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June 14th, 2014, 07:26 PM   #1
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Which Gaussian?

Hi all,

Suppose I had two probability distributions, P1 and P2, which are both normal, have the same standard deviation (say, 1), but different means (m1 and m2). Now, suppose a sample of size N is drawn from either P1 or P2, and we would like to know which. My question is:

"What is the probability that the sample was drawn from P2, given that the sample mean is m2?"

What I'm not asking is if it's more likely to have been drawn from P2 than from P1---of course it probably is. Instead, think of it this way. If I took many, many sample sets (each of size N) from P1, you'd expect their sample means to follow the standard error of the mean. Eventually, you might just get a sample whose mean equals m2. I want to know if there's any statistic that can distinguish between a sample of size N drawn from P1 whose mean happens to equal m2 and a sample of size N drawn from P2 whose mean equals m2.

Any help is appreciated!
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June 15th, 2014, 03:06 PM   #2
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Essentially it wll depend on how close the means are in comparison to the standard deviation.
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