List of probability distributions

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

Discrete distributions

Binomial distribution

Degenerate distribution

With finite support

The Bernoulli distribution, 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, 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 at x₀, where X is certain to take the value x₀. 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, 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 Poisson binomial distribution, which describes the number of successes in a series of independent Yes/No experiments with different success probabilities.
Fisher's noncentral hypergeometric distribution
Wallenius' noncentral hypergeometric distribution
Benford's law, which describes the frequency of the first digit of many naturally occurring data.

Poisson distribution

Skellam distribution

With infinite support

The beta negative binomial distribution
The Boltzmann distribution, a discrete distribution important in statistical physics which describes the probabilities of the various discrete energy levels of a system in thermal equilibrium. It has a continuous analogue. Special cases include:
- The Gibbs distribution
- The Maxwell–Boltzmann distribution
The Borel distribution
The extended negative binomial distribution
The extended hypergeometric distribution
The generalized log-series distribution
The geometric 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).
The logarithmic (series) distribution
The negative binomial distribution or Pascal distribution a generalization of the geometric distribution to the nth success.
The parabolic fractal distribution
The Poisson distribution, which describes a very large number of individually unlikely events that happen in a certain time interval. Related to this distributions 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 Poisson distribution 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 distribution of the difference between two independent Poisson-distributed random variables.
The skew elliptical distribution
The Yule–Simon distribution
The zeta distribution has uses in applied statistics and statistical mechanics, and perhaps may be of interest to number theorists. It is the Zipf distribution for an infinite number of elements.
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.

Continuous distributions

Beta distribution

Continuous uniform distribution

Supported on a bounded interval

The arcsine distribution on [a,b], which is a special case of the Beta distribution if a = 0 and b = 1.
The Beta 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.
The logitnormal distribution on (0,1).
The Dirac delta function although not strictly a function, is a limiting form of many continuous probability functions. It represents a discrete probability distribution concentrated at 0 — a degenerate distribution — but the notation treats it as if it were a continuous distribution.
The continuous uniform distribution on [a,b], where all points in a finite interval are equally likely.
- The rectangular distribution is a uniform distribution on [−1/2,1/2].
The Irwin–Hall distribution is the distribution of the sum of n i.i.d. U(0,1) random variables.
The Bates distribution is the distribution of the mean of n i.i.d. U(0,1) random variables.
The Kent distribution on the three-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 logarithmic distribution (continuous)
The Marchenko–Pastur distribution is important in the theory of random matrices.
The PERT distribution is a special case of the beta distribution
The raised cosine distribution on [ $\mu-s,\mu+s$ ]
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 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 Wigner semicircle distribution is important in the theory of random matrices.

Chi-squared distribution

Gamma distribution

Pareto distribution

Supported on intervals of length 2 $π$ – irectional distributions

The von Mises distribution
The wrapped normal distribution
The wrapped exponential distribution
The wrapped Lévy distribution
The wrapped Cauchy distribution
The wrapped Laplace distribution
The wrapped asymmetric Laplace distribution
The Dirac comb of period 2 π 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 2πn — a degenerate distribution — but the notation treats it as if it were a continuous distribution.

Supported on semi-infinite intervals, usually [0,∞)

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 F-distribution, which is the distribution of the ratio of two (normalized) chi-squared-distributed random variables, used in the analysis of variance. 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
Fisher's z-distribution
The folded normal distribution
The Fréchet distribution
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
The Generalized gamma distribution
The generalized Pareto distribution
The Gamma/Gompertz distribution
The Gompertz distribution
The half-normal distribution
Hotelling's T-squared distribution
The inverse Gaussian distribution, also known as the Wald distribution
The Lévy distribution
The log-Cauchy distribution
The log-Laplace distribution
The log-logistic distribution
The log-normal distribution, describing variables which can be modelled as the product of many small independent positive variables.
The Lomax distribution
The Mittag-Leffler distribution
The Nakagami distribution
The 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
The phased bi-exponential distribution is commonly used in pharmokinetics
The phased bi-Weibull distribution
The Rayleigh distribution
The Rayleigh mixture distribution
The Rice distribution
The shifted Gompertz distribution
The type-2 Gumbel distribution
The Weibull distribution 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 of particles generated by grinding, milling and crushing operations.

Cauchy distribution

Laplace distribution

Stable distribution

Supported on the whole real line

The Behrens–Fisher distribution, which arises in the Behrens–Fisher problem.
The Cauchy distribution, an example of a distribution which does not have an expected value or a variance. In physics it is usually called a Lorentzian profile, and is associated with many processes, including resonance energy distribution, impact and natural spectral line broadening and quadratic stark line broadening.
Chernoff's distribution
The Exponentially modified Gaussian distribution, a convolution of a normal distribution with an exponential distribution.
The Fisher–Tippett, extreme value, or log-Weibull distribution
Fisher's z-distribution
The skewed generalized t distribution
The generalized logistic distribution
The generalized normal distribution
The geometric stable distribution
The Gumbel distribution
The Holtsmark distribution, an example of a distribution that has a finite expected value but infinite variance.
The hyperbolic distribution
The hyperbolic secant distribution
The Johnson SU distribution
The Landau distribution
The Laplace distribution
The Lévy skew alpha-stable distribution or stable distribution is a family of distributions often used to characterize financial data and critical behavior; the Cauchy distribution, Holtsmark distribution, Landau distribution, Lévy distribution and normal distribution are special cases.
The Linnik distribution
The logistic distribution
The map-Airy distribution
The normal distribution, also called the Gaussian or the bell curve. It is ubiquitous in nature and statistics due to the central limit theorem: every variable that can be modelled as a sum of many small independent, identically distributed variables with finite mean and variance is approximately normal.
The Normal-exponential-gamma distribution
The Normal-inverse Gaussian distribution
The Pearson Type IV distribution (see Pearson distributions)
The skew normal distribution
Student's t-distribution, useful for estimating unknown means of Gaussian populations.
- The noncentral t-distribution
- The skew t 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 Cauchy distribution. It is found in spectroscopy when spectral line profiles are broadened by a mixture of Lorentzian and Doppler broadening mechanisms.
The Gaussian minus exponential distribution is a convolution of a normal distribution with (minus) an exponential distribution.
The Chen distribution.

With variable support

The generalized 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
The generalized Pareto distribution has a support which is either bounded below only, or bounded both above and below
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 random variables the probability density function of their joint distribution 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.
The Balding–Nichols model
The multinomial distribution, a generalization of the binomial distribution.
The multivariate normal distribution, a generalization of the normal distribution.
The multivariate t-distribution, a generalization of the Student's t-distribution.
The negative multinomial distribution, a generalization of the negative binomial distribution.
The generalized multivariate log-gamma distribution

Matrix-valued distributions

Non-numeric distributions

The categorical distribution

Miscellaneous distributions

The Cantor distribution
The generalized logistic distribution family
The Pearson distribution family
The phase-type distribution

Study design	Effect size Standard error Statistical power Sample size determination

Survey methodology	Sampling stratified cluster Opinion poll Questionnaire

Controlled experiments	Design control optimal Controlled trial Randomized Random assignment Replication Blocking Factorial experiment

Uncontrolled studies	Observational study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Confidence interval Testing hypotheses Power

Unbiased estimators	Mean unbiased minimum-variance Median unbiased

Biased estimators	Maximum likelihood Method of moments Minimum distance Density estimation

Parametric tests	Likelihood-ratio Wald Score

Specific tests

Z (normal) Student's t-test F Shapiro–Wilk Kolmogorov–Smirnov

Goodness of fit	Chi-squared G Sample source (Anderson–Darling) Sample normality (Shapiro–Wilk) Skewness / kurtosis normality (Jarque-Bera) Model comparison (Likelihood-ratio) Model quality (Akaike criterion)

Signed-rank	1-sample (Wilcoxon) 2-sample (Mann–Whitney U) 1-way anova (Kruskal–Wallis)

Bayesian inference

Correlation	Pearson product–moment Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity

Generalized linear model	Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality

Specific tests	Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey

Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR)

Frequency domain	Spectral density estimation Fourier analysis Wavelet

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time

Hazard function	Nelson–Aalen estimator

Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population statistics Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

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Probability distributions

List of probability distributions

Discrete univariate with finite support	Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Reciprocal Triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory-Schulz Fréchet Gamma Gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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