Random vector

A random vector is a measurable function $X:(\Omega,\mathcal F)\to(\mathbb R^d,\mathcal B(\mathbb R^d))$ defined on a probability space $(\Omega,\mathcal F,\mathbb P)$, where $\mathcal B(\mathbb R^d)$ is the Borel $\sigma$-algebra on $\mathbb R^d$.

Equivalently, $X=(X_1,\dots,X_d)$ where each coordinate $X_i$ is a random variable ; conversely, any $d$-tuple of random variables defines a random vector by bundling them into a single map.

Examples:

• Let $\Omega=[0,1]$ with $\mathbb P$ the uniform distribution, and define $X(\omega)=(\omega,\omega^2)\in\mathbb R^2$. Then $X$ is a random vector in $\mathbb R^2$.
• Roll two fair six-sided dice and let $X=(D_1,D_2)\in\{1,\dots,6\}^2$. This pair-valued map is a random vector (taking values in a finite subset of $\mathbb R^2$).