Sample data and its corresponding random variables

Question

Simple question but I can't find the logic behind that. In many texts I see expressions like

"Let $\{ x_1, \dots, x_n\}$ denote our sample data and $\{ X_1, \dots, X_n\}$ their corresponding random variables".

The above expression seems quite ambiguous to me since $X_i$ is in general a real-valued function and $x_i$ can be either a scalar $\in \mathbb{R}$, or a vector $\in \mathbb{R}^n$. So $X_i$ can be (for example) the $i$-th column of the matrix with columns $x_i, \forall i \in \{1,\dots,n\}$, or any other real-valued function as well. As a more concrete example, in the link

https://courses.cit.cornell.edu/econ620/reviewm8.pdf,

at the end of page 2, the author says "Let $t = t(y)$ be a function of the observations and let $T = t(Y)$ be the corresponding random variable". Definitely there is some underlying logic but wasn't able to find out from my bibliography. So can you please explain me what's the "silent" identification that happens between actual data and random variables? Cheers.

Answer 1

Consider a function $f: \mathcal X \rightarrow \mathcal X$. How would you distinguish between $f$ and a particular realization of it, $f(x)$? In general this is completely clear, the function is an object that lives on some abstract space while $f(x)$ is some value in $\mathcal X$.

A random variable is the exact same - $Y$ is in reality a function from a underlying probability space $\Omega$ into $\mathcal X$, which typically is the real numbers or similar. So when we write $Y$ we really mean the function $Y: \Omega \rightarrow \mathcal X$. A sample from $Y$ is just $Y(\omega)$, which lives in $\mathcal X$. So any sample is really just values of the function $Y$ evaluated at different $\omega$'s. It also that if we define $Z = f(Y)$, we have a composite function, which is just another random variable.