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Is there any known theorem about this?by JK423
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#1
Jan2413, 02:31 PM

P: 381

Suppose that you have a probability distribution P of a parameter which, for simplicity, it is twovalued. 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 NM results. BUT, we do the erasure with a completely random way. We calculate, again, the mean value with the NM 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. 


#2
Jan2413, 04:18 PM

Sci Advisor
P: 6,038

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.



#3
Jan2413, 04:49 PM

P: 381

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? 


#4
Jan2513, 02:46 PM

Sci Advisor
P: 6,038

Is there any known theorem about this?
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) 


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