- List of probability distributions
Many

probability distribution s are so important in theory or applications that they 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, 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.

* Thedegenerate distribution at "x"_{0}, where "X" is certain to take the value "x_{0}". 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.

* The discrete uniform distribution, where all elements of a finite set are equally likely. This is supposed to be the distribution of a balanced coin, an unbiased die, a casino roulette, or the first card of a well-shuffled deck. Also, one can use measurements of quantum states to generate uniform random variables. All these are "physical" or "mechanical" devices, subject to design flaws or perturbations, so the uniform distribution is only an approximation of their behaviour. In digital computers, pseudo-random number generators are used to produced a statistically random discrete uniform distribution.

* Thehypergeometric distribution , which describes the number of successes in the first "m" of a series of "n" independent Yes/No experiments, if the total number of successes is known.

*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.

* TheZipf-Mandelbrot law is a discrete power law distribution which is a generalization of theZipf distribution .

*Fisher's noncentral hypergeometric distribution

*Wallenius' noncentral hypergeometric distribution **With infinite support*** The

Boltzmann 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

** TheBose-Einstein distribution

** TheFermi-Dirac distribution

* Thegeometric distribution , a discrete distribution which describes the number of attempts needed to get the first success in a series of independent Yes/No experiments.

* The logarithmic (series) distribution

* Thenegative binomial distribution , a generalization of the geometric distribution to the "n"th success.

* Theparabolic fractal distribution

* ThePoisson distribution , which describes a very large number of individually unlikely events that happen in a certain time interval.

** TheConway-Maxwell-Poisson distribution , a two-parameter extension of the Poisson.

* TheSkellam distribution , the distribution of the difference between two independent Poisson-distributed random variables.

* 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.**Continuous distributions****upported on a bounded interval*** The

Beta distribution on [0,1] , of which the uniform distribution is a special case, and which is useful in estimating success probabilities.

* The continuous uniform distribution on ["a","b"] , where all points in a finite interval are equally likely.

** Therectangular distribution is a uniform distribution on [-1/2,1/2] .

* TheDirac 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 — adegenerate distribution — but the notation treats it as if it were a continuous distribution.

* TheIrwin-Hall distribution is the distribution of the sum of "n" i.i.d. U(0,1) random variables.

* TheKumaraswamy distribution is as versatile as the Beta distribution but has simple closed forms for both the cdf and the pdf.

* Thelogarithmic distribution (continuous)

* Thetriangular distribution on ["a", "b"] , a special case of which is the distribution of the sum of two uniformly distributed random variables (the "convolution" of two uniform distributions).

* Thevon Mises distribution

*TheWigner semicircle distribution is important in the theory ofrandom matrices .**upported on semi-infinite intervals, usually**[0,∞) * The

chi distribution

* Thenoncentral chi distribution

* Thechi-square 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 ingoodness-of-fit tests instatistics .

** Theinverse-chi-square distribution

** Thenoncentral chi-square distribution

** Thescale-inverse-chi-square distribution

* Theexponential distribution , which describes the time between consecutive rare random events in a process with no memory.

* TheF-distribution , which is the distribution of the ratio of two (normalized) chi-square distributed random variables, used in theanalysis of variance .

** Thenoncentral F-distribution

* TheGamma distribution , which describes the time until "n" 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

*Fisher's z-distribution

* Thehalf-normal distribution

* TheLévy distribution

* Thelog-logistic distribution

* Thelog-normal distribution , describing variables which can be modelled as the product of many small independent positive variables.

* ThePareto distribution , or "power law" distribution, used in the analysis of financial data and critical behavior.

* TheRayleigh distribution

* TheRice distribution

* Thetype-2 Gumbel distribution

* TheWald distribution

* TheWeibull distribution , of which the exponential distribution is a special case, is used to model the lifetime of technical devices.**upported on the whole real line*** The

Beta prime distribution

* TheCauchy distribution , an example of a distribution which does not have anexpected value or avariance . In physics it is usually called a Lorentzian profile, and is associated with many processes, includingresonance energy distribution, impact and naturalspectral line broadening and quadratic stark line broadening.

*Chernoff's distribution

* The Fisher-Tippett, extreme value, or log-Weibull distribution

** TheGumbel distribution , a special case of the Fisher-Tippett distribution

* Thegeneralized extreme value distribution

* Thehyperbolic secant distribution

* TheLandau distribution

* TheLaplace distribution

* TheLévy skew alpha-stable distribution is often used to characterize financial data and critical behavior.

* Themap-Airy distribution

* 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 variables is approximately normal.

* Theskew-normal distribution

*Student's t-distribution , useful for estimating unknown means of Gaussian populations.

** Thenoncentral t-distribution

* Thetype-1 Gumbel distribution

* The Voigt 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 of Lorentzian and Doppler broadening mechanisms.**Joint distributions**For any set of independent random variables the

probability density function of the joint distribution is the product of the individual ones.**Two or more random variables on the same sample space***

Dirichlet distribution , a generalization of thebeta distribution .

*TheEwens's sampling formula is a probability distribution on the set of all partitions of an integer "n", arising inpopulation genetics .

*multinomial distribution , a generalization of thebinomial distribution .

*multivariate normal distribution , a generalization of thenormal distribution .**Matrix-valued distributions***

Wishart distribution

*Inverse-Wishart distribution

*matrix normal distribution

*matrix t-distribution

*Hotelling's T-square distribution **Miscellaneous distributions*** The

Cantor distribution **See also***

copula (statistics)

*cumulative distribution function

*likelihood function

*list of statistical topics

*probability density function

*random variable

*histogram

*Truncated distribution

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