Multivariate Random Variable

In mathematics, probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose values 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 there may be correlations among them — often they represent different properties of an individual statistical unit (e.g. a particular person, event, etc.). 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, random process, etc.

More formally, a multivariate random variable is a column vector X = (X₁, ..., X_n)T (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space (Ω, P), where Ω is the sample space, is the sigma-algebra (the collection of all events), and P is the probability measure (a function returning every event's probability).