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.