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 )

 

[blue_raver22]

I am not aware of any specific theorem that guarantees the equality of <S> and <S'> for all cases and parameters. However, there are some principles and theorems in probability theory that can shed light on this situation.

Firstly, the Law of Large Numbers states that as the number of trials (N) increases, the sample mean (in this case, <S>) will approach the true mean of the population (in this case, the probability distribution P). This means that as N gets larger, <S> will become a more accurate representation of P.

Secondly, the Central Limit Theorem states that as the sample size (N-M) increases, the sampling distribution of the sample mean becomes approximately normal. This means that <S'> will approach a normal distribution with a mean of P and a standard deviation that decreases as (N-M) increases.

Based on these principles, we can infer that as N and (N-M) become larger, <S> and <S'> will become closer and closer to each other. However, the equality of <S> and <S'> is not guaranteed, as it ultimately depends on the specific values of N and M.

In summary, while there is no known theorem that guarantees the equality of <S> and <S'>, the Law of Large Numbers and the Central Limit Theorem suggest that <S> and <S'> will become increasingly closer as N and (N-M) increase. Further research and analysis may be needed to determine the extent of this closeness and any potential relationship between N and M that may guarantee equality.