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In books it is often written, Let random variables $X_1,\dots, X_n$ be a iid random sample from $f(x)$. What does it mean?

Are $X_1,X_2,\dots,X_n$ different values of one random variable $X$ which follows distribution $f(x)$, or all of them are different random variables?

If I have a dataset of 1000 persons' height (normally distributed) and $X$ is height of an individual and if I take a random sample then what are $X_1,X_2,\dots,X_n$? What does the term iid means?

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    $\begingroup$ A random sample from a finite population is not an iid sample unless you specify that the sampling is with replacement meaning (among other things) that it is possible to pick the same person more than once. In the standard meaning of "the phrase", $X_1, X_2, \ldots, X_n$ would the heights of 10 different people from the set of 1000 people whose heights have been recorded in your dataset. $\endgroup$
    – Dilip Sarwate
    Commented Mar 9, 2016 at 13:44
  • $\begingroup$ I am little bit confused, if X1,X2,…,Xn are realized value of a rv X~f(x) then how X1,X2,…,Xn are called random variable? $\endgroup$
    – Zed
    Commented Mar 9, 2016 at 13:54
  • $\begingroup$ The second paragraph of your question asks about something that is different from the query in the third paragraph. $\endgroup$
    – Dilip Sarwate
    Commented Mar 9, 2016 at 13:57
  • $\begingroup$ I am a little bit confused here regarding the second paragraph.. If X1,X2,…,Xn are realized value of a random variable X taken as a random sample from a population which have distribution f(x), then why books write these realized values as random variable X1,X2,…,Xn instead of x1,x2,...,xn (realized value)? how the realized value are random variable? $\endgroup$
    – Zed
    Commented Mar 9, 2016 at 14:43
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    $\begingroup$ Some textbooks do write realized values as for example $P(X = x_1), P(X = x_2), ... $. $\endgroup$
    – RobertF
    Commented Mar 9, 2016 at 14:49

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That's correct, $X_1, X_2, ..., X_n$ are $n$ independent and identically distributed (i.i.d.) values of the same variable drawn from the same distribution $f(x)$.

The term "i.i.d." means the values of $X$ are completely independent (the probability that $X$ assumes a particular value for record $i$ is unrelated to the values of $X$ for other records) and all values of $X$ are random pulls from the same distribution (e.g., the normal distribution with a given mean and variance).

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    $\begingroup$ It seems like one perhaps implicit issue raised by the OP is whether $X_i$ are random variables from a given distribution, or samples of a common random variable $X$. $\endgroup$
    – Antoni Parellada
    Commented Mar 9, 2016 at 13:34
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    $\begingroup$ @AntoniParellada - Yes the notation can be confusing. Sometimes $X_1, X_2, ...$ refers to completely separate variables (e.g., height, weight, etc.) from different distributions while in other cases $X_1, X_2, ...$ are specific values of a single random variable (and perhaps ought to be written as $X_1=x_1, X_2=x_2, ...$). One clue is the notation: $X_1, X_2, ..., X_k$ typically refers to $k$ different random variables while $X_1, X_2, ..., X_n$ refers to $n$ values of the same random variable. $\endgroup$
    – RobertF
    Commented Mar 9, 2016 at 14:43
  • $\begingroup$ RobertF I am now thoroughly confused. How can I tell whether $X_1,X_2,X_3$ refers to $k=3$ different random variables or $n=3$ values of the same random variable? $\endgroup$
    – Dilip Sarwate
    Commented Mar 9, 2016 at 15:11
  • $\begingroup$ @DilipSarwate - I look at context. For example, if you're reading a textbook or paper about multiple regression then it should be clearly stated that $X_1, X_2, ..., X_k$ are separate predictor variables. If you're studying sampling distributions with/without replacement from a univariate distribution (like height of an individual) then it ought to be clearly specified that $X_1, X_2, ..., X_n$ are random variables from the same distribution which may or may not be i.i.d. depending on whether you're sampling with/without replacement. $\endgroup$
    – RobertF
    Commented Mar 9, 2016 at 15:37
  • $\begingroup$ @RobertF Oh, come on! That is a nonsensical response. You said "One clue is the notation: $X_1, X_2, ..., X_k$ typically refers to $k$ different random variables while $X_1, X_2, ..., X_n$ refers to $n$ values of the same random variables." So I ask you again: How can I tell whether $X_1,X_2,X_3$ refers to $k=3$ different random variables or $n=3$ values of the same random variable? I have used your notation exactly. So, tell us, pretty please: what is the clue that your notation is providing to help us to distinguish between the two cases? $\endgroup$
    – Dilip Sarwate
    Commented Mar 10, 2016 at 3:38

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