This question is about notation. I would like to make clear to the students that one can approximate the mean by randomly drawing a subset of data. I came up with the following:
$$ \begin{align} & \frac{1}{N} \sum_{i=1}^{N} x_i \\ \approx & \frac{1}{n} \sum_{j \in J_n} x_j \end{align} $$ Where $J_n$ is a set of $n$ indices, drawn from $[1..N]$.
I wonder whether there is a cleaner way of expressing this approximation. Maybe I miss a standard notation for "a set of $n$ indices, 'randomly' drawn from $[1..N]$" ?
As a follow up, I'd also like to know how to properly express "a set of $n$ indices, 'randomly' drawn (without replacement) from $[1..N]$."
Thank you for suggestions and comments.
update (clarification in response to some of the comments below)
We are given a set of $N$ data: {$x_1, x_2, ... , x_N$}. From this set, I want to draw a subset of $n$ data.
Example:
For $n=3$, a possible subset of data is {$x_{43}, x_{19}, x_{87}$}. That is, I draw $n=3$ elements from the set of indices {$1, 2, ..., N$}. Using the notation in the post above, I use the symbol $J_{n=3}$ to denote some set of $3$ indices . All indices are equally likely to be drawn.
Optionally we can distinguish the cases "drawing with replacement" and "drawing without replacement"