List of multivariate and matrix-variable probability distributions
This list of multivariate and matrix-variant probability distributions contains multivariate probability distributions and matrix-variant probability distributions , divided into discrete multivariate distributions, absolutely continuous multivariate distributions and matrix-variant distributions.
notation
The following conventions apply:
- Is , so is and component-wise to be understood, i.e. exactly if for everyone
- Is , the order symbols designate the Loewner partial order , i.e. exactly when positive is definite and exactly when
- denotes the one vector of length and the - identity matrix .
Abbreviation is used
- , where is.
- .
for a . here denotes the trace of the matrix .
Discrete multivariate
Surname | carrier | parameter | Probability function | comment |
---|---|---|---|---|
Multinomial distribution , polynomial distribution | , | stochastic vector of | Generalization of the binomial distribution , is the multinomial coefficient | |
Negative Multinomial Distribution , Negative Multinomial Distribution, Negative Polynomial Distribution | , , , | : | Generalization of the negative binomial distribution | |
Multivariate hypergeometric distribution , general hypergeometric distribution | , | , , , | Generalization of the hypergeometric distribution | |
Polyhypergeometric distribution | , | , , , | Generalization of the multivariate hypergeometric distribution | |
Multivariate Poisson distribution | , | - | ||
Pólya / Eggenberger distribution | , | , , , , | - |
Absolutely multivariate
Surname | carrier | parameter | Probability density function | comment |
---|---|---|---|---|
Multi-dimensional normal distribution , multivariate normal distribution | , , . | : | Generalization of the normal distribution | |
Dirichlet distribution (of order ) | , , | , | - |
Matrix variat
Surname | carrier | parameter | Probability density function | comment |
---|---|---|---|---|
Matrixvariate normal distribution (English matrix variate normal distribution ) | , , , , | Generalization of the normal distribution | ||
Wishart distribution | , , Freedom | Here refers to the Multivariate gamma function . The Wishart distribution is the matrix-variant generalization of the chi-square distribution . | ||
Matrixvariate Student's t-distribution (English matrix variate t-distribution ) | , , , , Freedom |
|
Generalization of the Student t-distribution | |
Matrixvariate beta distribution (English matrix variate beta-type-I distribution ) | , | Generalization of the beta distribution | ||
Matrixvariate inverse beta distribution (English matrix variate beta-type-II distribution ) | , | Generalization of the inverse beta distribution |
See also
Individual evidence
- ↑ a b c d Klaus D. Schmidt: Measure and probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 296-300 , doi : 10.1007 / 978-3-642-21026-6 .
- ^ Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .
- ↑ Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 334 , doi : 10.1007 / 978-3-642-36018-3 .
- ↑ Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 562-563 , doi : 10.1007 / 978-3-642-36018-3 .
- ↑ a b c d e AK Gupta: Matrix variate distribution . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).