Multivariate random variable

In <a href="/facts/Probability_theory/mFBn51rE">probability</a>, and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, a multivariate random variable or random vector is a list or <a href="/facts/Vector_(mathematics)/U8mM7A5F">vector</a> of mathematical <a href="/facts/Variable_(mathematics)/VbQlrvKz">variables</a> each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. The individual variables in a random vector are grouped together because they are all part of a single mathematical system — often they represent different properties of an individual <a href="/facts/Statistical_unit/oIW2uEBb">statistical unit</a>. For example, while a given person has a specific age, height and weight, the representation of these features of an unspecified person from within a group would be a random vector. Normally each element of a random vector is a <a href="/facts/Real_number/R02gw5Pb">real number</a>.
Random vectors are often used as the underlying implementation of various types of aggregate <a href="/facts/Random_variable/TwTBXnLT">random variables</a>, e.g. a <a href="/facts/Random_matrix/Yqo1lwFW">random matrix</a>, <a href="/facts/Random_tree/5T6dH57r">random tree</a>, <a href="/facts/Random_sequence/z2uHfvK4">random sequence</a>, <a href="/facts/Stochastic_process/Ng1n1dnB">stochastic process</a>, etc.
Formally, a multivariate random variable is a <a href="/facts/Column_vector/pPuRr9Xk">column vector</a> 
 
 
 
 
 X
 
 =
 (
 
 X
 
 1
 
 
 ,
 …
 ,
 
 X
 
 n
 
 
 
 )
 
 
 T
 
 
 
 
 
 {\displaystyle \mathbf {X} =(X_{1},\dots ,X_{n})^{\mathsf {T}}}
 
 (or its <a href="/facts/Transpose/8wmsagGS">transpose</a>, which is a <a href="/facts/Row_vector/pPuRr9Xk">row vector</a>) whose components are <a href="/facts/Random_variable/TwTBXnLT">random variables</a> on the <a href="/facts/Probability_space/dEcv2VrU">probability space</a> 
 
 
 
 (
 Ω
 ,
 
 
 F
 
 
 ,
 P
 )
 
 
 {\displaystyle (\Omega ,{\mathcal {F}},P)}
 
, where 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 is the <a href="/facts/Sample_space/WVldkN4F">sample space</a>, 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 is the <a href="/facts/Sigma-algebra/8LJINLNc">sigma-algebra</a> (the collection of all events), and 
 
 
 
 P
 
 
 {\displaystyle P}
 
 is the <a href="/facts/Probability_measure/8BjhgoPR">probability measure</a> (a function returning each event's <a href="/facts/Probability/fgYvMhwb">probability</a>).

Multivariate random variable open-in-new

Multivariate random variable