Many

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...

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

Discrete distributions

With finite
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...

* The

Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probab ...

, 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 ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no qu ...

, 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 In probability theory and statistics, the beta-binomial distribution is a family of discrete probability distributions on a finite support of non-negative integers arising when the probability of success in each of a fixed or known number of Bern ...

, which describes the number of successes in a series of independent Yes/No experiments with heterogeneity in the success probability. * The

degenerate distribution In mathematics, a degenerate distribution is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point. By the latter ...

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 mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

. 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 a finite number of values are equally likely to be observed; every one of ''n'' values has equal probability 1/''n''. Ano ...

, where all elements of a finite set 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 discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, ''without' ...

, 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 In probability theory and statistics, the negative hypergeometric distribution describes probabilities for when sampling from a finite population without replacement in which each sample can be classified into two mutually exclusive categories li ...

, 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 Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small.Arno Berger and Theodore P ...

, which describes the frequency of the first digit of many naturally occurring data. * The ideal and robust soliton distributions. * Zipf's law or the Zipf distribution. A discrete

power-law In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one ...

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 * 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 ...

, a discrete distribution important in

statistical physics Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approxi ...

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 ...

. It has a continuous analogue. Special cases include: ** The

Gibbs 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 ...

** 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 P ...

* 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 the set \; ...

, 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 * The

negative binomial distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expr ...

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

compound Poisson distribution In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. T ...

* The parabolic fractal distribution * 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 or space if these events occur with a known ...

, 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 (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 distribution or the pos ...

, 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 Skew may refer to: In mathematics * Skew lines, neither parallel nor intersecting. * Skew normal distribution, a probability distribution * Skew field or division ring * Skew-Hermitian matrix * Skew lattice In abstract algebra, a skew lattice ...

* 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 for an infinite number of elements.

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 , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...

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

arcsine distribution In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root: :F(x) = \frac\arcsin\left(\sqrt x\right)=\frac+\frac for 0 ≤ ''x'' ...

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 is a special case of the

beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...

* The uniform distribution or rectangular distribution on 'a'',''b'' where all points in a finite interval are equally likely. * The

Irwin–Hall distribution In probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a nu ...

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 In probability theory, a logit-normal distribution is a probability distribution of a random variable whose logit has a normal distribution. If ''Y'' is a random variable with a normal distribution, and ''t'' is the standard logistic function, th ...

on (0,1). * The

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

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 or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives ...

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 in ...

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 In the mathematical theory of random matrices, the Marchenko–Pastur distribution, or Marchenko–Pastur law, describes the asymptotic behavior of singular values of large rectangular random matrices. The theorem is named after Ukrainian mathema ...

is important in the theory of random matrices. * The bounded

quantile-parameterized distribution Quantile-parameterized distributions (QPDs) are probability distributions that are directly parameterized by data. They were motivated by the need for easy-to-use continuous probability distributions flexible enough to represent a wide range of unce ...

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 * The triangular distribution 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 von Mises distribution (also known as the circular normal distribution or Tikhonov distribution) is a continuous probability distribution on the circle. It is a close approximation to the w ...

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-Bingh ...

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

Wigner semicircle distribution The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on minus;''R'', ''R''whose probability density function ''f'' is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0): :f(x)=\sq ...

is important in the theory of random matrices. * 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 if a coding scheme used for the set is ...

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 In probability theory and directional statistics, a wrapped Lévy distribution is a wrapped probability distribution that results from the "wrapping" of the Lévy distribution around the unit circle. Description The pdf of the wrapped Lévy ...

* The wrapped Cauchy distribution * The wrapped Laplace distribution * The wrapped asymmetric Laplace distribution * The

Dirac comb In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula \operatorname_(t) \ := \sum_^ \delta(t - k T) for some given period T. Here ''t'' is a real variable and t ...

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

* The

exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...

, which describes the time between consecutive rare random events in a process with no memory. * The exponential-logarithmic distribution * The Kaniadakis ''κ''-exponential distribution, which is a generalization of the exponential 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 distribution ...

, 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 collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...

. It is referred to as the beta prime distribution 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 In probability theory and statistics, the noncentral ''F''-distribution is a continuous probability distribution that is a noncentral generalization of the (ordinary) ''F''-distribution. It describes the distribution of the quotient (''X''/''n' ...

* The folded normal distribution * The Fréchet distribution * The

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

, which describes the time until ''n'' consecutive rare random events occur in a process with no memory. ** The Erlang distribution, 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 In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according t ...

*The Kaniadakis κ-Gamma distribution, which is a κ-deformation of the

generalized gamma distribution The generalized gamma distribution is a continuous probability distribution with two shape parameters (and a scale parameter). It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter). Since many di ...

. **The κ-Erlang distribution, which is a special case of the Kaniadakis κ-Gamma distribution. * The

* The generalized Pareto distribution Kaniadakis weibull pdf

* The Gamma/Gompertz distribution * The Gompertz distribution * The

half-normal distribution In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...

* 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 on (0,∞). Its probability density function is given by : f(x;\mu, ...

, also known as the Wald distribution * The Lévy distribution * The log-Cauchy distribution * The

log-Laplace distribution In probability theory and statistics, the log-Laplace distribution is the probability distribution of a random variable whose logarithm has a Laplace distribution. If ''X'' has a Laplace distribution with parameters ''μ'' and ''b'', then ''Y'' = ...

* The

log-logistic distribution In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative random variable. It is used in survival analysis as a parametric model for even ...

* 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

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 normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...

, describing variables which can be modelled as the product of many small independent positive variables. * The

Lomax distribution The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K. ...

* 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 second parameter controlling ...

* 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, actu ...

, 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 ...

, 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 the ...

* 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 determining the fate of substances administered ...

* 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 distribu ...

* 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 is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice R ...

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 The particle-size distribution (PSD) of a powder, or granular material, or particles dispersed in fluid, is a list of values or a mathematical function that defines the relative amount, typically by mass, of particles present according to size. Si ...

of particles generated by grinding, milling and crushing operations. * The Kaniadakis ''κ''-Weibull distribution. * The

Modified half-normal distribution In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...

distribution. The pdf of the distribution on the support

(0, \infty)

is spefied as

f(x)= \frac

, where

\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)

denotes the Fox-Wright Psi function. * 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 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) fu ...

, an example of a distribution which does not have an

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

or a

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...

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

resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...

energy distribution, impact and natural

spectral line A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...

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 and exponential random variables. An exGaussian random variable ''Z'' may be expressed as ...

, a convolution of a

normal distribution In 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 e^ The parameter \mu i ...

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 * 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 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. To ...

* The geometric stable distribution * 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. T ...

* The Holtsmark distribution, an example of a distribution that has a finite expected value but infinite variance. * The hyperbolic distribution * 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 * The Kaniadakis κ-Laplace distribution. * 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 expo ...

* 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 sta ...

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

, Holtsmark distribution, Landau distribution, Lévy distribution and

are special cases. * The

Linnik distribution Yuri Vladimirovich Linnik (russian: Ю́рий Влади́мирович Ли́нник; January 8, 1915 – June 30, 1972) was a Soviet mathematician active in number theory, probability theory and mathematical statistics. Linnik was born in ...

* The

logistic distribution Logistic may refer to: Mathematics * Logistic function, a sigmoid function used in many fields ** Logistic map, a recurrence relation that sometimes exhibits chaos ** Logistic regression, a statistical model using the logistic function ** Logit ...

* The map-Airy distribution * The

metalog distribution The metalog distribution is a flexible continuous probability distribution designed for ease of use in practice. Together with its transforms, the metalog family of continuous distributions is unique because it embodies ''all'' of following proper ...

, 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) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables thems ...

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

mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ari ...

and

is approximately normal. * The

normal-exponential-gamma distribution In probability theory and statistics, the normal-exponential-gamma distribution (sometimes called the NEG distribution) is a three-parameter family of continuous probability distributions. It has a location parameter \mu, scale parameter \theta ...

* The normal-inverse Gaussian distribution * The Pearson Type IV distribution (see Pearson distributions) * The

Quantile-parameterized distribution Quantile-parameterized distributions (QPDs) are probability distributions that are directly parameterized by data. They were motivated by the need for easy-to-use continuous probability distributions flexible enough to represent a wide range of unce ...

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 standard normal probability d ...

* Student's t-distribution, useful for estimating unknown means of Gaussian populations. ** The

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

** The

skew t distribution Skew may refer to: In mathematics * Skew lines, neither parallel nor intersecting. * Skew normal distribution, a probability distribution * Skew field or division ring * Skew-Hermitian matrix * Skew lattice * Skew polygon, whose vertices do n ...

Kaniadakis Gaussian Distribution Type II pdf

* 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 Tracy–Widom distribution is a probability distribution from random matrix theory introduced by . It is the distribution of the normalized largest eigenvalue of a random Hermitian matrix. The distribution is defined as a Fredholm determinant ...

* 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 Dopp ...

mechanisms. * The Chen distribution. * The Kaniadakis κ-Gaussian distribution, which is a generalization of the

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, Fréchet and Weibull families also known ...

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 generalized Pareto distribution has a support which is either bounded below only, or bounded both above and below * The

, 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 use ...

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 The Wakeby distribution is a five-parameter probability distribution defined by its quantile function, :W(p) =\xi + \frac(1 - (1-p)^) - \frac(1 - (1-p)^), and by its quantile density function, :W'(p) = w(p) = \alpha (1-p)^ + \gamma (1-p)^, w ...

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 ( ...

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 the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...

random variables the

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...

of their

joint distribution Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...

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 \bolds ...

, a generalization of the

. * 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 Ewe ...

is a probability distribution on the set of all partitions of an integer ''n'', arising in

population genetics Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and pop ...

. * 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 dice rolled ''n'' times. For ''n'' independent trials each of w ...

, 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 ...

, 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 applic ...

, a generalization of the Student's t-distribution. * The

negative multinomial distribution In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(''x''0, ''p'')) to more than two outcomes.Le Gall, F. The modes of a negative multinomial distributio ...

, a generalization of the

. * The

Dirichlet negative multinomial distribution In probability theory and statistics, the Dirichlet negative multinomial distribution is a multivariate distribution on the non-negative integers. It is a multivariate extension of the beta negative binomial distribution. It is also a generali ...

, a generalization of the beta negative binomial distribution. * 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 to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928. It is a family of probability distributions defin ...

* The inverse-Wishart distribution * The

Lewandowski-Kurowicka-Joe distribution In probability theory and Bayesian statistics, the Lewandowski-Kurowicka-Joe distribution, often referred to as the LKJ distribution, is a probability distribution over positive definite symmetric matrices with unit diagonals. Introduction T ...

* The matrix normal distribution * The matrix t-distribution * The Matrix Langevin distribution

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 ca ...

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

family * The Pearson distribution family * 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 ...

References

{{Reflist

Probability distributions In probability theory and statistics, a probability distribution is the mathematical Function (mathematics), function that gives the probabilities of occurrence of different possible outcomes for an Experiment (probability theory), experiment. ...

Discrete distributions

With finite support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...

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

With finite
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...