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http://www.arxiv.org/abs/0808.1808
...The conflation of a finite number of probability distributions P_1,..., P_n is a consolidation of those distributions into a single probability distribution Q=Q(P_1,..., P_n), where intuitively Q is the conditional distribution of independent random ...
dore P. Hill

http://www.arxiv.org/abs/1302.2361
...We use anticommuting variables to study probability distributions of random variables, that are solutions of Langevin s equation. We show that the probability density always enjoys worldpoint supersymmetry. The partition function, however, may not. W...
S. Nicolis, A. Zerkak

http://www.arxiv.org/abs/math/0501183
...utions are given, which exhibit the existence of a continuum of situations interpolating the extreme cases of infinitely and minimally divisible probability distributions....
S. Albeverio, H. Gottschalk, J. -L. Wu

http://www.arxiv.org/abs/cs/0506016
...We show how to store good approximations of probability distributions in small space....
Travis Gagie

http://www.arxiv.org/abs/0802.0695
...Probability distributions which can be obtained from superpositions of Gaussian distributions of different variances v = \sigma ^2 play a favored role in quantum theory and financial markets. Such superpositions need not necessarily obey the Chapman-...
Petr Jizba, Hagen Kleinert

http://www.arxiv.org/abs/1506.02198
...finition of stable distributions on $\mathbb Z$ our has algebraic character. Examples of corresponding limit theorems are given. Keywords: stable distributions; casual stable distribution; discrete stable distributions; limit theorems...
Lev B. Klebanov

http://www.arxiv.org/abs/1309.0595
...We prove that if $p\geq 1$ and $-1\leq r\leq p-1$ then the binomial sequence $\binom{np+r}{n}$, $n=0,1,...$, is positive definite and is the moment sequence of a probability measure $\nu(p,r)$, whose support is contained in $\left[0,p^p(p-1)^{1-p}\right]$...
Wojciech Mlotkowski, Karol A. Penson

http://www.arxiv.org/abs/0805.4168
...We study the passage (translocation) of a self-avoiding polymer through a membrane pore in two dimensions. In particular, we numerically measure the probability distribution Q(T) of the translocation time T, and the distribution P(s,t) of the translo...
Clément Chatelain, Yacov Kantor, Mehran Kardar

http://www.arxiv.org/abs/1008.3597
...We study the problem of quantization of discrete probability distributions, arising in universal coding, as well as other applications. We show, that in many situations this problem can be reduced to the covering problem for the unit simplex. This se...
Yuriy A. Reznik

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2050372
...Probability allows us to infer from a sample to a population. In fact, inference is a tool of probability theory. This paper looks briefly at the Binomial, Poisson, and Normal distributions. These are probability distributions, which are used extensi...
Antony Ugoni, Bruce F. Walker

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Dirichlet distribution

often used as prior distributions in Bayesian statistics, the conjugate prior of the categorical distribution and multinomial distribution

$\displaystyle{{f}_{{K}}{\left({x}_{{1}},\ldots,{x}_{{{K}}};\alpha_{{1}},\ldots\alpha_{{K}}\right)}}=\frac{{1}}{{{B}{\left(\alpha\right)}}}{\prod_{{{i}={1}}}^{{K}}}{{x}_{{i}}^{{\alpha_{{i}}-{1}}}}\text{on the open (K − 1)-dimensional simplex defined by:}{x}_{{1}},\ldots,{x}_{{{K}-{1}}}\gt{0}, {x}_{{1}}+..+{x}_{{{K}-{1}}}\lt{1}, {x}_{{K}}={1}-{x}_{{1}}-\ldots-{x}_{{{K}-{1}}},$

Balding–Nichols model

a statistical description of the allele frequencies in the components of a sub-divided population

$\displaystyle{f{{\left({x};{F},{p}\right)}}}=\frac{{{x}^{{\alpha-{1}}}{\left({1}-{x}\right)}^{{\beta-{1}}}}}{{{B}{\left(\alpha,\beta\right)}}}\ \text{ for }\ {x}\in{\left({0},{1}\right)},\text{where }\ \alpha=\frac{{{1}-{F}}}{{F}}{p}{\quad\text{and}\quad}\beta=\frac{{{1}-{F}}}{{F}}{\left({1}-{p}\right)},\ \text{ with }\ {F},{p}\in{\left({0},{1}\right)}$

Multinomial distribution

a generalization of the binomial distribution

$\displaystyle{f{{\left({x}_{{1}},\ldots,{x}_{{k}};{n},{p}_{{1}},\ldots,{p}_{{k}}\right)}}}=\frac{{{n}!}}{{{\prod_{{{i}={1}}}^{{k}}}{x}_{{i}}!}}{\prod_{{{i}={1}}}^{{k}}}{{p}_{{i}}^{{{x}_{{i}}}}}\ \text{ for }\ {x}_{{i}}\in{\left\lbrace{0},\ldots,{n}\right\rbrace}{\quad\text{and}\quad}{\sum_{{{i}={1}}}^{{k}}}{x}_{{i}}={n},\ \text{ where }\ {p}_{{i}}\ \text{ are event probabilities, thus }\ {\sum_{{{i}={1}}}^{{k}}}{p}_{{i}}={1}.$

Multivariate normal distribution

a generalization of the one-dimensional (univariate) normal distribution to higher dimensions

$\displaystyle{f{{\left(\vec{{x}};\vec{\mu},\Sigma\right)}}}={\left({2}\pi\right)}^{{-{k}{/}{2}}}{\left|\Sigma\right|}^{{-{1}{/}{2}}}{\exp{{\left(-\frac{{1}}{{2}}{\left({\left(\vec{{x}}-\vec{\mu}\right)}^{{T}}\Sigma^{{-{{1}}}}{\left(\vec{{x}}-\vec{\mu}\right)}\right)}\right)}}}\ \text{ for }\ \vec{{x}}\in\vec{\mu}+{s}{p}{a}{n}{\left(\Sigma\right)}\subseteq\mathbb{R}^{{k}},\ \text{ where }\ \vec{\mu}\in\mathbb{R}^{{k}}{\quad\text{and}\quad}\Sigma\in\mathbb{R}^{{{k}\times{k}}},\ \text{ the covariance matrix.}$

Negative multinomial distribution

a generalization of the negative binomial distribution (NB(r, p)) to more than two outcomes

$\displaystyle{f{{\left({k}_{{1}},\ldots{k}_{{m}};{k}_{{0}},\vec{{p}}\right)}}}=\Gamma{\left({\sum_{{{i}={0}}}^{{m}}}{k}_{{i}}\right)}\frac{{{{p}_{{0}}^{{{k}_{{0}}}}}}}{{\Gamma{\left({k}_{{0}}\right)}}}{\prod_{{{i}={1}}}^{{m}}}{\left(\frac{{{p}_{{i}}^{{{k}_{{i}}}}}}{{{k}_{{i}}!}}\right)}\ \text{ for }\ {k}_{{i}}\in\mathbb{N},{1}\le{i}\le{m},\text{where }\ {k}_{{0}}\in\mathbb{N},\text{the number of failures before the experiment is stopped, }\ {\quad\text{and}\quad}\vec{{p}}\in\mathbb{R}^{{m}}\text{, the m-vector of success probabilities.}$

Wishart distribution

a generalization to multiple dimensions of the chi-squared distribution

$\displaystyle{{f}_{{p}}{\left({X};{n},{V}\right)}}=\frac{{1}}{{{2}^{{\frac{{{n}{p}}}{{2}}}}{\left|{V}\right|}^{{\frac{{n}}{{2}}}}\Gamma_{{p}}{\left(\frac{{n}}{{2}}\right)}}}{\left|{X}\right|}^{{\frac{{{n}-{p}-{1}}}{{2}}}}{e}^{{-\frac{{1}}{{2}}{t}{r}{\left({V}^{{-{{1}}}}{X}\right)}}} \ \text{ for }\ {X}\in\mathbb{R}^{{{p}\times{p}}}\ \text{ positive-definite, where }\ {n}\gt{p}-{1},{V}\in\mathbb{R}^{{{p}\times{p}}}\ \text{ positive-definite, and }\ \Gamma_{{p}}\text{ is the multivariate gamma function.}$

Phase-type distribution

a probability distribution constructed by a convolution of exponential distributions

$\displaystyle{f{{\left({x};\vec{\alpha},{S}\right)}}}=\vec{\alpha}{\exp{{\left({S}{x}\right)}}}{S}^{{0}} \ \text{ for }\ {x}\in{\left[{0},\infty\right)},\ \text{ where }\ \vec{\alpha}\ \text{ the probability row vector, }\ {S}\in\mathbb{R}^{{{m}\times{m}}},\ \text{ the subgenerator matrix, and }\ {S}^{{0}}=-{S}\cdot\vec{{1}},\ \text{ where }\ \vec{{1}}\ \text{ represents an }\ {m}\times{1}\ \text{ vector.}$

Beta negative binomial distribution

the probability distribution of a discrete random variable X equal to the number of failures needed to get r successes in a sequence of independent Bernoulli trials where the probability p of success on each trial is constant within any given experiment but is itself a random variable following a beta distribution, varying between different experiments

$\displaystyle\text{}{\left({k}{\mid}\alpha,\beta,{r}\right)}=\frac{{\Gamma{\left({r}+{k}\right)}}}{{{k}!\Gamma{\left({r}\right)}}}\frac{{{B}{\left(\alpha+{r},\beta+{k}\right)}}}{{{B}{\left(\alpha,\beta\right)}}}$

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

$\displaystyle{f{{\left({s}{t}{a}{t}{e}\right)}}}\propto{e}^{{-\frac{{E}}{{{k}{T}}}}}$

Gibbs measure

a probability measure frequently seen in many problems of probability theory and statistical mechanics

$\displaystyle{P}{\left({X}={x}\right)}=\frac{{1}}{{{Z}{\left(\beta\right)}}}{\exp{{\left(-\beta{E}{\left({x}\right)}\right)}}}$

2. Maxwell–Boltzmann distribution

a probability distribution for the speed of a particle within the gas - the magnitude of its velocity

$\displaystyle{f{{\left({x}\right)}}}={\sqrt[{()}]{{\frac{{2}}{\pi}}}}\frac{{{x}^{{2}}{e}^{{-{x}^{{2}}{/}{\left({2}{a}^{{2}}\right)}}}}}{{{a}^{{3}}}}$

Borel distribution

a discrete probability distribution, arising in contexts including branching processes and queueing theory

$\displaystyle\text{}{\left({n};\mu\right)}={P}{r}{\left({X}={n}\right)}=\frac{{{e}^{{-\mu{n}}}{\left(\mu{n}\right)}^{{{n}-{1}}}}}{{{n}!}}$

Logistic distribution

a continuous probability distribution which the cumulative distribution function is the logistic function

$\displaystyle{f{{\left({x};\mu,{s}\right)}}}=\frac{{e}^{{-\frac{{{x}-\mu}}{{s}}}}}{{{s}{\left({1}+{e}^{{-\frac{{{x}-\mu}}{{s}}}}\right)}^{{2}}}}\ \text{ for }\ {x}\in\mathbb{R}\ \text{ where }\ {s}\gt{0}$

Poisson binomial distribution

the probability distribution of the number of successes in a sequence of n independent yes/no experiments with success probabilities p1, p2, ... , pn

$\displaystyle\text{}{\left({k};{n},{p}_{{i}}\right)}=\sum_{{{A}\in{F}_{{k}}}}\prod_{{{i}\in{A}}}{p}_{{i}}\prod_{{{j}\in{A}^{{c}}}}{\left({1}-{p}_{{j}}\right)}$

Normal distribution

also gauss- or gaussian distribution, often used in the natural and social sciences to represent real-valued random variables whose distributions are not known

$\displaystyle{f{{\left({x};\mu,\sigma\right)}}}=\frac{{1}}{\sqrt{{{2}\pi\sigma^{{2}}}}}{e}^{{-\frac{{\left({x}-\mu\right)}^{{2}}}{{{2}\sigma^{{2}}}}}}\ \text{ for }\ {x}\in\mathbb{R}$

Normal-exponential-gamma distribution

a three-parameter family of continuous probability distributions

$\displaystyle{f{{\left({x};\mu,{k},\theta\right)}}}\propto{\exp{{\left(\frac{{\left({x}-\mu\right)}^{{2}}}{{{4}\theta^{{2}}}}\right)}}}{D}_{{-{2}{k}-{1}}}{\left(\frac{{\left|{x}-\mu\right|}}{\theta}\right)}\ \text{ for }\ {x}\in\mathbb{R}$

Normal-inverse Gaussian distribution

a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution

$\displaystyle{f{{\left({x};\mu,\alpha,\beta,\delta\right)}}}=\frac{{\alpha\delta{K}_{{1}}{\left(\alpha\sqrt{{\delta^{{2}}+{\left({x}-\mu\right)}^{{2}}}}\right)}}}{{\pi\sqrt{{\delta^{{2}}+{\left({x}-\mu\right)}^{{2}}}}}} {e}^{{\delta\gamma+\beta{\left({x}-\mu\right)}}}\ \text{ for}{x}\in\mathbb{R}$

Skew normal distribution

a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness

$\displaystyle{f{{\left({x};\xi,\omega,\alpha\right)}}}=\frac{{1}}{{\omega\pi}}{e}^{{-\frac{{\left({x}-\xi\right)}^{{2}}}{{{2}\omega^{{2}}}}}}{\int_{{-{\infty}}}^{{\alpha\frac{{{x}-\xi}}{\omega}}}}{e}^{{-\frac{{t}^{{2}}}{{2}}}}{\left.{d}{t}\right.} \ \text{ for }\ {x}\in\mathbb{R}$

Weibull distribution

giving a distribution for which the failure rate is proportional to a power of time

$\displaystyle{f{{\left({x};\lambda,{k}\right)}}}=\frac{{k}}{\lambda}{\left(\frac{{x}}{\lambda}\right)}^{{{k}-{1}}}{e}^{{-{\left({x}{/}\lambda\right)}^{{k}}}}\ \text{ for }\ {x}\ge{0}$

Cauchy distribution

the distribution of the X-intercept of a ray issuing from (x0,γ) with a uniformly distributed angle

$\displaystyle{f{{\left({x};{x}_{{0}},\gamma\right)}}}=\frac{{1}}{{\pi\gamma}}\frac{\gamma^{{2}}}{{{\left({x}-{x}_{{0}}\right)}^{{2}}+\gamma^{{2}}}}\ \text{ for}{x}\in\mathbb{R}$

Exponentially modified Gaussian distribution

describing the sum of independent normal and exponential random variables

$\displaystyle{f{{\left({x};\mu,\sigma,\lambda\right)}}}=\frac{\lambda}{{2}}{e}^{{\frac{\lambda}{{2}}{\left({2}\mu+\lambda\sigma^{{2}}-{2}{x}\right)}}}{e}{r}{f}{c}{\left(\frac{{\mu+\lambda\sigma^{{2}}-{x}}}{{\sqrt{{2}}\sigma}}\right)} \ \text{ for }\ {x}\in\mathbb{R}$

Generalized extreme value distribution

the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables

$\displaystyle\text{f(x;mu,sigma,xi)=1/sigma t(x)^(xi_1)e^(-t(x)) }\ \text{ for }\ \text{}{x}\in{\left\lbrace{\left({\left[\mu-\frac{\sigma}{\xi},+\infty\right)} {w}{h}{e}{n} \xi\gt{0}\right)},{\left(\mathbb{R} {w}{h}{e}{n} \xi={0}\right)},{\left({\left(-\infty,\mu-\frac{\sigma}{\xi}\right]} {w}{h}{e}{n} \xi\lt{0}\right)}\right.}$

the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables

$\displaystyle\text{f(x;mu,sigma,xi)=1/sigma t(x)^(xi_1)e^(-t(x)) }\ \text{ for }\ \text{}{x}\in{\left\lbrace{\left({\left[\mu-\frac{\sigma}{\xi},+\infty\right)} {w}{h}{e}{n} \xi\gt{0}\right)},{\left(\mathbb{R} {w}{h}{e}{n} \xi={0}\right)},{\left({\left(-\infty,\mu-\frac{\sigma}{\xi}\right]} {w}{h}{e}{n} \xi\lt{0}\right)}\right.}$

Generalized normal distribution version 1

known also as the exponential power distribution, or the generalized error distribution

$\displaystyle{f{{\left({x};\mu,\alpha,\beta\right)}}}=\frac{\beta}{{{2}\alpha\Gamma{\left({1}{/}\beta\right)}}}{e}^{{-{\left({\left|{x}-\mu\right|}{/}\alpha\right)}^{\beta}}}\ \text{ for }\ {x}\in\mathbb{R}$

Generalized normal distribution version 2

a family of continuous probability distributions in which the shape parameter can be used to introduce skew

$\displaystyle{f{{\left({x};\xi,\alpha,\kappa\right)}}}=\frac{{\phi{\left({y}\right)}}}{{\alpha-\kappa{\left({x}-\xi\right)}}}\ \text{ for }\ {x}\in\mathbb{R}$

Type-1 Gumbel distribution

mainly used in the analysis of extreme values and in survival analysis (also known as duration analysis or event-history modelling)

$\displaystyle{f{{\left({x};{a},{b}\right)}}}={a}{b}{e}^{{-{\left({b}{e}^{{-{a}{x}}}+{a}{x}\right)}}} \ \text{ for }\ {x}\in\mathbb{R}$

Voigt profile

a line profile resulting from the convolution of two broadening mechanisms, one of which alone would produce a Gaussian profile, and the other would produce a Lorentzian profile

$\displaystyle{f{{\left({x};\sigma,\gamma\right)}}}=\frac{{{R}{e}{\left[{w}{\left({z}\right)}\right]}}}{{\sigma\sqrt{{{2}\pi}}}} \ \text{ for x in }\ \mathbb{R}$

Generalized Pareto distribution

used to model the tails of another distribution

$\displaystyle{f{{\left({x};\mu,\sigma,\xi\right)}}}=\frac{{1}}{\sigma}{\left({1}+\xi{z}\right)}^{{-{\left({1}+{1}{/}\xi\right)}}}\ \text{ for }\ {x}\ge\mu\ \text{ when }\ \xi\ge{0},\ \text{ and for }\ {x}\in{\left[\mu,\mu-\frac{\sigma}{\xi}\right]}\ \text{ when }\ \xi\lt{0},$

used to model the tails of another distribution

$\displaystyle\text{f(x;mu,sigma,xi)=1/sigma (1+xiz)^(-(1+1//xi)) }\ \ \text{ for }\ \text{x >=mu}\ \text{ when }\ \text{xi>=0, }\ \ \text{ and for }\ \ \text{ x in[mu,mu-sigma/xi] }\ \ \text{ when }\ \text{}{i}\lt{0},$

Rectified Gaussian distribution

a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier)

$\displaystyle{f{{\left({x};\mu,\sigma\right)}}}=\Phi{\left(-\frac{\mu}{\sigma}\right)}\delta{\left({x}\right)}+\frac{{1}}{\sqrt{{{2}\pi\sigma^{{2}}}}}{e}^{{-\frac{{\left({x}-\mu\right)}^{{2}}}{{{2}\sigma^{{2}}}}}}{U}{\left({x}\right)}\ \text{ for }\ {x}\in\mathbb{R},$

Bernoulli distribution

the probability distribution of a random variable which takes value 1 with success probability p and value 0 with failure probability

$\displaystyle{f{{\left({k};{p}\right)}}}={\left\lbrace\matrix{{p}&{\quad\text{if}\quad}{k}={1}\\{1}-{p}&{\quad\text{if}\quad}{k}={0}}\right.}$

Extended negative binomial distribution

a discrete probability distribution extending the negative binomial distribution; a truncated version of the negative binomial distribution for which estimation methods have been studied

$\displaystyle{f{{\left({k};{m},{r},{p}\right)}}}={\left\lbrace\matrix{{0}&{\quad\text{if}\quad}{k}\in{\left\lbrace{0},{1},\ldots,{m}-{1}\right\rbrace}\\\frac{{{\left(\matrix{{k}+{r}-{1}\\{k}}\right)}{p}^{{k}}}}{{{\left({1}-{p}\right)}^{{-{r}}}-{\sum_{{{j}={0}}}^{{{m}-{1}}}}{\left(\matrix{{j}+{r}-{1}\\{j}}\right)}{p}^{{j}}}}&{\quad\text{if}\quad} {k}\in\mathbb{N}\ \text{ with }\ {k}\ge{m}}\right.}$

Geometric distribution (success)

The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}

$\displaystyle{f{{\left({k}\right)}}}={P}{r}{\left({X}={k}\right)}={\left({1}-{p}\right)}^{{{k}-{1}}}{p}$

Geometric distribution (failures)

The probability distribution of the number Y = X − 1 of failures before the first success, supported on the set { 0, 1, 2, 3, ... }

$\displaystyle{f{{\left({k}\right)}}}={P}{r}{\left({Y}={k}\right)}={\left({1}-{p}\right)}^{{k}}{p}$

Logarithmic distribution

a discrete probability distribution derived from the Maclaurin series expansion

$\displaystyle{f{{\left({k}\right)}}}=\frac{{-{1}}}{{{\ln{{\left({1}-{p}\right)}}}}}\frac{{{p}^{{k}}}}{{k}}$

Negative binomial distribution

a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs

$\displaystyle{f{{\left({k};{r},{p}\right)}}}={P}{r}{\left({X}={k}\right)}={\left(\matrix{{k}+{r}-{1}\\{k}}\right)}{p}^{{k}}{\left({1}-{p}\right)}^{{r}}$

Geometric stable distribution

a type of leptokurtic probability distribution

$\displaystyle\text{}\text{not analytically expressible in general}\text{}$

a type of leptokurtic probability distribution

$\displaystyle\text{}\text{not analytically expressible}\text{}$

Hyperbolic distribution

a continuous probability distribution characterized by the logarithm of the probability density function being a hyperbola

$\displaystyle{f{{\left({x};\mu,\alpha,\beta,\delta\right)}}}=\frac{\gamma}{{{2}\alpha\delta{K}_{{1}}{\left(\delta\gamma\right)}}}{\exp{{\left(-\alpha\sqrt{{\delta^{{2}}+{\left({x}-\mu\right)}^{{2}}}}+\beta{\left({x}-\mu\right)}\right)}}}\ \text{ for }\ {x}\in\mathbb{R}$

Hyperbolic secant distribution

a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function

$\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{2}}{\sech{{\left(\frac{\pi}{{2}}{x}\right)}}}\ \text{ for }\ {x}\in\mathbb{R}$

Johnson S_U distribution

a transformation of the normal distribution

$\displaystyle{f{{\left({x};\gamma,\xi,\delta,\lambda\right)}}}=\frac{\delta}{{\lambda\sqrt{{{2}\pi}}}}\frac{{1}}{\sqrt{{{1}+{\left(\frac{{{x}-\xi}}{\lambda}\right)}^{{2}}}}}{e}^{{-\frac{{1}}{{2}}{\left(\gamma+\delta{{\text{sinh}}^{{-{1}}}{\left(\frac{{{x}-\xi}}{\lambda}\right)}}\right)}^{{2}}}}\ \text{ for }\ {x}\in\mathbb{R}$

Landau distribution

a special case of the stable distribution.

$\displaystyle{f{{\left({x};{c}\right)}}}=\frac{{1}}{{{2}\pi{i}}}{\int_{{{c}-{i}\infty}}^{{{c}+{i}\infty}}}{e}^{{{s}{\ln{{s}}}+{x}{s}}}{d}{s}\ \text{ for x in RR }\$

Laplace distribution

governing the difference between two independent identically distributed exponential random variables

$\displaystyle{f{{\left({x};\mu,{b}\right)}}}=\frac{{1}}{{{2}{b}}}{\exp{{\left(-\frac{{\left|{x}-\mu\right|}}{{b}}\right)}}}\ \text{ for }\ {x}\in\mathbb{R}\ \text{ where }\ {b}\gt{0}$

Inverse-Wishart distribution

used as the conjugate prior for the covariance matrix of a multivariate normal distribution.

$\displaystyle{{f}_{{p}}{\left({X};\nu,{P}{s}{i}\right)}}=\frac{{{\left|{P}{s}{i}\right|}^{{\frac{\nu}{{2}}}}}}{{{2}^{{\frac{{\nu{p}}}{{2}}}}\Gamma_{{p}}{\left(\frac{\nu}{{2}}\right)}}}{\left|{X}\right|}^{{-\frac{{\nu+{p}+{1}}}{{2}}}}{e}^{{-\frac{{1}}{{2}}{t}{r}{\left({P}{s}{i}{X}^{{-{{1}}}}\right)}}} \ \text{ for }\ {X}\in\mathbb{R}^{{{p}\times{p}}}\ \text{ positive-definite, where }\ \nu\gt{p}-{1},{P}{s}{i}\in\mathbb{R}^{{{p}\times{p}}}\ \text{ positive-definite, and }\ \Gamma_{{p}}\text{ is the multivariate gamma function.}$

Matrix normal distribution

a generalization of the multivariate normal distribution to matrix-valued random variables

$\displaystyle{{f}_{{{n}\times{p}}}{\left({X};{M},{U},{V}\right)}}=\frac{{\exp{{\left(-\frac{{1}}{{2}}{t}{r}{\left[{V}^{{-{{1}}}}{\left({X}-{M}\right)}^{{T}}{U}^{{-{{1}}}}{\left({X}-{M}\right)}\right]}\right)}}}}{{{\left({2}\pi\right)}^{{{n}{p}{/}{2}}}{\left|{V}\right|}^{{{n}{/}{2}}}{\left|{U}\right|}^{{{p}{/}{2}}}}} \ \text{ for }\ {X}\in\mathbb{R}^{{{n}\times{p}}}\text{, where }\ {M}\in\mathbb{R}^{{{n}\times{p}}}\text{, }\ {U}\in\mathbb{R}^{{{n}\times{n}}}\text{, positive-definite, and }\ {V} \in\mathbb{R}^{{{p}\times{p}}},\ \text{ positive-definite.}$

Matrix t-distribution

the generalization of the multivariate t-distribution from vectors to matrices

$\displaystyle{{f}_{{{n}\times{p}}}{\left({X};\nu,{M},\Sigma,\Omega\right)}}={K}\times{\left|{I}_{{n}}+\Sigma^{{-{{1}}}}{\left({X}-{M}\right)}\Omega^{{-{{1}}}}{\left({X}-{M}\right)}^{{T}}\right|}^{{-\frac{{\nu+{n}+{p}+{1}}}{{2}}}} \ \text{ for }\ {X}\in\mathbb{R}^{{{n}\times{p}}}\text{, where }\ {M}\in\mathbb{R}^{{{n}\times{p}}}\text{, }\ \Sigma\in\mathbb{R}^{{{n}\times{n}}}\text{, positive-definite, and }\ \Omega\in\mathbb{R}^{{{p}\times{p}}},\ \text{ positive-definite;}$

Cantor distribution

the probability distribution whose cumulative distribution function is the Cantor function

$\displaystyle\text{}\text{none}\text{}$

Pearson distribution

a system originally devised in an effort to model visibly skewed observations

$\displaystyle\text{}\text{Any valid solution to the differential equation: }\ \text{}$

a system originally devised in an effort to model visibly skewed observations

$\displaystyle\text{}\text{Any valid solution to the differential equation: }\ \text{}$

Rayleigh distribution

observed when the overall magnitude of a vector is related to its directional components

$\displaystyle{f{{\left({x};\sigma\right)}}}=\frac{{x}}{\sigma^{{2}}}{e}^{{-{x}^{{2}}{/}{\left({2}\sigma^{{2}}\right)}}}\ \text{ for }\ {x}\ge{0}\ \text{ where }\ \sigma\gt{0}$

Rayleigh mixture distribution

a weighted mixture of multiple probability distributions where the weightings are equal to the weightings of a Rayleigh distribution

$\displaystyle{f{{\left({x};\sigma,{n}\right)}}}={\int_{{0}}^{\infty}}\frac{{{r}{e}^{{-{r}^{{2}}{/}{\left({2}\sigma^{{2}}\right)}}}}}{\sigma^{{2}}}\tau{\left({x},{r};{n}\right)}{d}{r} \ \text{ for }\ {x}\ge{0}\ \text{ where }\ \sigma\gt{0}$

Rice distribution

the probability distribution of the magnitude of a circular bivariate normal random variable with potentially non-zero mean

$\displaystyle{f{{\left({x};\nu,\sigma\right)}}}=\frac{{x}}{\sigma^{{2}}}{\exp{{\left(-\frac{{{x}^{{2}}+\nu^{{2}}}}{{{2}\sigma^{{2}}}}\right)}}}{I}_{{0}}{\left(\frac{{{x}\nu}}{\sigma^{{2}}}\right)}\ \text{ for }\ {x}\ge{0}\ \text{ where }\ \nu,\sigma\ge{0}$

Shifted Gompertz distribution

the distribution of the largest of two independent random variables one of which has an exponential distribution and the other has a Gumbel distribution

$\displaystyle{f{{\left({x};{b},\eta\right)}}}={b}{e}^{{-{b}{x}}}{e}^{{-\eta{e}^{{-{b}{x}}}}}{\left[{1}+\eta{\left({1}-{e}^{{-{b}{x}}}\right)}\right]}\ \text{ for }\ {x}\ge{0}\ \text{ where }\ {b},\eta\ge{0}$

Type-2 Gumbel distribution

similar to the Weibull distributions

$\displaystyle{f{{\left({x};{a},{b}\right)}}}={a}{b}{x}^{{-{a}-{1}}}{e}^{{-{b}{x}^{{-{a}}}}}\ \text{ for }\ {x}\gt{0}$

Beta-binomial distribution

the binomial distribution in which the probability of success at each trial is not fixed but random and follows the beta distribution

$\displaystyle{f{{\left({k}{\mid}{n},\alpha,\beta\right)}}}={\left(\matrix{{n}\\{k}}\right)}\frac{{{B}{\left({k}+\alpha,{n}-{k}+\beta\right)}}}{{{B}{\left(\alpha,\beta\right)}}}$

Kumaraswamy distribution

similar to the Beta distribution, but much simpler to use especially in simulation studies due to the simple closed form of both its probability density function and cumulative distribution function

$\displaystyle{f{{\left({x};{a},{b}\right)}}}={a}{b}{x}^{{{a}-{1}}}{\left({1}-{x}^{{a}}\right)}^{{{b}-{1}}}\ \text{ for }\ {x}\in{\left[{0},{1}\right]}$

Raised cosine distribution

a continuous probability distribution with a cosine-formed probability density function

$\displaystyle{f{{\left({x};\mu,{s}\right)}}}=\frac{{1}}{{{2}{s}}}{\left[{1}+{\cos{{\left(\frac{{{x}-\mu}}{{s}}\pi\right)}}}\right]}\ \text{ for }\ {x}\in{\left[\mu-{s},\mu+{s}\right]}$

Reciprocal distribution

characterised by its probability density function, within the support of the distribution, being proportional to the reciprocal of the variable.

$\displaystyle{f{{\left({x};{a},{b}\right)}}}=\frac{{1}}{{{x}{\left[{\ln{{\left({b}\right)}}}-{\ln{{\left({a}\right)}}}\right]}}}\ \text{ for }\ {x}\in{\left[{a},{b}\right]}\ \text{ where }\ {a}\gt\gt{0}$

Triangular distribution

a continuous probability distribution with triangle-formed probability density function, where the triangle is limited by lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b

$\displaystyle{f{{\left({x};{a},{b},{c}\right)}}}={\left\lbrace{\left(\frac{{{2}{\left({x}-{a}\right)}}}{{{\left({b}-{a}\right)}{\left({c}-{a}\right)}}}\ \text{ for }\ {x}\in{\left[{a},{c}\right)}\right)},{\left(\frac{{2}}{{{b}-{a}}}\text{ for }\ {x}={c}\right)},{\left(\frac{{{2}{\left({b}-{x}\right)}}}{{{\left({b}-{a}\right)}{\left({b}-{c}\right)}}}\ \text{ for }\ {x}\in{\left({c},{b}\right]}\right)}\right.}\ \text{ for }\ {x}\in{\left[{a},{b}\right]}\ \text{ where }\ {a}\lt{b}{\quad\text{and}\quad}{a}\le{c}\le{b}$

Truncated normal distribution

the probability distribution of a normally distributed random variable whose value is either bounded below or above (or both)

$\displaystyle{f{{\left({x};\mu,\sigma,{a},{b}\right)}}}=\frac{{\phi{\left(\frac{{{x}-\mu}}{\sigma}\right)}}}{{\Phi{\left(\frac{{{b}-\mu}}{\sigma}\right)}-\Phi{\left(\frac{{{a}-\mu}}{\sigma}\right)}}}\ \text{ for }\ {a}\le{x}\le{b}\text{, where }\ \phi\ \text{ is the probability density function of the standard normal distribution and }\ \Phi\ \text{ is its cumulative distribution function.}$

a continuous probability distribution defined by a unique quadratic function with lower limit a and upper limit b

$\displaystyle{f{{\left({x};{a},{b},\alpha,\beta\right)}}}=\alpha{\left({x}-\beta\right)}^{{2}}\ \text{ for }\ {x}\in{\left[{a},{b}\right]}$

Von Mises distribution

a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution

$\displaystyle{f{{\left({x};\mu,\kappa\right)}}}=\frac{{e}^{{\kappa{\cos{{\left({x}-\mu\right)}}}}}}{{{2}\pi{I}_{{0}}{\left(\kappa\right)}}}\ \text{ for }\ {x}\in\text{any interval of length }\ {2}\pi$

Von Mises–Fisher distribution

a probability distribution on the (p-1)-dimensional sphere in p-dimensional R-vectorial space, applied, for example, to model the interaction of electric dipoles in an electric field

$\displaystyle{{f}_{{p}}{\left(\vec{{x}};\vec{\mu},\kappa\right)}}={C}_{{p}}{\left(\kappa\right)}{\exp{{\left(\kappa\vec{\mu}^{{T}}\vec{{x}}\right)}}} \ \text{ for the random p-dimensional unit vector }\ \vec{{x}},$

Degenerate distribution

the probability distribution of a random variable which only takes a single value

$\displaystyle{f{{\left({k}_{{0}}\right)}}}=\delta{\left({x}-{k}_{{0}}\right)}$

Wigner semicircle distribution

the probability distribution supported on the interval [−R, R] the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized

$\displaystyle{f{{\left({x};{R}\right)}}}=\frac{{2}}{{\pi{R}^{{2}}}}\sqrt{{{R}^{{2}}-{x}^{{2}}}} \ \text{ for }\ {x}\in{\left[-{R},{R}\right]}$

Beta prime distribution

the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in odds

$\displaystyle{f{{\left({x};\alpha,\beta\right)}}}=\frac{{{x}^{{\alpha-{1}}}{\left({1}+{x}\right)}^{{-\alpha-\beta}}}}{{{B}{\left(\alpha,\beta\right)}}}\ \text{ for }\ {x}\gt{0}$

Beta distribution

a distribution which model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines: allele frequencies in population genetics; time allocation in project management; sunshine data; variability of soil properties; heterogeneity in the probability of HIV transmission

$\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{{B}{\left(\alpha,\beta\right)}}}{x}^{{\alpha-{1}}}{\left({1}-{x}\right)}^{{\beta-{1}}}\ \text{ for }\ {x}\in{\left[{0},{1}\right]}$

Logit-normal distribution

a probability distribution of a random variable whose logit has a normal distribution

$\displaystyle{f{{\left({x};\mu,\sigma\right)}}}=\frac{{1}}{{\sigma\sqrt{{{2}\pi}}}}\frac{{1}}{{{x}{\left({1}-{x}\right)}}}{e}^{{-\frac{{{\left({\log{{i}}}{t}{\left({x}\right)}-\mu\right)}^{{2}}}}{{{2}\sigma^{{2}}}}}}\ \text{ for }\ {x}\in{\left[{0},{1}\right]}$

Irwin–Hall distribution

a distribution for a random variable defined as sum of a number of independent random variables, each having a uniform distribution

$\displaystyle{f{{\left({x};{n}\right)}}}=\frac{{1}}{{{2}{\left({n}-{1}\right)}!}}{\sum_{{{k}={0}}}^{{{n}}}}{\left(-{1}\right)}^{{k}}{\left(\matrix{{n}\\{k}}\right)}{\left({x}-{k}\right)}^{{{n}-{1}}}{s}{g}{n}{\left({x}-{k}\right)}\ \text{ for }\ {x}\in{\left[{0},{n}\right]}$

Bates distribution

a probability distribution of the mean of a number of statistically independent uniformly distributed random variables on the unit interval.

$\displaystyle{f{{\left({x};{n}\right)}}}=\frac{{n}}{{{2}{\left({n}-{1}\right)}!}}{\sum_{{{k}={0}}}^{{n}}}{\left(-{1}\right)}^{{k}}{\left(\matrix{{n}\\{k}}\right)}{\left({n}{x}-{k}\right)}^{{{n}-{1}}}{s}{g}{n}{\left({n}{x}-{k}\right)}\ \text{ for }\ {x}\in{\left[{a},{b}\right]}$

Kent distribution

a probability distribution on the two-dimensional unit sphere, an analogue on the two-dimensional unit sphere of the bivariate normal distribution with an unconstrained covariance matrix

$\displaystyle{f{{\left(\vec{{x}};\vec{\gamma}_{{1}},\vec{\gamma}_{{2}},\vec{\gamma}_{{3}},\kappa,\beta\right)}}}=\frac{{1}}{{\vec{{c}}{\left(\kappa,\beta\right)}}}{\exp{{\left\lbrace\kappa\vec{\gamma}_{{1}}\cdot\vec{{x}}+\beta{\left[{\left(\vec{\gamma}_{{2}}\cdot\vec{{x}}\right)}^{{2}}-{\left(\vec{\gamma}_{{3}}\cdot\vec{{x}}\right)}^{{2}}\right]}\right\rbrace}}} \ \text{ for }\ \vec{{x}}\ \text{ a three-dimensional unit vector}$

Gamma/Gompertz distribution

used as an aggregate-level model of customer lifetime and a model of mortality risks

$\displaystyle{f{{\left({x};{b},{s},\beta\right)}}}=\frac{{{b}{s}{e}^{{{b}{x}}}{e}\beta^{{s}}}}{{\left(\beta-{1}+{e}^{{{b}{x}}}\right)}^{{{s}+{1}}}}\ \text{ for }\ {x}\ge{0} \ \text{ where }\ {b},{s},\beta&\gt{0}$

Gompertz distribution

applied to describe the distribution of adult lifespans by demographers and actuaries

$\displaystyle{f{{\left({x};\eta,{b}\right)}}}={b}\eta{e}^{{{b}{x}}}{e}^{\eta}{\exp{{\left(-\eta{e}^{{{b}{x}}}\right)}}}\ \text{ for }\ {x}\ge{0} \ \text{ where }\ \eta,{b}\gt{0}$

Hypergeometric distribution

a discrete probability distribution that describes the probability of k successes in n draws, without replacement, from a finite population of size N that contains exactly K successes, wherein each draw is either a success or a failure

$\displaystyle\text{}{\left({k};{n},{N},{K}\right)}=\frac{{{\left(\matrix{{K}\\{k}}\right)}{\left(\matrix{{N}-{K}\\{n}-{k}}\right)}}}{{{\left(\matrix{{N}\\{n}}\right)}}}$

Half-normal distribution

a special case of the folded normal distribution

$\displaystyle{f{{\left({x};\sigma\right)}}}=\frac{\sqrt{{{2}}}}{{\sigma\sqrt{{\pi}}}}{\exp{{\left(-\frac{{x}^{{2}}}{{{2}\sigma^{{2}}}}\right)}}}\ \text{ for }\ {x}\gt{0}$

Inverse Gaussian distribution

the distribution of the time a Brownian Motion with positive drift takes to reach a fixed positive level

$\displaystyle{f{{\left({x};\mu,\lambda\right)}}}={\left(\frac{\lambda}{{{2}\pi{x}^{{3}}}}\right)}^{{{1}{/}{2}}}{\exp{{\left(\frac{{-\lambda{\left({x}-\mu\right)}^{{2}}}}{{{2}\mu^{{2}}{x}}}\right)}}}\ \text{ for }\ {x}\gt{0}\ \text{ where }\ \mu,\lambda\gt{0}$

Birnbaum–Saunders distribution

a probability distribution used extensively in reliability applications to model failure times

$\displaystyle{f{{\left({x};\gamma,\mu,\beta\right)}}}=\frac{{\sqrt{{\frac{{{x}-\mu}}{\beta}}}+\sqrt{{\frac{\beta}{{{x}-\mu}}}}}}{{{2}\gamma{\left({x}-\mu\right)}}}\phi{\left(\frac{{\sqrt{{\frac{{{x}-\mu}}{\beta}}}-\sqrt{{\frac{\beta}{{{x}-\mu}}}}}}{{\gamma}}\right)}\ \text{ for }\ {x}\gt\mu$

Chi distribution

a continuous probability distribution. It is the distribution of the square root of the sum of squares of independent random variables having a standard normal distribution

$\displaystyle{f{{\left({x};{k}\right)}}}=\frac{{{2}^{{{1}-\frac{{k}}{{2}}}}{x}^{{{k}-{1}}}{e}^{{-\frac{{x}^{{2}}}{{2}}}}}}{{\Gamma{\left({k}{/}{2}\right)}}}\ \text{ for }\ {x}\ge{0}$

Chi-squared (Chi2) distribution

the distribution of a sum of the squares of k independent standard normal random variables

$\displaystyle{f{{\left({x};{k}\right)}}}=\frac{{{x}^{{\frac{{k}}{{2}}-{1}}}{e}^{{-\frac{{x}}{{2}}}}}}{{{2}^{{\frac{{k}}{{2}}}}\Gamma{\left({k}{/}{2}\right)}}}\ \text{ for }\ {x}\ge{0}$

Dagum distribution

a special case of the Generalized Beta II (GB2) distribution (a generalization of the Beta prime distribution)

$\displaystyle{f{{\left({x};{a},{b},{p}\right)}}}=\frac{{{a}{p}}}{{x}}\frac{{{\left({x}{/}{b}\right)}^{{{a}{p}}}}}{{{\left({\left({x}{/}{b}\right)}^{{a}}+{1}\right)}^{{{p}+{1}}}}}\ \text{ for }\ {x}\gt{0}$

Uniform distribution (discrete)

a symmetric probability distribution whereby a finite number of values are equally likely to be observed

$\displaystyle{f{{\left({k};{n}\right)}}}=\frac{{1}}{{n}}$

Exponential distribution

the probability distribution that describes the time between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate

$\displaystyle{f{{\left({x};\lambda\right)}}}=\lambda{e}^{{-\lambda{x}}} \ \text{ for }\ {x}\ge{0}$

Parabolic fractal distribution

a type of discrete probability distribution in which the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank

$\displaystyle{f{{\left({n};{b},{c}\right)}}}\propto{n}^{{-{b}}}{\exp{{\left(-{c}{\left({\log{{n}}}\right)}^{{2}}\right)}}}$

Poisson distribution

a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event

$\displaystyle{f{{\left({k};\lambda\right)}}}=\frac{{\lambda^{{k}}{e}^{{-\lambda}}}}{{{k}!}}$

Conway–Maxwell–Poisson distribution

a solution to handling queueing systems with state-dependent service rates

$\displaystyle{f{{\left({x};\lambda,\nu\right)}}}={P}{r}{\left({X}={x}\right)}=\frac{{\lambda^{{x}}}}{{{\left({x}!\right)}^{\nu}}}\frac{{1}}{{{Z}{\left(\lambda,\nu\right)}}}$

Zero-truncated Poisson distribution

a certain discrete probability distribution whose support is the set of positive integers

$\displaystyle\text{}{\left({k};\lambda\right)}={P}{\left({X}={k}{\mid}{k}\gt{0}\right)}=\frac{{{f{{\left({k};\lambda\right)}}}}}{{{1}-{f{{\left({0};\lambda\right)}}}}}=\frac{{\lambda^{{k}}}}{{{\left({e}^{\lambda}-{1}\right)}{k}!}}$

a discrete probability distribution where a random variate X has a 50% chance of being either +1 or -1

$\displaystyle{f{{\left({k}\right)}}}={\left\lbrace\matrix{{1}{/}{2}&{\quad\text{if}\quad}{k}=-{1}\\{1}{/}{2}&{\quad\text{if}\quad}{k}=+{1}\\{0}&\ \text{ otherwise}}\right.}$

Skellam distribution

the discrete probability distribution of the difference n1-n2 of two statistically independent random variables N1 and N2 each having Poisson distributions with different expected values mu1 and mu2

$\displaystyle\text{}{\left({k};\mu_{{1}},\mu_{{2}}\right)}={e}^{{-{\left(\mu_{{1}}+\mu_{{2}}\right)}{\left(\frac{\mu_{{1}}}{\mu_{{2}}}\right)}^{{{k}{/}{2}}}{I}_{{k}}{\left({2}\sqrt{{\mu_{{1}}\mu_{{2}}}}\right)}}}$

Yule–Simon distribution

a preferential attachment process

$\displaystyle{f{{\left({k};\rho\right)}}}=\rho{B}{\left({k},\rho+{1}\right)}$

Zeta distribution

the normalization of the Zipf distribution

$\displaystyle{f{{\left({k};{s}\right)}}}={k}^{{-{s}}}{/}\zeta{\left({s}\right)}$

Zipf–Mandelbrot law

a power-law distribution on ranked data

$\displaystyle\text{}{\left({k};{N},{q},{s}\right)}=\frac{{{1}{/}{\left({k}+{q}\right)}^{{s}}}}{{{H}_{{{N},{q},{s}}}}}$

Binomial distribution

the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p

$\displaystyle{f{{\left({k};{n},{p}\right)}}}={\left(\matrix{{n}\\{k}}\right)}{p}^{{k}}{\left({1}-{p}\right)}^{{{n}-{k}}}$

Arcsine distribution

a special case of the beta distribution with α = β = 1/2

$\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{\pi\sqrt{{{x}{\left({1}-{x}\right)}}}}}\ \text{ for }\ {x}\in{\left[{0},{1}\right]}$

F-distribution

arising frequently as the null distribution of a test statistic, most notably in the analysis of variance

$\displaystyle{f{{\left({x};{d}_{{1}},{d}_{{2}}\right)}}}=\frac{{\sqrt{{\frac{{{\left({d}_{{1}}{x}\right)}^{{{d}_{{1}}}}{\left({d}_{{2}}\right)}^{{{d}_{{2}}}}}}{{{\left({d}_{{1}}{x}+{d}_{{2}}\right)}^{{{d}_{{1}}+{d}_{{2}}}}}}}}}}{{{x}{B}{\left({d}_{{1}}{/}{2},{d}_{{2}}{/}{2}\right)}}} \ \text{ for }\ {x}\ge{0}$

Folded normal distribution

a probability distribution related to the normal distribution, of which the probability mass to the left of the x = 0 is "folded" over by taking the absolute value

$\displaystyle\text{the random variable Y = |X| has a folded normal distribution: }\ {{f}_{{Y}}{\left({x};\mu,\sigma\right)}}=\frac{{1}}{{\sigma\sqrt{{{2}\pi}}}}{e}^{{-\frac{{\left({x}-\mu\right)}^{{2}}}{{{2}\sigma^{{2}}}}}}+ \frac{{1}}{{\sigma\sqrt{{{2}\pi}}}}{e}^{{-\frac{{\left({x}+\mu\right)}^{{2}}}{{{2}\sigma^{{2}}}}}} \ \text{ for }\ {x}\ge{0}$

Fréchet distribution

a special case of the generalized extreme value distribution

$\displaystyle{f{{\left({x};\alpha,{s},{m}\right)}}}=\frac{\alpha}{{s}}{\left(\frac{{{x}-{m}}}{{s}}\right)}^{{-{1}-\alpha}}{e}^{{-{\left(\frac{{{x}-{m}}}{{s}}\right)}^{{-{\alpha}}}}} \ \text{ for }\ {x}\gt{m}$

Gamma distribution

the maximum entropy probability distribution for a random variable

$\displaystyle{f{{\left({x};\alpha,\beta\right)}}}=\frac{\beta^{\alpha}}{{\Gamma{\left(\alpha\right)}}}{x}^{{\alpha-{1}}}{e}^{{-\beta{x}}} \ \text{ for }\ {x}\gt{0}$

a continuous probability distribution of a random variable whose logarithm is normally distributed

$\displaystyle\text{f(x;mu,sigma)=1/(x sigma sqrt(2pi))e^(-(lnx-mu)^2/(2sigma^2)) }\ \ \text{ for }\ \text{x>0 }\ \ \text{ where }\ \text{sigma>0}$

Generalized gamma distribution

a generalization of the two-parameter gamma distribution

$\displaystyle{f{{\left({x};{a},{d},{p}\right)}}}=\frac{{{p}{/}{a}^{{d}}}}{{\Gamma{\left({d}{/}{p}\right)}}}{x}^{{{d}-{1}}}{e}^{{-{\left({x}{/}{a}\right)}^{{p}}}} \ \text{ for }\ {x}\gt{0},\ \text{ where }\ {a}\gt{0},{d}\gt{0}{\quad\text{and}\quad}{p}\gt{0}$

Lévy distribution

a van der Waals profile, with frequency as the dependent variable, a special case of the inverse-gamma distribution

$\displaystyle{f{{\left({x};\mu,{c}\right)}}}=\sqrt{{\frac{{c}}{{{2}\pi}}}}\frac{{e}^{{-\frac{{c}}{{{2}{\left({x}-\mu\right)}}}}}}{{\left({x}-\mu\right)}^{{{3}{/}{2}}}}\ \text{ for }\ {x}\ge\mu\ \text{ where }\ {c}\gt{0}$

Log-Cauchy distribution

a probability distribution of a random variable whose logarithm is distributed in accordance with a Cauchy distribution

$\displaystyle{f{{\left({x};\mu,\sigma\right)}}}=\frac{{1}}{{{x}\pi\sigma{\left[{1}+{\left(\frac{{{\ln{{x}}}-\mu}}{\sigma}\right)}^{{2}}\right]}}}\ \text{ for}\gt{0}\ \text{ where }\ {s}{i}{g}{m}\gt{0}$

Log-Laplace distribution

the probability distribution of a random variable whose logarithm has a Laplace distribution

$\displaystyle{f{{\left({x};\mu,{b}\right)}}}=\frac{{1}}{{{2}{b}{x}}}{\exp{{\left(-\frac{{\left|{\ln{{x}}}-\mu\right|}}{{b}}\right)}}}$

Log-logistic distribution

the probability distribution of a random variable whose logarithm has a logistic distribution

$\displaystyle{f{{\left({x};\alpha,\beta\right)}}}=\frac{{{\left(\beta{/}\alpha\right)}{\left({x}{/}\alpha\right)}^{{\beta-{1}}}}}{{\left({1}+{\left({x}{/}\alpha\right)}^{\beta}\right)}^{{2}}}\ \text{ for }\ {x}\ge{0}\ \text{ where }\ \alpha,\beta\gt{0}$

Nakagami distribution

a probability distribution related to the gamma distribution, used to model attenuation of wireless signals traversing multiple paths

$\displaystyle{f{{\left({x};{m},\Omega\right)}}}=\frac{{{2}{m}^{{m}}}}{{\Gamma{\left({m}\right)}\Omega^{{m}}}}{x}^{{{2}{m}-{1}}}{\exp{{\left(-\frac{{m}}{\Omega}{x}^{{2}}\right)}}}\ \text{ for }\ \gt{0}\ \text{ where }\ {m}\ge\frac{{1}}{{2}}{\quad\text{and}\quad}\Omega\gt{0}$

Pareto distribution

a power law probability distribution that is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena

$\displaystyle{f{{\left({x};\alpha,{x}_{{m}}\right)}}}=\frac{{\alpha{{x}_{{m}}^{\alpha}}}}{{x}^{{\alpha+{1}}}}\ \text{ for }\ {x}\ge{x}_{{m}}\ \text{ where }\ {x}_{{m}}\ge{0}{\quad\text{and}\quad}\alpha\gt{0}$

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