Is there any known theorem about this?

JK423

Suppose that you have a probability distribution P of a parameter which, for simplicity, it is two-valued. For example, it could be a coin where we denote heads with "+" and tails with "-". Suppose that we throw the coin N times, and the results tend to follow the probability ditribution P for large enough N (which naturally could be 50%-50%).
We then calculate the mean value of all these results, which are a series of "+" and "-" (N in total), and find a number <S>.
Suppose now, that, we erase M of these results (with M<N) from the total N, and we are left with the rest N-M results. BUT, we do the erasure with a completely random way. We calculate, again, the mean value with the N-M results, <S'>.

QUESTION
Will the mean values <S> and <S'> be equal?

The answer will surely depend on the numbers N, M. Is there any known theorem about this, that guarantees the equality for appropriate N, M?

Edit: For finite N, M ofcourse the equality is impossible. What i mean is whether <S'> can approach <S> very very close, for appropriate N, M.

mathman

Is your question about the sample mean or the theoretical mean? For the sample means, they could be different. For the theoretical means, they are equal.

JK423

Hmm, what is the difference? Perhaps, you mean that the sample mean includes a finite N while the theoretical an infinite N?

I am looking for a theorem on sample mean with finite N, so that it's applicable in real applications..
I am glad that the theoretical means are equal though :). Is there any proof of this to your knowledge?

mathman

The theoretical mean is determined from the probability distribution. It has nothing to do with sample size. There is a theorem (law of large numbers) which states (under the proper conditions) that the sample mean -> theoretical mean as N becomes infinite.

I don't know what kind of theorem you are looking for. Perhaps the following may help.
http://en.wikipedia.org/wiki/Sampling_(statistics)