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After looking for a dozen of questions on cross validated, I've decided to write my own. We know the following mapping is called a random variable $$X:\Omega\to \mathbb{R}$$ where $\Omega$ is set of all possible outcomes. For simple stochastic experiments like coin tossing we know the sample space explicitly is $\Omega= \text{\{Head, Tail\}}$. However, in most cases when we use random variables and we only know the distribution or not even that; we certainly ignore the sample space let alone its interpretation. I am concerned about the interpretations of some "real life" random variables. As an example let $H$ be the distribution representing the height of the person. assume $X \sim H$. Let me also define the experiment: picking a person randomly and getting his/her height.

Question

How can we interpret $\Omega = \{\omega_1,\omega_2, \omega_3,...\}$

each $\omega_i$ in my intuition cannot be simply the all possible 7 billion people on earth, I guess it must all possible 'types' of people, it is like a latent feature that describes a particular type of a possible person in some sense? Therefore, it takes into account even people how are not even born? Or is it possible to have a discrete $\Omega$ for this problem whatsoever? And if yes what is that mysterious $w_i$.

P.S. If possible, also tell whether H must have any hypothetical parameters.

If there is some clear why to explain this mystery, I beg your help.

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    $\begingroup$ In standard usage, the meanings of the arrows $\to,\mapsto$ are different. For that reason I changed $X:\Omega\mapsto\mathbb R$ to $X:\Omega\to\mathbb R. \qquad$ $\endgroup$
    – Michael Hardy
    Feb 4, 2019 at 20:50
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    $\begingroup$ The second arrow is used in things like $x\mapsto x^3,$ which means the function whose output is the cube of its input. $\endgroup$
    – Michael Hardy
    Feb 4, 2019 at 20:51
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    $\begingroup$ The question is using $H$ with two different meanings. $\endgroup$
    – Xi'an
    Feb 5, 2019 at 6:06
  • $\begingroup$ If you want to study the probability theory from this aspect, maybe be you should study the measure theory first. need tens of pages to explain/answer your question. $\endgroup$
    – user158565
    Feb 7, 2019 at 3:33
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    $\begingroup$ @user158565 I'm having a hard time seeing how measure theory has anything at all to say about interpreting statistical concepts. $\endgroup$
    – whuber ♦
    Feb 7, 2019 at 22:16

1 Answer 1

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Suppose $\Omega$ is some population of persons and $\omega\in\Omega,$ i.e. $\omega$ is one of those persons. Further suppose each person has some probability of being the one who is chosen. Then $X(\omega)$ is a random variable whose distribution depends on the probabilities of various persons being chosen and also on the values of $X$ at the various values of $\omega.$

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