Multivariate random variable

In probability and statistics, a multivariate random variable or random vector is a list or vector 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, stochastic process, etc.

Formally, a multivariate random variable is a column vector ${\displaystyle \mathbf {X} =(X_{1},\dots ,X_{n})^{\mathsf {T}}}$ (or its transpose, which is a row vector) whose components are random variables on the probability space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$, 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).

Every random vector gives rise to a probability measure on ${\displaystyle \mathbb {R} ^{n}}$ with the Borel algebra as the underlying sigma-algebra. This measure is also known as the joint probability distribution, the joint distribution, or the multivariate distribution of the random vector.

The distributions of each of the component random variables ${\displaystyle X_{i}}$ are called marginal distributions. The conditional probability distribution of ${\displaystyle X_{i}}$ given ${\displaystyle X_{j}}$ is the probability distribution of ${\displaystyle X_{i}}$ when ${\displaystyle X_{j}}$ is known to be a particular value.

The cumulative distribution function ${\displaystyle F_{\mathbf {X} }:\mathbb {R} ^{n}\mapsto [0,1]}$ of a random vector ${\displaystyle \mathbf {X} =(X_{1},\dots ,X_{n})^{\mathsf {T}}}$ is defined as

where ${\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n})^{\mathsf {T}}}$.

Random vectors can be subjected to the same kinds of algebraic operations as can non-random vectors: addition, subtraction, multiplication by a scalar, and the taking of inner products.