Multivariate random variable

In mathematics, probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables 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 variables, e.g. a random matrix, random tree, random sequence, , etc.

More formally, a multivariate random variable is a column vector ${\displaystyle \mathbf {X} =(X_{1},...,X_{n})^{T}}$ (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space as each other, ${\displaystyle (\Omega ,{\mathcal {F}},P)}$, where ${\displaystyle \Omega }$ is the sample space, ${\displaystyle {\mathcal {F}}}$ is the sigma-algebra (the collection of all events), and ${\displaystyle P}$ is the probability measure (a function returning each event's probability).

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