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The '''Mittag-Leffler distributions''' are two families of [[probability distributions]] on the half-line <math>[0,\infty)</math>. They are parametrized by a real For any complex <math>\alpha</math> whose real part is positive, the series ...

...nts are ring-valued, it is not necessarily possible to multiply two formal distributions. ...s values in a space of [[endomorphism algebra|endomorphism]]-valued formal distributions.<ref name="kac">{{cite book |last1=Kac |first1=Victor G. |title=Vertex alge ...

name =Complex inverse Wishart Distribution| *<math>\mathcal{C}\Gamma_p</math> is the complex [[multivariate gamma function]] ...

{{about|limits of sequences of generalized functions|limits of probability distributions|Convergence of random variables#Convergence in distribution}} ...in the theory of [[generalized function]]s, the '''limit of a sequence of distributions''' is the distribution that sequence approaches. The distance, suitably qua ...

<math>\Gamma(X) </math> the set of Gaussian distributions on the group <math>X</math>, and by <math>\Gamma^s(X) </math> the set of sy ...tps://bookstore.ams.org/mmono-116/ G.M. Feldman. Arithmetic of probability distributions and characterization problems on Abelian groups, Transl. Math. Monographs. ...

...stitution|Jacobian change of variables]]. The number ''m'' can be real or complex. ...ious [[power function]]s. In addition to the power functions, homogeneous distributions on '''R''' include the [[Dirac delta function]] and its derivatives. ...

| name = Complex Wishart ...cal{C}\widetilde{\mathbf{\Gamma}}_p</math> is the <math>p</math>-variate [[complex multivariate gamma function]] ...

...nce'' (WCD) extends this idea to more complex problems, such as non-normal distributions and multiple specifications. Although the strict mathematical formalism is complex, in a simplified interpretation, WCD is the maximum of all possible ''perfo ...

...gram |last2=Rubin |first2=Herman |date=1964-03-01 |title=Multivariate Beta Distributions and Independence Properties of the Wishart Distribution |url=http://project ...ally derived by.<ref name="olkinrubin1964" /> When considering independent distributions, ...

...for [[scale parameter]]s. In the [[algebra of random variables]], inverse distributions are special cases of the class of [[ratio distribution]]s, in which the num With ''k'' = 1, the distributions of ''X'' and 1&nbsp;/&nbsp;''X'' are identical (''X'' is then [[Cauchy dist ...

...|325px|Symmetric stable distributions]]<br /><small>Symmetric ''α''-stable distributions with unit scale factor; ''α''=1.5 (blue line) represents the Holtsmark dist ...= [[File:Levy distributionCDF.svg|325px|CDF's for symmetric ''α''-stable distributions; ''α''=3/2 represents the Holtsmark distribution]] ...

== Generalized Definition for Complex Index Sets == ...stationary increments can be generalized to stochastic processes with more complex index sets <math> T </math>. ...

{{Short description|Statistical distribution of complex random variables}} | name = Complex normal ...

...rienne W. |last3=Kotz |first3=Samuel |date=2005 |title=Univariate Discrete Distributions |publisher=Wiley |isbn=0-471-27246-9}} §6.2.2 (p.253–254)</ref> with parame which holds for any complex-values <math>m</math> and <math>n</math> and any non-negative integer <math ...

==Probability distributions== ...enphases'''. In the CSE each of these ''n'' eigenvalues appears twice. The distributions have [[probability density function|densities]] with respect to the eigenan ...

{{Short description|Family of distributions that generalize the multivariate normal distribution}} ...icularly for spherical distributions (which are defined below). Elliptical distributions are also used in [[robust statistics]] to evaluate proposed multivariate-st ...

===Models with different distributions and link functions=== ...l generalized linear model. In hierarchical generalized linear models, the distributions of random effect <math>u</math> do not necessarily follow [[normal distribu ...

* Vectorial distributions<ref name="Marconi_2003"/> ...le of describing solid motions as well as fluids and gases: thus unlocking complex physics to be solved such as [[Fluid–structure interaction|fluid-structure ...

...moderate amounts of uncertainty in a large system may allow the resulting distributions to contain values beyond the slack bus's margins. ...es and the sum of the complex powers specified at the remaining buses. The complex power allocated to this bus is computed as part of the solution. In order f ...

...rs=1 |title=Null Beamsteering Using Distributed Arrays and Shared Aperture Distributions |journal=IEEE Transactions on Antennas and Propagation |volume=68 |issue=7 ...ion is a zero order [[Bessel function]] of the first kind, multiplied by a complex exponential, given by <math>e^{it\frac{b+a}{2}}J_0(\frac{b-a}{2}t)</math>. ...