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In most mathematical statistic textbook problems, a question always ask: Given you have $X_1, X_2, \ldots, X_n$ iid from a random sample with pdf:(some pdf). My question is why can't the sample come from one random variable such as $X_1$ since $X_1$ itself is a random variable. Why do you need the sample to come from multiple iid random variables?

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The samples do come from the same random variable, but the different samples need different names, that is, it makes no sense to say something like "Let $X_1$, $X_1$,$\ldots,$ $X_1$ denote a random sample with pdf $f_X(x)$". –  Dilip Sarwate Oct 3 '11 at 1:17
    
@Henning: that sure looks like an answer to me... :) –  J. M. Oct 3 '11 at 1:28

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up vote 4 down vote accepted

A random variable is something that has one definite value each time you do the experiment (whatever you define "the experiment" to be), but possibly a different value each time you do it. If you collect a sample of several random values, the production of all those random values must -- in order to fit the structure of the theory -- be counted as part of one single experiment. Therefore, if you had only one variable, there couldn't be any different values in your sample.

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