Conventional notation for samples of random variables? Is there any conventional notation to separate population values from sample values in statistics?
For example, how would one differentiate $\mathbb{E}(X) = \mu$, the population mean, from $\mathbb{E}(X) = \bar{x}$, the sample mean? Would you use a small $x$ like this: $\mathbb{E}(x) = \bar{x}$ ?
 A: $\newcommand{\v}{\operatorname{var}}\newcommand{\e}{\operatorname{E}}$Some aspects of notation in this regard are chaotic, but nobody uses such a notation as $\e(X)$ to denote a sample mean.
If the sample is $X_1,\ldots,X_n$ and these are i.i.d. random variables then $\e(X_1)=\cdots=\e(X_n)$ is the population mean, and the sample mean $(X_1+\cdots+X_n)/n$ is often denoted $\overline X$ or $\overline X_n,$ and that is random variable in its own right, having the same expected value as it any of the observations in the sample, i.e. $\e(\overline X) = \e(X_1).$
Very often Greek letters are used, thus: $\mu=\e(X_1) = \cdots =\e(X_n),$ $\sigma^2 = \v(X_1) = \cdots = \v(X_n).$
Then we have
\begin{align}
\mu & = \e(X_1) = \cdots = \e(X_n) = \text{the population mean} \\[10pt]
\sigma^2 & = \v(X_1) = \cdots = \v(X_n) = \text{the population variance} \\[10pt]
\overline X & = \frac{X_1+\cdots+X_n} n = \text{the sample mean} \\[10pt]
S^2 & = \frac { (X_1-\overline X)^2 + \cdots + (X_n - \overline X)^2} {n-1} = \text{the sample variance}
\end{align}
And $S^2$ is also a random variable in its own right.
One also often writes $\widehat \mu$ or $\widehat{\sigma^2}$ to refer to functions of $X_1,\ldots,X_n$ used as estimators of the unobservable population mean and population variance. These are also random variables in their own right.
In this context, the word "random" often in effect means depending on the sample.
