Why are samples always taken from iid random variables? 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? 
 A: 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.
A: Suppose the sample came from a single random variable $X$, say a dice with sample $[3,5,1,3]$. To compute the probability of observing this sample, you'd start by writing down something like 
$$\text{Pr}\,(X=3\wedge X=5 \wedge X=1 \wedge X=3).$$ But this is zero, since the events $X=3$, $X=5$ and $X=2$ are mutually exclusive.
Taking the math seriously you might even come to the conclusion that the philosophical underpinnings of frequentism make no sense. There is no such thing as a single random variable, which can be repeatedly sampled and produce different outcomes.
Not sure if this argument is what Henning Makholm meant by "in order to fit the structure of the theory".
