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A random variable is a measurable function from $E$ to $F$, where $(E,\mathscr E, \mathbb P)$ is a probability space and $(F,\mathscr F)$ is a measurable space.

I've seen in many sites that $E$ is the "unknown space". I don't know why it is said "unknown". For example, if the experiment consists in throwing a coin, the $E$ space has only two elements ("head" or "tails"), and it is pretty obvious that they are known.

I've been reading some answers on stats.SE and apparently every experiment (even the previous coin experiment) has to be visualized in this fashion:

  • $E$ is a box full of coins
  • Each coin is either heads or tails
  • one picks randomly one coin from the box

The point here is that there are many, maybe infinitely many, coins and we are picking one.

My question is then, how many elements are there in $E$ in the coin throwing experiment: just two (one corresponding to "heads", the other to "tails") or many, many ones ?

If the answer is there are many one, then it would be consistent with the fact that $E$ is called the "unknown space".

niobium
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    In the end, what the sample space is for an experiment is a choice. We as the mathematicians have a choice to make as to how to describe and model the particular scenario. We make such a choice so as to have the events we are interested in describing can be described unambiguously as subsets of our sample space. If all we cared about was whether or not a coin landed on heads or tails, we have the choice of being able to say that that is the only piece of information we care about. If we also cared about what date each coin was minted, or the monetary value.. then we need more specificity – JMoravitz Nov 15 '22 at 12:49
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    That is not to say that the more specific sample space keeping all of the information about the types of coins in addition to whether we flipped heads or tails was wrong to use per se... but it will be more annoying to use certainly. Also, to emphasize... it is incorrect to blindly assume that $\Pr(A)=\dfrac{|A|}{|S|}$. If some of your coins in your box were biased, then you will want to be more careful. – JMoravitz Nov 15 '22 at 12:51
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    Abstraction works fine. A visualisation of the coin experiment might be helpful but is not really necessary. Sufficient is the existence of iid random variables that take values in ${0,1}$ with suitable probabilities. On that base we can apply mathematics. Actually we do not even have to wonder how to construct the underlying probability space. The fact that such a space can be constructed is enough on its own. Have a look at this question. – drhab Nov 15 '22 at 13:19

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