It is well known that
Random variable is a function from sample space of a random experiment to $\mathbb{R}$.
Consider the following sentences from deep learning book
Let $\{x^{(1)}, \cdots , x^{(m)}\}$ be a set of $m$ independent and identically distributed data points. A point estimator or statistic is any function of the data: $$\hat{θ}_m = g(x(1), \cdots , x(m))$$. ........................................................................................................................................................... For now, we take the frequentist perspective on statistics. That is, we assume that the true parameter value $θ$ is fixed but unknown, while the point estimate $\hat{θ}$ is a function of the data. Since the data is drawn from a random process, any function of the data is random. Therefore $\hat{θ}$ is a random variable.
Now my doubts are
1) Input to a random variable is an outcome of a random experiment. Then is it true that each data point is an outcome of a random experiment? If yes, then what may be that random experiment and who are running it?
2) If dataset is a subset of or itself a sample space, then what is the reason for many statements stating every dataset has an underlying probability distribution? instead of saying that each dataset is a subset of a sample space of a random experiment?
3) Output of a random variable is a real number. But in general parameter can be a vector of real number. How to understand it?