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I know this has been asked and answered, but many even have contradictory definitions as to which represents the mean of a probability distribution. I would like to cite: https://stats.stackexchange.com/a/137397/296060.
Regarding this, I would like to know: In the second case of un-fair die we have only taken a sample of 6 outcomes {1, 2, 3, 4, 5, 6}, therefore Sample Mean and E[X] are different. If we had taken a larger sample, the probability of each term would act out and, the Sample Mean and E[X] would again be the same, am I right?
Eg: A sample of 15 outcomes instead of 6 would have given {1, 1, 2, 2, 3, 3, 4, ,4, 4, 5, 5, 5, 6, 6, 6}, right ?
And then Sample Mean and E[X] would be same = 3.80.
Am I correct ?

kjetil b halvorsen
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Divyansh Singh
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  • Welcome to Cross Validated! What is $X$ in your mind? Traditionally, we would consider the sample mean to be an estimate of the expected value of a random variable $X$, where $X$, loosely speaking, is the population distribution. – Dave Sep 10 '20 at 03:00
  • Yeah. And so, can we say that a good sample is one where the E[X] is or almost equal to sample mean ? – Divyansh Singh Sep 10 '20 at 03:16
  • I guess, but we don’t get to know the population mean. After all, if we knew it, why would we go through the trouble of estimating it? // I’d also phrase it as “the sample mean is almost equal to E[X]” since the E[X] is a fixed value. – Dave Sep 10 '20 at 03:18
  • Ok. And is my conclusion correct considering the example I gave, that whether the E[X] and $\bar{X}$ turn out same or different depends on the sample we take and frequency of occurrence of elements in the sample ? – Divyansh Singh Sep 10 '20 at 03:23
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    E[X] does not depend on the sample. Certainly $\bar{x}$ depends on the sample, however. – Dave Sep 10 '20 at 03:25
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    Yeah, and so whether the Sample Mean turns out to be same as E[X] is just a matter of chance of sample we took. They don't have to necessarily be the same always – Divyansh Singh Sep 10 '20 at 03:29
  • I sure hope not, because that would create some difficulties for a sample of one family in a country with a mean of 2.5 children per family! – whuber Sep 10 '20 at 16:38
  • That's what I am implying, it depends on the sample you take, whether the Sample Mean will be same as expected value – Divyansh Singh Sep 10 '20 at 16:47

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