List of probability distributions

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

Discrete distributions[edit]

With finite support[edit]

With infinite support[edit]

Absolutely continuous distributions[edit]

Supported on a bounded interval[edit]

Supported on intervals of length 2π – directional distributions[edit]

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

Supported on the whole real line[edit]

With variable support[edit]

  • 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[edit]

Joint distributions[edit]

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[edit]

Distributions of matrix-valued random variables[edit]

Non-numeric distributions[edit]

Miscellaneous distributions[edit]

See also[edit]


  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: 1–23. 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: 1–22. doi:10.1080/00949655.2022.2067853.