Editing Multivariate random variable (section)

{{Short description|Random variable with multiple component dimensions}}{{Probability fundamentals}}
{{broader|Multivariate statistics}}

In [[probability theory|probability]], and [[statistics]], a '''multivariate random variable''' or '''random vector''' is a list or [[vector (mathematics)|vector]] of mathematical [[Variable (mathematics)|variable]]s 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 [[statistical unit]]. 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 [[real number]].

Random vectors are often used as the underlying implementation of various types of aggregate [[random variable]]s, e.g. a [[random matrix]], [[random tree]], [[random sequence]], [[stochastic process]], etc.

Formally, a multivariate random variable is a [[column vector]] <math> \mathbf{X} = (X_1,\dots,X_n)^\mathsf{T} </math> (or its [[transpose]], which is a [[row vector]]) whose components are [[random variable]]s on the [[probability space]] <math>(\Omega, \mathcal{F}, P)</math>, where <math>\Omega</math> is the [[sample space]], <math>\mathcal{F}</math> is the [[sigma-algebra]] (the collection of all events), and <math>P</math> is the [[probability measure]] (a function returning each event's [[probability]]).