Definition:Random Vector

Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $n \in \N$.

Let $\struct {S_1, \Sigma_1}$, $\struct {S_2, \Sigma_2}$, $\ldots$, $\struct {S_n, \Sigma_n}$ be measurable spaces.

Let:

$\ds S = \prod_{i \mathop = 1}^n S_i$

For each integer $1 \le i \le n$, let $X_i$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S_i, \Sigma_i}$.

Define a function $\mathbf X : \Omega \to S$ by:

$\map {\mathbf X} \omega = \tuple {\map {X_1} \omega, \map {X_2} \omega, \ldots, \map {X_n} \omega}$

for each $\omega \in \Omega$.

We call $\mathbf X$ a random vector.

A random vector is also known as a multivariate random variable.

Let $\EE$ be the experiment of throwing a standard $6$-sided die $4$ times.

Then the outcome of $\EE$ can be expressed as a random vector $\tuple {x_1, x_2, x_3, x_4}$, where each $x_i$ is uniformly distributed over the sample space $\set {1, 2, 3, 4, 5, 6}$.

If $\EE$ is performed with the results $5$, $2$, $5$ and $6$, then the outcome would be reported as $\tuple {5, 2, 5, 6}$.

Random Vector is Random Variable shows that $\mathbf X$ is $\Sigma/\Sigma'$-measurable, where $\ds \Sigma' = \bigotimes_{i \mathop = 1}^n \Sigma_i$ is the product $\sigma$-algebra of $\Sigma_1, \Sigma_2, \ldots, \Sigma_n$.
Vector is Random Vector iff Components are Random Variables shows that $\tuple {X_1, X_2, \ldots, X_n}$ is a random vector if and only if $X_i$ is a random variable for each $i$.