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

• TheBernoulli distribution, which takes value 1 with probabilityp and value 0 with probabilityq = 1 −p.
• TheRademacher distribution, which takes value 1 with probability 1/2 and value −1 with probability 1/2.
• Thebinomial distribution, which describes the number of successes in a series of independent Yes/No experiments all with the same probability of success.
• Thebeta-binomial distribution, which describes the number of successes in a series of independent Yes/No experiments with heterogeneity in the success probability.
• Thedegenerate distribution atx₀, whereX is certain to take the valuex₀. This does not look random, but it satisfies the definition ofrandom variable. This is useful because it puts deterministic variables and random variables in the same formalism.
• Thediscrete uniform distribution, where all elements of a finiteset 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.
• Thehypergeometric distribution, which describes the number of successes in the firstm of a series ofn consecutive Yes/No experiments, if the total number of successes is known. This distribution arises when there is no replacement.
• Thenegative hypergeometric distribution, a distribution which describes the number of attempts needed to get thenth success in a series of Yes/No experiments without replacement.
• ThePoisson 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 robustsoliton distributions.
• Zipf's law or the Zipf distribution. A discretepower-law distribution, the most famous example of which is the description of the frequency of words in the English language.
• TheZipf–Mandelbrot law is a discrete power law distribution which is a generalization of the Zipf distribution.

• Thebeta negative binomial distribution
• TheBoltzmann distribution, a discrete distribution important instatistical physics which describes the probabilities of the various discrete energy levels of a system inthermal equilibrium. It has a continuous analogue. Special cases include:
  • TheGibbs distribution
  • TheMaxwell–Boltzmann distribution
• TheBorel distribution
• Thediscrete phase-type distribution, a generalization of the geometric distribution which describes thefirst hit time of the absorbing state of a finite terminatingMarkov chain.
• Theextended negative binomial distribution
• The generalized log-series distribution
• TheGauss–Kuzmin distribution
• Thegeometric distribution, 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).
• TheHermite distribution
• Thelogarithmic (series) distribution
• Themixed Poisson distribution
• Thenegative binomial distribution or Pascal distribution, a generalization of the geometric distribution to thenth success.
• The discretecompound Poisson distribution
• Theparabolic fractal distribution
• ThePoisson distribution, 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: thedisplaced Poisson, the hyper-Poisson, the general Poisson binomial and the Poisson type distributions.
  • TheConway–Maxwell–Poisson distribution, a two-parameter extension of thePoisson distribution with an adjustable rate of decay.
  • Thezero-truncated Poisson distribution, for processes in which zero counts are not observed
• ThePolya–Eggenberger distribution
• TheSkellam distribution, the distribution of the difference between two independent Poisson-distributed random variables.
• Theskew elliptical distribution
• TheYule–Simon distribution
• Thezeta distribution has uses in applied statistics and statistical mechanics, and perhaps may be of interest to number theorists. It is theZipf distribution for an infinite number of elements.
• TheHardy distribution, which describes the probabilities of the hole scores for a given golf player.

• TheBeta distribution on [0,1], 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.
  • Thefour-parameter Beta distribution, a straight-forward generalization of the Beta distribution to arbitrary bounded intervals${\displaystyle [a,b]}$.
• Thearcsine distribution on [a,b], which is a special case of the Beta distribution ifα = β = 1/2,a = 0, andb = 1.
• ThePERT distribution is a special case of the four-parameterbeta distribution.
• Theuniform 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.
• TheIrwin–Hall distribution is the distribution of the sum ofn independent random variables, each of which having the uniform distribution on [0,1].
• TheBates distribution is the distribution of the mean ofn independent random variables, each of which having the uniform distribution on [0,1].
• Thelogit-normal distribution on (0,1).
• TheDirac delta function, although not strictly a probability distribution, is a limiting form of many continuous probability functions. It represents adiscrete probability distribution concentrated at 0 — adegenerate distribution — it is aDistribution (mathematics) in the generalized function sense; but the notation treats it as if it were a continuous distribution.
• TheKent distribution on the two-dimensional sphere.
• TheKumaraswamy distribution is as versatile as the Beta distribution but has simple closed forms for both the cdf and the pdf.
• Thelogit metalog distribution, which is highly shape-flexible, has simple closed forms, and can be parameterized with data using linear least squares.
• TheMarchenko–Pastur distribution is important in the theory ofrandom matrices.
• The boundedquantile-parameterized distributions, which are highly shape-flexible and can be parameterized with data using linear least squares (seeQuantile-parameterized distribution#Transformations)
• Theraised cosine distribution on [${\displaystyle \mu -s,\mu +s}$]
• Thereciprocal distribution
• Thetriangular distribution on [a,b], a special case of which is the distribution of the sum of two independent uniformly distributed random variables (theconvolution of two uniform distributions).
• Thetrapezoidal distribution
• Thetruncated normal distribution on [a,b].
• TheU-quadratic distribution on [a,b].
• Thevon Mises–Fisher distribution on theN-dimensional sphere has thevon Mises distribution as a special case.
• TheBingham distribution on theN-dimensional sphere.
• TheWigner semicircle distribution is important in the theory ofrandom matrices.
• Thecontinuous Bernoulli distribution is a one-parameterexponential family that provides a probabilistic counterpart to the binarycross-entropy loss.

• TheHenyey–Greenstein phase function
• TheMie phase function
• Thevon Mises distribution
• Thewrapped normal distribution
• Thewrapped exponential distribution
• Thewrapped Lévy distribution
• Thewrapped Cauchy distribution
• Thewrapped Laplace distribution
• Thewrapped asymmetric Laplace distribution
• TheDirac comb of period 2π, although not strictly a function, is a limiting form of many directional distributions. It is essentially a wrappedDirac delta function. It represents adiscrete probability distribution concentrated at 2πn — adegenerate distribution — but the notation treats it as if it were a continuous distribution.

• TheBeta prime distribution
• TheBirnbaum–Saunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times.
• Thechi distribution
  • Thenoncentral chi distribution
• Thechi-squared distribution, which is the sum of the squares ofn independent Gaussian random variables. It is a special case of the Gamma distribution, and it is used ingoodness-of-fit tests instatistics.
  • Theinverse-chi-squared distribution
  • Thenoncentral chi-squared distribution
  • Thescaled inverse chi-squared distribution
• TheDagum distribution
• Theexponential distribution, which describes the time between consecutive rare random events in a process with no memory.
• Theexponential-logarithmic distribution
• TheF-distribution, which is the distribution of the ratio of two (normalized) chi-squared-distributed random variables, used in theanalysis of variance. It is referred to as thebeta 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.
  • Thenoncentral F-distribution
• Thefolded normal distribution
• TheFréchet distribution
• TheGamma distribution, which describes the time untiln consecutive rare random events occur in a process with no memory.
  • TheErlang distribution, which is a special case of the gamma distribution with integral shape parameter, developed to predict waiting times inqueuing systems
  • Theinverse-gamma distribution
• Thegeneralized gamma distribution
• Thegeneralized Pareto distribution
• TheGamma/Gompertz distribution
• TheGompertz distribution
• Thehalf-normal distribution
• TheHartman–Watson distribution
• Hotelling's T-squared distribution
• Theinverse Gaussian distribution, also known as the Wald distribution
• TheLévy distribution
• Thelog-Cauchy distribution
• Thelog-Laplace distribution
• Thelog-logistic distribution
• Thelog-metalog distribution, which is highly shape-flexile, has simple closed forms, can be parameterized with data using linear least squares, and subsumes thelog-logistic distribution as a special case.
• Thelog-normal distribution, describing variables which can be modelled as the product of many small independent positive variables.
• TheLomax distribution
• TheMittag-Leffler distribution
• TheNakagami distribution
• ThePareto distribution, or "power law" distribution, used in the analysis of financial data and critical behavior.
• ThePearson Type III distribution
• Thephase-type distribution, used inqueueing theory
• Thephased bi-exponential distribution is commonly used inpharmacokinetics
• Thephased bi-Weibull distribution
• The semi-boundedquantile-parameterized distributions, which are highly shape-flexible and can be parameterized with data using linear least squares (seeQuantile-parameterized distribution § Transformations
• TheRayleigh distribution
• TheRayleigh mixture distribution
• TheRice distribution
• Theshifted Gompertz distribution
• Thetype-2 Gumbel distribution
• TheWeibull distribution or Rosin Rammler distribution, of which theexponential distribution is a special case, is used to model the lifetime of technical devices and is used to describe theparticle size distribution of particles generated by grinding,milling andcrushing operations.
• Themodified half-normal distribution.^[1]
• ThePolya-Gamma distribution^[2]
• Themodified Polya-gamma distribution.^[3]

• TheBehrens–Fisher distribution, which arises in theBehrens–Fisher problem.
• TheCauchy distribution, an example of a distribution which does not have anexpected value or avariance. In physics it is usually called aLorentzian profile, and is associated with many processes, includingresonance energy distribution, impact and naturalspectral line broadening and quadraticstark line broadening.
• Thecentralized inverse-Fano distribution, which is the distribution representing the ratio of independent normal and gamma-difference random variables.
• Chernoff's distribution
• Theexponentially modified Gaussian distribution, a convolution of anormal distribution with anexponential distribution, and theGaussian minus exponential distribution, a convolution of a normal distribution with the negative of an exponential distribution.
• Theexpectile distribution, which nests the Gaussian distribution in the symmetric case.
• TheFisher–Tippett, extreme value, or log-Weibull distribution
• Fisher's z-distribution
• Theskewed generalized t distribution
• Thegamma-difference distribution, which is the distribution of the difference of independent gamma random variables.
• Thegeneralized logistic distribution
• Thegeneralized normal distribution
• Thegeometric stable distribution
• TheGumbel distribution
• TheHoltsmark distribution, an example of a distribution that has a finite expected value but infinite variance.
• Thehyperbolic distribution
• Thehyperbolic secant distribution
• TheJohnson SU distribution
• TheLandau distribution
• TheLaplace distribution
• TheLévy skew alpha-stable distribution orstable distribution is a family of distributions often used to characterize financial data and critical behavior; theCauchy distribution,Holtsmark distribution,Landau distribution,Lévy distribution andnormal distribution are special cases.
• TheLinnik distribution
• Thelogistic distribution
• Themap-Airy distribution
• Themetalog distribution, which is highly shape-flexible, has simple closed forms, and can be parameterized with data using linear least squares.
• Thenormal distribution, also called the Gaussian or the bell curve. It is ubiquitous in nature and statistics due to thecentral limit theorem: every variable that can be modelled as a sum of many small independent, identically distributed variables with finitemean andvariance is approximately normal.
• Thenormal-exponential-gamma distribution
• Thenormal-inverse Gaussian distribution
• ThePearson Type IV distribution (seePearson distributions)
• TheQuantile-parameterized distributions, which are highly shape-flexible and can be parameterized with data using linear least squares.
• Theskew normal distribution
• Student's t-distribution, useful for estimating unknown means of Gaussian populations.
  • Thenoncentral t-distribution
  • Theskew t distribution
• TheChampernowne distribution
• Thetype-1 Gumbel distribution
• TheTracy–Widom distribution
• TheVoigt distribution, or Voigt profile, is the convolution of anormal distribution and aCauchy distribution. It is found in spectroscopy whenspectral line profiles are broadened by a mixture ofLorentzian andDoppler broadening mechanisms.
• TheChen distribution.

• Thegeneralized extreme value distribution 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
• Thegeneralized Pareto distribution has a support which is either bounded below only, or bounded both above and below
• Themetalog 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.
• TheTukey 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.
• TheWakeby distribution

• Therectified Gaussian distribution replaces negative values from anormal distribution with a discrete component at zero.
• Thecompound poisson-gamma or Tweedie distribution is continuous over the strictly positive real numbers, with a mass at zero.

For any set ofindependent random variables theprobability density function of theirjoint distribution is the product of their individual density functions.

• TheDirichlet distribution, a generalization of thebeta distribution.
• TheEwens's sampling formula is a probability distribution on the set of allpartitions of an integern, arising inpopulation genetics.
• TheBalding–Nichols model
• Themultinomial distribution, a generalization of thebinomial distribution.
• Themultivariate normal distribution, a generalization of thenormal distribution.
• Themultivariate t-distribution, a generalization of theStudent's t-distribution.
• Thenegative multinomial distribution, a generalization of thenegative binomial distribution.
• TheDirichlet negative multinomial distribution, a generalization of thebeta negative binomial distribution.
• Thegeneralized multivariate log-gamma distribution
• TheMarshall–Olkin exponential distribution
• Thecontinuous-categorical distribution, anexponential family supported on thesimplex that generalizes thecontinuous Bernoulli distribution.

• TheWishart distribution
• Theinverse-Wishart distribution
• TheLewandowski-Kurowicka-Joe distribution
• Thematrix normal distribution
• Thematrix t-distribution
• TheMatrix Langevin distribution
• Thematrix variate beta distribution
• TheUniform distribution on a Stiefel manifold

• Thecategorical distribution

• TheCantor distribution
• Thegeneralized logistic distribution family
• Themetalog distribution family
• ThePearson distribution family
• Thephase-type distribution

• Mixture distribution
• Relationships among probability distributions
• ProbOnto

• Statistics-related lists
• Probability distributions
• Types of probability distributions

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