Let $X$ be the outcome of a chance experiment with $E(X) = \mu$ and $V(X) = \sigma^2$. When $\mu$ and $\sigma^2$ can be estimated by repeating the experiment $n$ times with outcomes $x_1, x_2,...,x_n$, sample mean and variance are given by: $$\bar x = \frac{1}{n}\sum_{i=1}^{n}x_i$$ $$ s^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i-\bar x)^2 $$
This is followed by problems asking to prove $E(\bar x)=\mu$, $E((\bar x-\mu)^2)=\sigma^2/n$, $E(s^s)=\frac{n-1}{n}\sigma^2$. My question is notation related, should I be looking at $x_1, x_2,...x_n$ and $\bar x$ as random variables (which are typically denoted by capital letters in my book) each with $\mu$, $\sigma^2$, not actual outcomes of a specific sample (lower case letters are typically used to represent values i.e. $P(X = x)$)? This is a bit ambiguous since from the equations for $\bar x$ and $s^2$ I think (?) it could be interpreted both ways but in the proof it doesn't make sense if we are looking at a specific sample.
Yes, the $x_i$ are independent random variables having the distribution of $X$. Thus $\overline{x}$, which is defined in terms of the $x_i$’s, is a random variable, and so is $s^2$.