beta-Kappa distribution) is a continuous probability distribution defined over positive real numbers. It is named after Camilo Dagum, who proposed it in a series of papers in the 1970s. The Dagum distribution ar ...

*The

exponential distribution In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuousl ...

, which describes the time between consecutive rare random events in a process with no memory. *The exponential-logarithmic distribution *The

F-distribution In probability theory and statistics, the ''F''-distribution or ''F''-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribut ...

, which is the distribution of the ratio of two (normalized) chi-squared-distributed random variables, used in the

analysis of variance Analysis of variance (ANOVA) is a family of statistical methods used to compare the Mean, means of two or more groups by analyzing variance. Specifically, ANOVA compares the amount of variation ''between'' the group means to the amount of variati ...

. It is referred to as the

beta prime distribution In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kindJohnson et al (1995), p 248) is an absolutely continuous probability distribution. If p\in ,1/math ...

when it is the ratio of two chi-squared variates which are not normalized by dividing them by their numbers of degrees of freedom. **The noncentral F-distribution *The

folded normal distribution The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...

*The

Fréchet distribution The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function :\ \Pr(\ X \le x\ ) = e^ ~ \text ~ x > 0 ~. where is a shape para ...

*The

Gamma distribution In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the g ...

, which describes the time until ''n'' consecutive rare random events occur in a process with no memory. **The

Erlang distribution The Erlang distribution is a two-parameter family of continuous probability distributions with Support (mathematics), support x \in [0, \infty). The two parameters are: * a positive integer k, the "shape", and * a positive real number \lambda, ...

, which is a special case of the gamma distribution with integral shape parameter, developed to predict waiting times in queuing systems **The inverse-gamma distribution *The generalized gamma distribution *The

generalized Pareto distribution In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions. It is often used to model the tails of another distribution. It is specified by three parameters: location \mu, scale \sigma, and shap ...

*The Gamma/Gompertz distribution *The Gompertz distribution *The half-normal distribution *The Hartman–Watson distribution * Hotelling's T-squared distribution *The

inverse Gaussian distribution In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support (mathematics), support on (0,∞). Its probability density function is ...

, also known as the Wald distribution *The

Lévy distribution In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is k ...

*The

log-Cauchy distribution In probability theory, a log-Cauchy distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Cauchy distribution. If ''X'' is a random variable with a Cauchy distribution, then ''Y'' = e ...

*The log-Laplace distribution *The log-logistic distribution *The log-metalog distribution, which is highly shape-flexile, has simple closed forms, can be parameterized with data using linear least squares, and subsumes the log-logistic distribution as a special case. *The

log-normal distribution In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...

, describing variables which can be modelled as the product of many small independent positive variables. *The Lomax distribution *The Mittag-Leffler distribution *The

Nakagami distribution The Nakagami distribution or the Nakagami-''m'' distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter m\geq 1/2 and a scale parameter \Omega > 0. ...

*The

Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial scien ...

, or "power law" distribution, used in the analysis of financial data and critical behavior. *The Pearson Type III distribution *The

phase-type distribution A phase-type distribution is a probability distribution constructed by a convolution or mixture of exponential distributions. It results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. The sequence i ...

, used in

queueing theory Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because th ...

*The phased bi-exponential distribution is commonly used in

pharmacokinetics Pharmacokinetics (from Ancient Greek ''pharmakon'' "drug" and ''kinetikos'' "moving, putting in motion"; see chemical kinetics), sometimes abbreviated as PK, is a branch of pharmacology dedicated to describing how the body affects a specific su ...

*The phased bi-Weibull distribution *The semi-bounded

s, which are highly shape-flexible and can be parameterized with data using linear least squares (see *The

Rayleigh distribution In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distributi ...

*The Rayleigh mixture distribution *The Rice distribution *The shifted Gompertz distribution *The type-2 Gumbel distribution *The

Weibull distribution In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It models a broad range of random variables, largely in the nature of a time to failure or time between events. Examples are maximum on ...

or Rosin Rammler distribution, of which the

is a special case, is used to model the lifetime of technical devices and is used to describe the

particle size distribution In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...

of particles generated by grinding,

milling Milling may refer to: * Milling (minting), forming narrow ridges around the edge of a coin * Milling (grinding), breaking solid materials into smaller pieces by grinding, crushing, or cutting in a mill * Milling (machining), a process of using ro ...

and crushing operations. *The

modified half-normal distribution In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...

. *The Polya-Gamma distribution *The modified Polya-gamma distribution. JohnsonSU

Supported on the whole real line

*The Behrens–Fisher distribution, which arises in the Behrens–Fisher problem. *The

Cauchy distribution The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) ...

, an example of a distribution which does not have an

expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...

or a

variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...

. In physics it is usually called a Lorentzian profile, and is associated with many processes, including

resonance Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration whose frequency matches a resonant frequency (or resonance frequency) of the system, defined as a frequency that generates a maximu ...

energy distribution, impact and natural

spectral line A spectral line is a weaker or stronger region in an otherwise uniform and continuous spectrum. It may result from emission (electromagnetic radiation), emission or absorption (electromagnetic radiation), absorption of light in a narrow frequency ...

broadening and quadratic stark line broadening. *The centralized inverse-Fano distribution, which is the distribution representing the ratio of independent normal and gamma-difference random variables. * Chernoff's distribution *The

exponentially modified Gaussian distribution In probability theory, an exponentially modified Gaussian distribution (EMG, also known as exGaussian distribution) describes the sum of independent Normal distribution, normal and Exponential distribution, exponential random variables. An exGau ...

, a convolution of a

normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...

with an

, and the Gaussian minus exponential distribution, a convolution of a normal distribution with the negative of an exponential distribution. *The expectile distribution, which nests the Gaussian distribution in the symmetric case. *The Fisher–Tippett, extreme value, or log-Weibull distribution *

Fisher's z-distribution Fisher's ''z''-distribution is the statistical distribution of half the logarithm of an ''F''-distribution variate: : z = \frac 1 2 \log F It was first described by Ronald Fisher in a paper delivered at the International Mathematical Con ...

*The skewed generalized t distribution *The gamma-difference distribution, which is the distribution of the difference of independent gamma random variables. *The generalized logistic distribution *The

generalized normal distribution The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. ...

*The

geometric stable distribution A geometric stable distribution or geo-stable distribution is a type of leptokurtic probability distribution. Geometric stable distributions were introduced in Klebanov, L. B., Maniya, G. M., and Melamed, I. A. (1985). A problem of Zolotarev and ...

*The

Gumbel distribution In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. Thi ...

*The Holtsmark distribution, an example of a distribution that has a finite expected value but infinite variance. *The

hyperbolic distribution The hyperbolic distribution is a continuous probability distribution characterized by the logarithm of the probability density function being a hyperbola. Thus the distribution decreases exponentially, which is more slowly than the normal distrib ...

*The

hyperbolic secant distribution In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic se ...

*The Johnson SU distribution *The

Landau distribution Landau (), officially Landau in der Pfalz (, ), is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990), a lon ...

*The

Laplace distribution In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponen ...

*The Lévy skew alpha-stable distribution or

stable distribution In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be st ...

is a family of distributions often used to characterize financial data and critical behavior; the

, Holtsmark distribution,

and

are special cases. *The Linnik distribution *The

logistic distribution In probability theory and statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It rese ...

*The map-Airy distribution *The metalog distribution, which is highly shape-flexible, has simple closed forms, and can be parameterized with data using linear least squares. *The

, also called the Gaussian or the bell curve. It is ubiquitous in nature and statistics due to the

central limit theorem In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...

: every variable that can be modelled as a sum of many small independent, identically distributed variables with finite

mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...

and

is approximately normal. *The normal-exponential-gamma distribution *The normal-inverse Gaussian distribution *The Pearson Type IV distribution (see

Pearson distribution The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics. History The Pearson syste ...

s) *The

Quantile-parameterized distribution A quantile-parameterized distribution (QPD) is a probability distributions that is directly parameterized by data. They were created to meet the need for easy-to-use continuous probability distributions flexible enough to represent a wide range of u ...

s, which are highly shape-flexible and can be parameterized with data using linear least squares. *The

skew normal distribution In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness. Definition Let \phi(x) denote the Normal distribution, standard ...

Student's t-distribution In probability theory and statistics, Student's distribution (or simply the distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...

, useful for estimating unknown means of Gaussian populations. **The

noncentral t-distribution The noncentral ''t''-distribution generalizes Student's t-distribution, Student's ''t''-distribution using a noncentrality parameter. Whereas the central probability distribution describes how a test statistic ''t'' is distributed when the diff ...

**The skew t distribution *The

Champernowne distribution In statistics, the Champernowne distribution is a symmetric, continuous probability distribution, describing random variables that take both positive and negative values. It is a generalization of the logistic distribution that was introduced by D. ...

*The type-1 Gumbel distribution *The Tracy–Widom distribution *The Voigt distribution, or Voigt profile, is the convolution of a

and a

. It is found in spectroscopy when

profiles are broadened by a mixture of Lorentzian and

Doppler broadening In atomic physics, Doppler broadening is broadening of spectral lines due to the Doppler effect caused by a distribution of velocities of atoms or molecules. Different velocities of the emitting (or absorbing) particles result in different Doppl ...

mechanisms. *The Chen distribution.

With variable support

*The

generalized extreme value distribution In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel distribution, Gumbel, Fréchet distribution, F ...

has a finite upper bound or a finite lower bound depending on what range the value of one of the parameters of the distribution is in (or is supported on the whole real line for one special value of the parameter *The

has a support which is either bounded below only, or bounded both above and below *The metalog distribution, which provides flexibility for unbounded, bounded, and semi-bounded support, is highly shape-flexible, has simple closed forms, and can be fit to data using linear least squares. *The

Tukey lambda distribution Formalized by John Tukey, the Tukey lambda distribution is a continuous, symmetric probability distribution defined in terms of its quantile function. It is typically used to identify an appropriate distribution (see the comments below) and not us ...

is either supported on the whole real line, or on a bounded interval, depending on what range the value of one of the parameters of the distribution is in. *The Wakeby distribution

Mixed discrete/continuous distributions

*The

rectified Gaussian distribution In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (co ...

replaces negative values from a

with a discrete component at zero. *The compound poisson-gamma or Tweedie distribution is continuous over the strictly positive real numbers, with a mass at zero.

Joint distributions

For any set of

independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...

random variables the

probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...

of their

joint distribution A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...

is the product of their individual density functions.

Two or more random variables on the same sample space

*The

Dirichlet distribution In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted \operatorname(\boldsymbol\alpha), is a family of continuous multivariate probability distributions parameterized by a vector of pos ...

, a generalization of the

beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval

or (0, 1) in terms of two positive Statistical parameter, parameters, denoted by ''alpha'' (''α'') an ...

. *The

Ewens's sampling formula In population genetics, Ewens's sampling formula describes the probabilities associated with counts of how many different alleles are observed a given number of times in the sample. Definition Ewens's sampling formula, introduced by Warren Ewen ...

is a probability distribution on the set of all

partitions of an integer In number theory and combinatorics, a partition of a non-negative integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same ...

''n'', arising in

population genetics Population genetics is a subfield of genetics that deals with genetic differences within and among populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as Adaptation (biology), adaptation, s ...

. *The Balding–Nichols model *The

multinomial distribution In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a ''k''-sided die rolled ''n'' times. For ''n'' statistical independence, indepen ...

, a generalization of the

. *The

multivariate normal distribution In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One d ...

, a generalization of the

. *The

multivariate t-distribution In statistics, the multivariate ''t''-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's ''t''-distribution, which is a distribution applica ...

, a generalization of the

. *The negative multinomial distribution, a generalization of the

. *The Dirichlet negative multinomial distribution, a generalization of the

. *The generalized multivariate log-gamma distribution *The Marshall–Olkin exponential distribution *The continuous-categorical distribution, an

supported on the

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

that generalizes the continuous Bernoulli distribution.

Distributions of matrix-valued random variables

*The

Wishart distribution In statistics, the Wishart distribution is a generalization of the gamma distribution to multiple dimensions. It is named in honor of John Wishart (statistician), John Wishart, who first formulated the distribution in 1928. Other names include Wi ...

*The

inverse-Wishart distribution In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the cov ...

*The Lewandowski-Kurowicka-Joe distribution *The

matrix normal distribution In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables. Definition The probability density ...

*The

matrix t-distribution In statistics, the matrix ''t''-distribution (or matrix variate ''t''-distribution) is the generalization of the multivariate ''t''-distribution from vectors to matrices.Zhu, Shenghuo and Kai Yu and Yihong Gong (2007)"Predictive Matrix-Variate ...

*The Matrix Langevin distribution *The matrix variate beta distribution *The Uniform distribution on a Stiefel manifold

Non-numeric distributions

*The

categorical distribution In probability theory and statistics, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution) is a discrete probability distribution that describes the possible results of a random variable that can ...

Miscellaneous distributions

*The

Cantor distribution The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. This distribution has neither a probability density function nor a probability mass function, since although its cumulative ...

*The generalized logistic distribution family *The metalog distribution family *The

family *The

References

{{Statistics

Probability distributions In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spac ...

Discrete distributions

With finite support

With infinite support

Absolutely continuous distributions

Supported on a bounded interval

Supported on intervals of length 2 – directional distributions

Supported on semi-infinite intervals, usually ,∞)

Supported on the whole real line

With variable support

Mixed discrete/continuous distributions

Joint distributions

Two or more random variables on the same sample space

Distributions of matrix-valued random variables

Non-numeric distributions

Miscellaneous distributions

See also

References