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I've been studying statistics on my own and I'm having a hard time understanding some notations. On this page: http://scipp.ucsc.edu/~haber/ph116C/iid.pdf, specifically on the second paragraph, the author writes:

$X_1, X_2, \dots, X_n$ are iid random variables.

My question is:

Are $X_1, X_2, \dots, X_n$ numbers? I mean, there is a random variable $X$ and $X_1, X_2, \dots, X_n$ are just outcomes of $X$? If not, how come can I take an average of $X_1, X_2, \dots, X_n$ if they are not numbers?

kjetil b halvorsen
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RafaelSantiago94
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    Does this answer your question? [What is meant by a "random variable"?](https://stats.stackexchange.com/questions/50/what-is-meant-by-a-random-variable) – Pitouille Nov 03 '21 at 13:36
  • @Pitouille I already read these answer (witch is great, btw), but I'm still confused on how do you get an average of random numbers. – RafaelSantiago94 Nov 03 '21 at 13:39
  • Hi: They become numbers when the random variable is realized. Say it was inches of rainfall each day. Then, $X_1$ could be monday, $X_2$ tuesday etc. But next monday, they would all be realized so you could take the average. – mlofton Nov 03 '21 at 14:29
  • Suppose you throw two fair dice and take their average value: before throwing, each die's value is a random variable with $1,2,3,4,5,6$ being equally likely. You could imagine throwing a $3$ and a $6$ so an average of $4.5$. In fact you could get any average from $1,1.5,2,2.5,\ldots,6$, though getting an average of $3.5$ is six times more likely than getting an average of $1$. So in this sense $\bar X=\frac{X_1+X_2}2$ is also a random variable with a distribution. – Henry Nov 03 '21 at 14:44
  • @mlofton so, basically, X1, X2,..., Xn are outcomes (realizations) of the same random variable X, right? As an another example, if I had a dataset, let's say, of heights of n individuals, then my random variable (X) would be height and X1 would be the height of the first individual and so on? – RafaelSantiago94 Nov 03 '21 at 14:45
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    $X_1$ is the random variable designating the height of the first person before measurement. Once you have the accurate measurement, it stops being random and the observation is more usually written $x_1$ – Henry Nov 03 '21 at 14:47
  • @Henry I got your ideas. It's just seems weird to me now how come different random variables could have the same expected value, as in the article that I posted. It' s just because they are iid? – RafaelSantiago94 Nov 03 '21 at 17:01
  • Identically distributed random variables do have the same expected value (if that exists) – Henry Nov 03 '21 at 23:33
  • RafaelSantiago94: Yes, that's correct. – mlofton Nov 04 '21 at 15:04

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