If there is a sample with sequence $ x_1 , x_2 , ... , x_n$ for example, that is taken randomly from a population, what exactly are these elements $ x_1 , x_2 , ... , x_n$? What do they represent? Every time I search for an explanation, all they do is take some numbers $ 2,5,3,1,6,3$ for example and estimate the mean $\mu$ of the population, but what do these elements $ 2,5,3,1,6,3$ represent?
I'm asking this question because I'm trying to prove the expected value of the sample mean $\bar x$
$ \bar x = \frac {\sum x_i}{n} = \frac {x_1 + x_2 + ... + x_n}{n}$
$E( \bar x) = \frac {1}{n} [E(x_1) + E(x_2) + ... + E(x_n)]$
I'm stuck here because I don't know what $x_1,x_2,...,x_n$ represent and what does it mean to take the expected value of them, I know each expected value of them should be equal to $\mu$ but I don't know why, hopefully someone enlightens me, thank you.
A lecturer wants to know the average age of the people in a large auditorium. It's obviously impractical to ask everyone his age age, so he selects $10$ people at random and asks them their ages. The $10$ people chosen are the elements of a random sample. The average of their ages is the sample mean.
EDIT
Suppose we have a set of $1000$ distinct integers. We select $10$ integers at random and compute their average. The $10$ integers chosen are the elements of the sample, and their average is the sample mean. But we could perform this experiment again, choosing another random sample of $10$ numbers (after replacing the first sample) and computing their average. Say we do this repeatedly. Each time we do this, we get a value of a random variable $X.$ You are being asked to show that $E(X)$ is the average of all the numbers in the population, so that taking the sample mean is actually a reasonable way of estimating the population average.
In answer to your second comment, $x_1$ is a random variable whose value is the first number selected at random, so it makes perfect sense to ask for its expectation. When we say $x_1=3$ we mean that in a particular experiment, the first number selected was $3,$ not that $x_1$ is a symbol representing the integer $3.$