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.


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.

  • 2
    $\begingroup$ In standard usage, the meanings of the arrows $\to,\mapsto$ are different. For that reason I changed $X:\Omega\mapsto\mathbb R$ to $X:\Omega\to\mathbb R. \qquad$ $\endgroup$ Feb 4, 2019 at 20:50
  • 1
    $\begingroup$ The second arrow is used in things like $x\mapsto x^3,$ which means the function whose output is the cube of its input. $\endgroup$ Feb 4, 2019 at 20:51
  • 3
    $\begingroup$ 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
  • 1
    $\begingroup$ @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


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 $\omega.$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.