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Everyplace I read about the central limit theorem (CLT), a sample (size n) is thought of as a sequence of random variables $(X_1,X_2,...,X_n )$, where each value comes from a random variable. These random variables could even have different distributions (Lindberg CLT).

A random variable is a mapping from some $\Omega$ to a value. i.e. one outcome in $\Omega$ is mapped to one value, and another outcome in $\Omega$ is mapped to another value.

Then why is, all of a sudden, every value in a sample viewed as a random variable "in itself" (viewed as a sequences of random variables). Isn't it more intuitive to instead just think of all the values in the sample as values of one random variable. The distribution of that one random variable would have the same distribution as the population, and the CLT could be used in regard to the distribution of that one random variable. I see this might be tricky if each value in the sample comes from a different distribution, but I am curious about the case of a sample being view as iid random variables.

cjkilimanjaro
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  • Check this question. –  Dec 12 '19 at 16:56
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    Only one outcome in $\Omega$ “happens” and it corresponds to a realization of the entire sequence. It doesn’t make sense to have the $X_n$ be all the same random variable. If that were the case then the sequence would always be constant, so certainly we can’t model an iid sequence of non degenerate variables in this way. – spaceisdarkgreen Dec 12 '19 at 16:56
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    That said, you can consider the sequence as a random sequence, i.e. a map from $\Omega$ to the set of sequences in $\mathbb R$, rather than a sequence of maps $\Omega\to \mathbb R$. – spaceisdarkgreen Dec 12 '19 at 17:02
  • @d.k.o. Great link! But, even though I understand how a sample is represented as a sequence of random variable, I fail to see why it can't also be considered as multiple outcomes/values of one random variable that represents the population distribution? Any ideas? – cjkilimanjaro Dec 12 '19 at 18:41
  • @cjkilimanjaro Try to make a formal statement from that sentence (i.e. "multiple outcomes of one random variable"). –  Dec 12 '19 at 18:48
  • @d.k.o. Sure; we have sample {$\omega_i,\ i \in {1,2,...,30}$} i.e. sample size = 30. Instead of saying every $\omega_i$ represent one outcome of random variable $X_i$ i.e. that there are 30 random variables one for each $\omega_i$. Why can't we instead say every $\omega_i$ is one (out of 30) outcomes/realization of one random variable $X$ i.e. sample = ${ \omega_1,\omega_2,...,\omega_{30}}$ = ${X(\omega_1),X(\omega_2),...,X(\omega_{30}) }$, where every $\omega_i$ is an outcome in the sample space of $X$ (sample space of $X$ = population). – cjkilimanjaro Dec 12 '19 at 19:31
  • What is a probability space here, i.e. $(\Omega,\mathcal{F},\mathsf{P})$? –  Dec 12 '19 at 19:55
  • @d.k.o. probability space $(\Omega, \mathcal{F,P})$, where $\Omega$ is the population (i.e. all possible outcomes). And the *one random variable $X$ is defined on that same probability space (i.e. $\Omega$ is "domain" of X). – cjkilimanjaro Dec 12 '19 at 20:34

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