Is it different to sample a random variable $n$ times than it is to sample an $n$-vector once?

Question

Let $X:\Omega\longrightarrow \mathbb{R}$ be a random variable. Suppose we take $n$ "observations" (aka. "samples") from $X$, and we put them into an $n$-tuple $(a_1, .., a_n)$, where $a_i$ is the $i$-th observation.

Now suppose we take an $n$-tuple $(X_, ..., X)$, made of $n$ copies of our random variable $X$, and we sample it once, resulting in an $n$-tuple $(b_1, ..., b_n)$.

1. Are these 2 processes mathematically equivalent?
2. If equivalent, are there any mathematical advantages of taking one viewpoint over the other?
3. Does it make sense to build vectors of random variables? Can we even call them vectors? Do they form a vector space?

Answer 1

Random variables are just functions of the space of states.

Sampling is just evaluating them.

Having $X(w_1), X(w_2), ..., X(w_n)$ is the same as having $(X(w_1), X(w_2), ..., X(w_n))$.

What is not the same is doing things like $(X(w), X(w), ..., X(w))$.