I have a small doubt in the way estimators are defined. We have a sequence of data points $X^{(1)}, X^{(2)}, ... , X^{(n)}$. We then define an estimator as a function of these data points. When calculating the bias of an estimator we find the expectation of this function of "random variables".
But are $X^{(1)}, X^{(2)} ... $ all random variables? These are just observed values. I know the definition says they are samples coming from some unknown distribution (at least the parameter of the distribution is not known). How can we possibly compute $E[X^{(i)}]$ when we do not know what the distribution is? Also I tend to see each $X^{(i)}$ as an observed data value of the random variable and not a random variable. What is it that I am missing?
Many proofs that I have seen (like the one that proves $\bar{X}$ is unbiased) end up using the iid assumption to get rid of the $E[X^{(i)}]$. None of them talks about calculating expectation explicitly. Any more examples to show what is meant by calculating the expectation of values of random variables and saying that these are iid could be helpful in understanding the concepts.
Update:
One way I am beginning to think of it is the following:
We have a set of n data points corresponding to a random variable. Now instead of looking at these as n distinct samples of one random variable, define n iid random variables having the same distribution as the original random variables and consider these to be the n values of these n random variables. Am I thinking in the right direction?