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I have a question about one single random variable vs. a bunch of random variables:

If there are n observations from a single RV (call it $X$), and there are n observations in total from n i.i.d RVs (call them $Y_1,Y_2,...,Y_n$), 1 observation from each of the $n$ RVs. The distribution of $Y_n$ is the same as $X$.

Do the two groups of observations have the same distribution?

(If strictly speaking, two groups of observations cannot have distribution because they are not RVs and only RVs can have distribution. So the question is do the two groups of observation have the same mean and variance (or any other statistical parameters) when $n\to+\infty$)

Maybe due to my understanding of RV is not deep enough, I could not find the correct direction to solve the problem. Neither do I know if this question is meaningful or asked correctly.

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    $\begingroup$ Your question is not clear. Mean and variance of which variables? Can you give more concrete details about what these variables are? Also, if this is a homework question, please add the self-study tag and read its wiki. $\endgroup$ Jan 6, 2017 at 14:43
  • $\begingroup$ In short, $n$ independent observations from a random variable $X$ means the same as one observation of the vector $(X_1,\ldots,X_n)$ made of independent replications of the random variable $X$. $\endgroup$
    – Xi'an
    Jan 6, 2017 at 14:46
  • $\begingroup$ @MarquisdeCarabas Sorry. I meant the variance and mean of the two groups of n observations each drawn from X and Y1...Yn. This is not homework, just some unclear theory background for myself. $\endgroup$
    – intelliarm
    Jan 6, 2017 at 16:53
  • $\begingroup$ @Xi'an I would expect that. Could you give some reference or point me to some theory which can lead to this conclusion? $\endgroup$
    – intelliarm
    Jan 6, 2017 at 16:54
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    $\begingroup$ This seems like a legitimate question to me, &, in light of the upvoted answer, not too unclear to be answerable. I'm voting to leave open. $\endgroup$ Jan 6, 2017 at 21:40

2 Answers 2

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When modelling a sample $(x_1,\ldots,x_n)$ as an $i.i.d$ sample from a given distribution $F$, the correct way of modelling is to see this sample as the realisation of $n$ random variables $(X_1,\ldots,X_n)$ made of $n$ independent random variables identically distributed from $F$:

$$(x_1,\ldots,x_n)=(X_1,\ldots,X_n)(\omega)\qquad\omega\in\Omega$$

The concept of $n$ realizations of a single random variable is a shortcut that is not well-defined because one cannot handle independence with a single random variable.

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I think it depends on the theoretical and mainstream approach. This approach could not be proved, but it is such an axiom. I looked in my books on math. stat., authors do not mention this question.

Some points about that:

  1. Math. stat. theory approach based on tests and theory derived from i.i.d. condition. With this condition and assumption that the random sample contains n observations for n random variables we know how to get unbiased, consistent and effective estimates. In case when we assign n observation got from one rv, we cannot apply CLT, MLE, F, chi square etc.

  2. As mentioned above, it is hard to incorporate independence in statistical approach if we consider all n observations from one random variable. For example, in average, temperature rises in the summer, i.e. previous observation could influence the next level of temperature. I.e. cumulative distribution could change through the time. If we have only one rv, we could not explain this influence. Also, there are some “structural breaks” in time series or non time sample, this means that we could have different distributions.

  1. Remind definition:

For a given sample space S of some experiment, a random variable (rv) is any rule that associates a number with each outcome in S. In mathematical language, a random variable is a function whose domain is the sample space and whose range is the set of real numbers

Could you, please, prove that the set of real numbers does not change through the time or does not depend on smth? But in the case when there are only 1 observation for each rv this question drops.

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