Many probability distributions that are important in theory or applications have been given specific names.

Discrete distributions

With finite support

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

Rademacher distribution In probability theory and statistics, the Rademacher distribution (which is named after Hans Rademacher) is a discrete probability distribution where a random variate ''X'' has a 50% chance of being +1 and a 50% chance of being -1. A series ( ...

, which takes value 1 with probability 1/2 and value −1 with probability 1/2. * The binomial distribution, 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 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 d ...

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. This is useful because it puts deterministic variables and random variables in the same formalism. * The discrete uniform distribution, 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 lik ...

, 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 ...

, 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 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 random variable X equal to the number of failures needed to get r successes in a sequence of independent Bernoulli trials. The probabi ...

* 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 th ...

, 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 approxim ...

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 i ...

. 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 probabilit ...

** The Maxwell–Boltzmann distribution * The Borel distribution * The extended negative binomial distribution * The generalized log-series distribution * The

Gauss–Kuzmin distribution In mathematics, the Gauss–Kuzmin distribution is a discrete probability distribution that arises as the limit probability distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed in (0,&nbs ...

* 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 A mixed Poisson distribution is a Univariate distribution, 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 P ...

* The negative binomial distribution 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. ...

* 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 co ...

, 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, for processes in which zero counts are not observed * The Polya–Eggenberger distribution * The

Skellam distribution The Skellam distribution is the discrete probability distribution of the difference N_1-N_2 of two statistically independent random variables N_1 and N_2, each Poisson distribution, Poisson-distributed with respective expected values \mu_1 and \mu ...

, the distribution of the difference between two independent Poisson-distributed random variables. * The skew elliptical distribution * The Yule–Simon distribution * The

zeta distribution In probability theory and statistics, the zeta distribution is a discrete probability distribution. If ''X'' is a zeta-distributed random variable with parameter ''s'', then the probability that ''X'' takes the integer value ''k'' is given by t ...

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 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 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 distribution is 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-parameter beta ...

is a special case of the beta distribution * 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 and statistics, the Irwin–Hall distribution, named after Joseph Oscar Irwin and Philip Hall, is a probability distribution for a random variable defined as the sum of a number of independent random variables, each having a unifo ...

is the distribution of the sum of ''n'' independent random variables, each of which having the uniform distribution on ,1 * The

Bates distribution In probability and business statistics, the Bates distribution, named after Grace Bates, is a probability distribution of the mean of a number of statistically independent uniformly distributed random variables on the unit interval. This dist ...

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 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 d ...

in the generalized function sense; but the notation treats it as if it were a continuous distribution. * The Kent distribution 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 elements are random variables. Many important properties of physical systems can be represented mathemat ...

. * The bounded quantile-parameterized distributions, which are highly shape-flexible and can be parameterized with data using linear least squares (see Quantile-parameterized distribution#Transformations) * The

raised cosine distribution In probability theory and statistics, the raised cosine distribution is a continuous probability distribution supported on the interval mu-s,\mu+s/math>. The probability density function (PDF) is :f(x;\mu,s)=\frac \left +\cos\left(\frac\,\pi\rig ...

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 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 In probability theory and statistics, the trapezoidal distribution is a continuous probability distribution whose probability density function graph resembles a trapezoid. Likewise, trapezoidal distributions also roughly resemble mesas or plateau ...

* 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 on the ''N''-dimensional sphere has the von Mises distribution as a special case. * The Bingham distribution 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

. * 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 von Mises distribution * The

wrapped normal distribution In probability theory and directional statistics, a wrapped normal distribution is a wrapped probability distribution that results from the "wrapping" of the normal distribution around the unit circle. It finds application in the theory of Brownia ...

* The

wrapped exponential distribution In probability theory and directional statistics, a wrapped exponential distribution is a wrapped probability distribution that results from the "wrapping" of the exponential distribution around the unit circle. Definition The probability den ...

* The wrapped Lévy distribution * The

wrapped Cauchy distribution In probability theory and directional statistics, a wrapped Cauchy distribution is a wrapped probability distribution that results from the "wrapping" of the Cauchy distribution around the unit circle. The Cauchy distribution is sometimes known a ...

* 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 th ...

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

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* 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 exponential distribution, 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 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. Definitions ...

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 ''α'' > 0 is a ...

* The Gamma distribution, 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 to ...

*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 dis ...

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

* 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 In probability and statistics, the Gompertz distribution is a continuous probability distribution, named after Benjamin Gompertz. The Gompertz distribution is often applied to describe the distribution of adult lifespans by demographers and ac ...

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

, 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 Mittag-Leffler distributions are two families of probability distributions on the half-line ,\infty). They are parametrized by a real \alpha \in (0, 1/math> or \alpha \in , 1/math>. Both are defined with the Mittag-Leffler function, named afte ...

* The Nakagami distribution * The Pareto distribution, or "power law" distribution, used in the analysis of financial data and critical behavior. * The Pearson Type III distribution * The Phase-type distribution, 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 * The phased bi-Weibull distribution * The semi-bounded quantile-parameterized distributions, 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 distribut ...

* The Rayleigh mixture distribution * The

Rice distribution Rice is the seed of the grass species '' Oryza sativa'' (Asian rice) or less commonly ''Oryza glaberrima'' (African rice). The name wild rice is usually used for species of the genera '' Zizania'' and '' Porteresia'', both wild and domesticate ...

* The

shifted Gompertz distribution The shifted Gompertz distribution is the distribution of the larger of two independent random variables one of which has an exponential distribution with parameter b and the other has a Gumbel distribution with parameters \eta and b . In its o ...

* The

type-2 Gumbel distribution In probability theory, the Type-2 Gumbel probability density function is :f(x, a,b) = a b x^ e^\, for :0 < x < \infty. For

0 the

* 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 Re ...

or Rosin Rammler distribution, of which the exponential distribution 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. Sig ...

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 hal ...

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 In statistics, the Behrens–Fisher distribution, named after Ronald Fisher and Walter Behrens, is a parameterized family of probability distributions arising from the solution of the Behrens–Fisher problem proposed first by Behrens and severa ...

, 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) fun ...

, an example of a distribution which does not have an expected value 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 numbe ...

. 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 function, periodic force (or a Fourier analysis, Fourier component of it) is equal or close to a natural frequency of the system ...

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 ...

, a convolution of a normal distribution with an exponential distribution, 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 dis ...

* 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 * The

Holtsmark distribution The (one-dimensional) Holtsmark distribution is a continuous probability distribution. The Holtsmark distribution is a special case of a stable distribution with the index of stability or shape parameter \alpha equal to 3/2 and the skewness parame ...

, 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 distribu ...

* 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 sec ...

* The Johnson SU distribution * The

Landau distribution In probability theory, the Landau distribution is a probability distribution named after Lev Landau. Because of the distribution's "fat" tail, the moments of the distribution, like mean or variance, are undefined. The distribution is a particular ...

* The Kaniadakis κ-Laplace distribution. * The Laplace distribution * 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

and normal distribution are special cases. * The Linnik distribution * 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, which is highly shape-flexible, has simple closed forms, and can be parameterized with data using linear least squares. * The normal distribution, 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 themsel ...

: 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 a ...

* The

normal-inverse Gaussian distribution The normal-inverse Gaussian distribution (NIG) is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. The NIG distribution was noted by Blaesild i ...

* 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 system ...

s) * The Quantile-parameterized distributions, 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 skew t distribution Kaniadakis Gaussian Distribution Type II pdf

Kaniadakis Gaussian Distribution Type II pdf

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

. It is found in spectroscopy when

profiles are broadened by a mixture of Lorentzian and Doppler broadening mechanisms. * The Chen distribution. * The Kaniadakis κ-Gaussian distribution, which is a generalization of the normal 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, Fréchet and Weibull families also known a ...

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 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 replaces negative values from a normal distribution 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 * Independ ...

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) ca ...

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, a generalization of the beta distribution. * The Ewens's sampling formula 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 (biology), adaptation, ...

. * 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 wh ...

, a generalization of the binomial distribution. * 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 normal distribution. * 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 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 distribution ...

, a generalization of the negative binomial distribution. * 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 generaliz ...

, a generalization of the

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

supported on the simplex that generalizes the continuous Bernoulli distribution.

Distributions of matrix-valued random variables

* The Wishart distribution * 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 co ...

* 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. The matrix ''t''-distribution shares the same relationship with the multi ...

* 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 can ...

Miscellaneous distributions

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

family * The phase-type distribution

References

{{Reflist

Probability distributions 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 phenomenon ...

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