List of probability distributions

Many probability distributions that are important in theory or applications 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 negative hypergeometric distribution, a distribution which describes the number of attempts needed to get the nth success in a series of Yes/No experiments without replacement.
  • The Poisson binomial distribution, which describes the number of successes in a series of independent Yes/No experiments with different success probabilities.
  • Fisher's noncentral hypergeometric distribution
  • Wallenius' noncentral hypergeometric distribution
  • Benford's law, which describes the frequency of the first digit of many naturally occurring data.
  • The ideal and robust soliton distributions.
  • Zipf's law 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.
Conway–Maxwell–Poisson distribution
Poisson distribution
Skellam distribution

With infinite support

Absolutely continuous distributions
Beta distribution
Kumaraswamy distribution
Continuous uniform distribution

Supported on a bounded interval
  • 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 four-parameter Beta distribution, a straight-forward generalization of the Beta distribution to arbitrary bounded intervals .
  • 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 is a special case of the four-parameter beta distribution.
  • The uniform distribution or rectangular distribution on [a,b], where all points in a finite interval are equally likely, is a special case of the four-parameter Beta distribution.
  • The Irwin–Hall distribution is the distribution of the sum of n independent random variables, each of which having the uniform distribution on [0,1].
  • The Bates distribution is the distribution of the mean of n independent random variables, each of which having the uniform distribution on [0,1].
  • 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 degenerate distribution — it is a Distribution (mathematics) 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.
  • 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 on []
  • 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 Bingham distribution on the N-dimensional sphere.
  • The Wigner semicircle distribution is important in the theory of random matrices.
  • The continuous Bernoulli distribution is a one-parameter exponential family that provides a probabilistic counterpart to the binary cross-entropy loss.
Chi-squared distribution
Gamma distribution
Pareto distribution

Supported on intervals of length 2π – directional distributions

Supported on semi-infinite intervals, usually [0,∞)
Cauchy distribution
Johnson SU distribution
Laplace distribution
Stable distribution

Supported on the whole real line

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 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

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

Distributions of matrix-valued random variables

Non-numeric distributions

Miscellaneous distributions

See also

References
  1. Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). "The Modified-Half-Normal distribution: Properties and an efficient sampling scheme". Communications in Statistics - Theory and Methods. 52 (5): 1591–1613. doi:10.1080/03610926.2021.1934700. ISSN 0361-0926. S2CID 237919587.
  2. Polson, Nicholas G.; Scott, James G.; Windle, Jesse (2013). "Bayesian Inference for Logistic Models Using Pólya–Gamma Latent Variables". Journal of the American Statistical Association. 108 (504): 1339–1349. arXiv:1205.0310. doi:10.1080/01621459.2013.829001. ISSN 0162-1459. JSTOR 24247065. S2CID 2859721. Retrieved 11 July 2021.
  3. Pal, Subhadip; Gaskins, Jeremy (23 May 2022). "Modified Pólya-Gamma data augmentation for Bayesian analysis of directional data". Journal of Statistical Computation and Simulation. 92 (16): 3430–3451. doi:10.1080/00949655.2022.2067853. S2CID 249022546.
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