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I am trying to understand some basic ideas of econometrics (and mathematical statistics) from the precise, mathematical point of view, avoiding vague explanations. I am beginning to learn about these notions from the introductory, undergraduate level, but I have a good mathematical background. Could you please explain precisely some basic notions?
What I mean is the concept of a random sample, first. If there are some mistakes or other inaccuracy, correct it, please.
Suppose we have some statistical population. Precisely, a population is a probability space $X$ plus some random variable, say $\xi$. We may imagine that $X$ is a set of people living in our city and $\xi$ maps each person to his or her height. It is okay, isn't it?
Well, what is the random sample of size $n$? I understand it just intuitively. What about the precise definition? "A random sample of size $n$ is a set of $n$ random variables, which are well distributed..." - I read in Wikipedia and many other sources. That is what is very strange. Suppose $\theta$ is some random variable defined on $X$. How can I build the corresponding sample of size $1$? I think that the sample of size $1$ is just a point of $X$, without any random variables. Is it possible to build such a correspondence generally?
I know that my question was raised here many times in different forms, but I'd like to reformulate it, trying to get the underlying principles of statistics without the loss of mathematical rigour. So, could you tell me if there are precise definitions in statistics?