# Unbiased sampling and subsampling

Suppose I have a distribution $\mathbb{F}$ 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 $\mathbb{F}$. As a result, all $x_1, ..., x_n$ are independent with identical distribution.

We know that $\mathbb{E}[X]=M$.

Now suppose I derive another set of subsamples without replacement of size $m$ from $X$ where $m\leq n$. Let's call this new subsamples $Y=\{y_1, y_2, ..., y_m\}$.

Now can I say $\mathbb{E}[Y]=M$?

Based on the rule of total expectation, we know that $\mathbb{E}[Y]=\mathbb{E}[\mathbb{E}[Y|X]]$. 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.

• It depends on how you resample, but you have the right idea of looking at iterated expectations – Taylor Jul 23 '18 at 16:23
• Can you explain more? – Infintyyy Jul 23 '18 at 16:24
• The inner expectation requires the conditional probabilities, the ones that you haven’t specified. – Taylor Jul 23 '18 at 16:25
• Just assume a general subsampling without replacement. For example, the case where each time we sample a ball uniformly from a bag without replacement. The balls of this bag are already sampled i.i.d from a distribution $\mathbb{F}$. I hope this example makes the question more clear. – Infintyyy Jul 23 '18 at 16:29
• If the observations in $Y$ are sampled randomly from $X$, i.e. without any dependence on the values of the $x_i \in X$, then $Y$ is a random sample from $\mathbb{F}$ as well. – Jim Jul 23 '18 at 21:00