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 ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle x>0}$ ${\displaystyle x\geq 0}$ ${\displaystyle x\geq 0}$ ${\displaystyle x_{i}\geq 0}$ ${\displaystyle i}$
• Is , the order symbols designate the Loewner partial order , i.e. exactly when positive is definite and exactly when ${\displaystyle X\in \mathbb {R} ^{n\times n}}$ ${\displaystyle X>0}$ ${\displaystyle X}$ ${\displaystyle X>Y}$ ${\displaystyle X-Y>0}$
• ${\displaystyle \mathbf {1} _{k}}$denotes the one vector of length and the - identity matrix . ${\displaystyle k}$ ${\displaystyle \mathbf {1} _{k\times k}}$ ${\displaystyle k\times k}$

Abbreviation is used

${\displaystyle a_{1}+a_{2}+\dots +a_{k}=\mathbf {1} _{k}^{T}a}$, where is. ${\displaystyle a=(a_{1},a_{2},\dots ,a_{k})}$

${\displaystyle \operatorname {etr} (A)=\exp(\operatorname {spur} (A))}$.

for a . here denotes the trace of the matrix . ${\displaystyle A\in \mathbb {R} ^{n\times n}}$ ${\displaystyle \operatorname {spur} (A)}$ ${\displaystyle A}$

Discrete multivariate

Surname	carrier	parameter	Probability function	comment
Multinomial distribution , polynomial distribution	${\displaystyle x\in \mathbb {N} _{0}^{k}}$, ${\displaystyle \mathbf {1} _{k}^{T}x=n}$	${\displaystyle p}$ stochastic vector of ${\displaystyle \mathbb {R} ^{k}}$	${\displaystyle f(x)={n \choose x}\;\prod _{i=1}^{k}p_{i}^{x_{i}}}$	Generalization of the binomial distribution , is the multinomial coefficient ${\displaystyle {n \choose x}}$
Negative Multinomial Distribution , Negative Multinomial Distribution, Negative Polynomial Distribution	${\displaystyle x\in \mathbb {N} _{0}^{k}}$	${\displaystyle p\in \mathbb {R} ^{k}}$, , , ${\displaystyle p_{i}\in (0,1)}$ ${\displaystyle \mathbf {1} _{k}^{T}p\leq 1}$ ${\displaystyle \alpha \in (0,+\infty )}$	: ${\displaystyle f(x)={\frac {\Gamma (\alpha +\mathbf {1} _{k}^{T}x)(1-\mathbf {1} _{k}^{T}p)^{\alpha }}{\Gamma (\alpha )\prod _{i=1}^{k}x_{i}!}}\prod _{i=1}^{k}p_{i}^{x_{i}}}$	Generalization of the negative binomial distribution
Multivariate hypergeometric distribution , general hypergeometric distribution	${\displaystyle x\in \mathbb {N} _{0}^{k}}$, ${\displaystyle x_{1}+\dots +x_{k}=n}$	${\displaystyle B\in \mathbb {N} _{0}^{k}}$, , , ${\displaystyle \mathbf {1} _{k}^{T}B=N}$ ${\displaystyle n,N\in \mathbb {N} }$ ${\displaystyle n\leq N}$	${\displaystyle f(x)={\binom {N}{n}}^{-1}\prod _{i=1}^{k}{\binom {B_{i}}{x_{i}}}}$	Generalization of the hypergeometric distribution
Polyhypergeometric distribution	${\displaystyle x\in \mathbb {N} _{0}^{k}}$, ${\displaystyle \mathbf {1} _{k}^{T}x\leq n}$	${\displaystyle n,N\in \mathbb {N} }$, , , ${\displaystyle n\leq N}$ ${\displaystyle K\in \{1,\dots ,N-1\}^{k}}$ ${\displaystyle \mathbf {1} _{k}^{T}K\leq N}$	${\displaystyle f(x)={\binom {n}{M}}^{-1}{\binom {1-\mathbf {1} _{k}^{T}K}{n-\mathbf {1} _{k}^{T}x}}\prod _{i=1}^{k}{\binom {K_{i}}{x_{i}}}}$	Generalization of the multivariate hypergeometric distribution
Multivariate Poisson distribution	${\displaystyle \mathbb {N} _{0}^{k}}$	${\displaystyle \alpha \in \mathbb {R} ^{k}}$, ${\displaystyle \alpha >0}$	${\displaystyle f(x):=\prod _{i=1}^{k}e^{-\alpha _{i}}{\frac {\alpha _{i}^{x_{i}}}{x_{i}!}}}$	-
Pólya / Eggenberger distribution	${\displaystyle x\in \mathbb {N} _{0}^{k}}$, ${\displaystyle \mathbf {1} _{k}^{T}x\leq n}$	${\displaystyle n\in \mathbb {N} }$, , , , ${\displaystyle a\in (0,+\infty )}$ ${\displaystyle b\in \mathbb {R} ^{k}}$ ${\displaystyle b>0}$ ${\displaystyle \mathbf {1} _{k}^{T}b\leq a}$	${\displaystyle f(x)={\frac {{\binom {a+n-1-\mathbf {1} _{k}^{T}(b-x)}{n+\mathbf {1} _{k}^{T}x}}\prod _{i=1}^{k}{\binom {b_{i}+x_{i}-1}{x_{i}}}}{\binom {a+n-1}{n}}}}$	-

Absolutely multivariate

Surname	carrier	parameter	Probability density function	comment
Multi-dimensional normal distribution , multivariate normal distribution	${\displaystyle \mathbb {R} ^{n}}$	${\displaystyle \mu \in \mathbb {R} ^{n}}$, , . ${\displaystyle \Sigma \in \mathbb {R} ^{n\times n}}$ ${\displaystyle \Sigma >0}$	: ${\displaystyle f(x)={\frac {1}{\sqrt {(2\pi )^{p}\det(\Sigma )}}}\exp \left(-{\frac {1}{2}}(x-\mu )^{T}\Sigma ^{-1}(x-\mu )\right)}$	Generalization of the normal distribution
Dirichlet distribution (of order ) ${\displaystyle K}$	${\displaystyle x\in \mathbb {R} ^{K}}$, , ${\displaystyle \mathbf {1} _{K}^{T}x=1}$ ${\displaystyle x\geq 0}$	${\displaystyle \alpha \in \mathbb {R} ^{n}}$, ${\displaystyle \alpha >0}$	${\displaystyle f(x)={\frac {\Gamma \left(\sum _{i=1}^{K}\alpha _{i}\right)}{\prod _{i=1}^{K}\Gamma (\alpha _{i})}}\prod _{i=1}^{K}x_{i}^{\alpha _{i}-1}}$	-

Matrix variat

Surname	carrier	parameter	Probability density function	comment
Matrixvariate normal distribution (English matrix variate normal distribution )	${\displaystyle \mathbb {R} ^{n\times p}}$	${\displaystyle M\in \mathbb {R} ^{n\times p}}$, , , , ${\displaystyle U\in \mathbb {R} ^{n\times n}}$ ${\displaystyle V\in \mathbb {R} ^{p\times p}}$ ${\displaystyle U>0}$ ${\displaystyle V>0}$	${\displaystyle f(X)={\frac {\operatorname {etr} \left(-{\frac {1}{2}}\left[V^{-1}(X-M)^{T}U^{-1}(X-M)\right]\right)}{(2\pi )^{np/2}\det(V)^{n/2}\det(U)^{p/2}}}}$	Generalization of the normal distribution
Wishart distribution	${\displaystyle \mathbb {R} ^{p\times p}}$	${\displaystyle V\in \mathbb {R} ^{p\times p}}$, , Freedom ${\displaystyle V>0}$ ${\displaystyle n\geq p}$	${\displaystyle f(X)={\frac {\det(X)^{\tfrac {n-p-1}{2}}\operatorname {etr} \left(-{\tfrac {1}{2}}V^{-1}X\right)}{2^{\tfrac {np}{2}}\det(V)^{\tfrac {n}{2}}\Gamma _{p}({\tfrac {n}{2}})}}}$	Here refers to the Multivariate gamma function . The Wishart distribution is the matrix-variant generalization of the chi-square distribution . ${\displaystyle \Gamma _{p}(\cdot )}$
Matrixvariate Student's t-distribution (English matrix variate t-distribution )	${\displaystyle \mathbb {R} ^{n\times p}}$	${\displaystyle M\in \mathbb {R} ^{n\times p}}$, , , , Freedom ${\displaystyle \Omega \in \mathbb {R} ^{p\times p}}$ ${\displaystyle \Sigma \in \mathbb {R} ^{n\times n}}$ ${\displaystyle \Omega >0,\Sigma >0}$ ${\displaystyle \nu }$	${\displaystyle f(X)={\frac {\Gamma _{p}\left({\tfrac {\nu +n+p-1}{2}}\right)}{(\pi )^{\tfrac {np}{2}}\Gamma _{p}\left({\frac {\nu +p-1}{2}}\right)}}\det(\Omega )^{-{\tfrac {n}{2}}}\det(\Sigma )^{-{\tfrac {p}{2}}}}$ ${\displaystyle \cdot \det \left(\mathbb {1} _{n\times n}+\Sigma ^{-1}(X-M)\Omega ^{-1}(X-M)^{T}\right)^{-{\frac {\nu +n+p-1}{2}}}}$	Generalization of the Student t-distribution
Matrixvariate beta distribution (English matrix variate beta-type-I distribution )	${\displaystyle X\in \mathbb {R} ^{p\times p}}$, ${\displaystyle 0<X<\mathbb {1} _{p\times p}}$	${\displaystyle a>{\tfrac {1}{2}}(p-1),\,b>{\tfrac {1}{2}}(p-1)}$	${\displaystyle f(X)={\frac {1}{\beta _{p}(a,b)}}\det(X)^{a-(p+1)/2}\det(\mathbb {1} _{p\times p}+X)^{b-(p+1)/2}}$	Generalization of the beta distribution
Matrixvariate inverse beta distribution (English matrix variate beta-type-II distribution )	${\displaystyle X\in \mathbb {R} ^{p\times p}}$, ${\displaystyle 0<X}$	${\displaystyle a>{\tfrac {1}{2}}(p-1),\,b>{\tfrac {1}{2}}(p-1)}$	${\displaystyle f(X)={\frac {1}{\beta _{p}(a,b)}}\det(X)^{a-(p+1)/2}\det(\mathbb {1} _{p\times p}+X)^{-(a+b)}}$	Generalization of the inverse beta distribution

See also

• List of univariate probability distributions

Individual evidence

1. ↑ ^{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 .
2. ^ 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 .
3. ↑ 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 .
4. ↑ 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 .
5. ↑ ^{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 ).