# Interpreting sample space of a unknown random variable

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.

• In standard usage, the meanings of the arrows $\to,\mapsto$ are different. For that reason I changed $X:\Omega\mapsto\mathbb R$ to $X:\Omega\to\mathbb R. \qquad$ – Michael Hardy Feb 4 '19 at 20:50
• The second arrow is used in things like $x\mapsto x^3,$ which means the function whose output is the cube of its input. – Michael Hardy Feb 4 '19 at 20:51
• The question is using $H$ with two different meanings. – Xi'an Feb 5 '19 at 6:06
• 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. – user158565 Feb 7 '19 at 3:33
• @user158565 I'm having a hard time seeing how measure theory has anything at all to say about interpreting statistical concepts. – whuber Feb 7 '19 at 22:16

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 $$\omega.$$