Suppose I have a distribution with mean M. Also, assume we have a set of i.i.d samples of size n denoted by X=$\{x_1,x_2,...,x_n\}$ from . As a result, all $x_1,...,x_n$ are independent with identical distribution.
We know that [X]=M.
Now suppose I derive another set of subsamples without replacement of size m from X where m ≤ n. Let's call this new subsamples Y=$\{y_1,y_2,...,y_m\}$.
Now can I say [Y]=M?
Based on the rule of total expectation, we know that [Y]=[[Y|X]]. I am guessing that using this law we may be able to answer yes to the previous question as the set X is not fixed.
If my claim is not correct, please give a counterexample or prove why we cannot show $ \mathbb{E}[X]\neq \mathbb{E}[Y]$