If $X_1$ and $X_2$ are random variables, then how is the random variable $X=(X_1,X_2)$ defined?

Question

If $X_1$ and $X_2$ are (we can suppose real-valued) random variables, then how is the random variable $X=(X_1,X_2)$ defined? Is there a name for this concept so I can look it up on i.e. Wikipedia? (I have come across this notation in a Decision Theory textbook).

A specific question I have in mind: if $ϕ∈Ω$ (i.e., our sample space) and $X_1(ϕ)=1$ and $X_2(ϕ)=2$, then what is $X(ϕ)$ in this context?

Answer 1

In your example, $X(\phi)$ is the ordered pair $(1,2).$

Throw a die and get a number in the set $\{1,2,3,4,5,6\}.$ Call the outcome $X_1.$

Throw another die. Call the outcome $X_2.$

If $X_1=1$ and $X_2=5$ then $(X_1,X_2)$ is the ordered pair $(1,5).$

"Ordered" means $(1,5)$ is not the same as $(5,1).$

Answer 2

A real random variable is a bona fide function $X:\Omega\to{\mathbb R}$ defined on some probability space $\Omega$. If $X_1$ and $X_2$ are two such functions then $X:=(X_1,X_2)$ is an ${\mathbb R}^2$-valued random variable. You then can ask (and hopefully answer) questions like "What is the probability that the point $X=(X_1,X_2)$ lies in the unit disc?" or "What is the probability that $X_2\geq X_1^2$?".

If for a given $\omega\in\Omega$ one has $X_1(\omega)=a$ and $X_2(\omega)=b$ then $X(\omega)=(a,b)$.

What is "random" about the objects $X_1$, $X_2$, or $(X_1,X_2)$ is that in the intended view of things "fate" selects the point $\omega\in\Omega$ where these functions are evaluated.