A note on notation: Random variables will always be denoted with uppercase letters and the realized values of the variable (or its range) will be denoted by the corresponding lowercase letters. Thus, the random variable X can take the value x.
For example, consider an experiment designed to gain information about some health characteristic of a population of people... the body weights of several people in the population might be measured. These different weights would be observations on different random variables, one for each person measured.
The random variables $X_1, ... , X_n$ are called a random sample of size n from the population f(x) if $X_1, ... , X_n$ are mutually independent random variables and the marginal pdf or pmf of each $X_i$ is the same function f(x).
Page 207 - I think the sentence below is incorrect:
Under the random sampling model each $X_i$ is an observation on the same variable and each $X_i$ has a marginal distribution given by f(x).
First of all, given the authors' notation convention, we should rewrite this statement as follows, where the first $X_i$ has been replaced with $x_i$ because we are referring to a realized value of the random variable (an observation) and not the random variable itself.
Under the random sampling model each $x_i$ is an observation on the same variable and each $X_i$ has a marginal distribution given by f(x).
Secondly, as the authors themselves explained in the quoted text from page 139, $X_i$ is a different random variable from $X_j$ for all i$\neq$j. $X_1, ... , X_n$ each have the same distribution function (are identically distributed) but these are different random variables. For example, $X_1$ is the weight of the first person; $X_2$ is the weight of the second person, etc. It seems to me that in this sentence the authors have fallen into the trap of mistaking a random variable for its distribution. Or did I interpret things incorrectly? If this is a mistake, as I think it is, it is very unfortunate as it occurs in the paragraph explaining what a random sample is and this is precisely the place where the student is trying to get clarity on observations/random variables/distributions and how they interrelate in the case of a random sample.