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For a population that is uniformly distributed , if samples are taken do they necessarily have a uniform distribution as well ?

The thought experiment is like this . If you have a bag of 70 balls with 10 balls of each the colours of the rainbow ( VIBGYOR) . From this if you select a set of 14 balls. What can you tell about the distribution of the sample . After repeated sampling with the distribution tend to 2 of each colour ?

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Loosely speaking, the Glivenko-Cantelli theorem says that the empirical distribution converges to the population distribution. This is how the world should work: if we keep drawing from a population, we should be pretty close to the population. Maybe it takes ten draws to get "close" and maybe it takes ten-trillion-bazillion draws, but we should get close.

However, we do not get a uniform distribution with a finite sample size. For a continuous uniform distribution, this cannot ever happen, as the empirical distribution is finite and therefore discrete. (Note that being discrete does not require finite-ness, however.) For a discrete uniform distribution, the empirical distribution can happen to be distributed the same way as the population distribution, but sometimes values will be over-represented or under-represented.

After repeated sampling with the distribution tend to 2 of each colour ?

No, though you will wind up averaging two of each color.

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In asymptotica, this is ensured by the Glivenko-Cantelli Theorem

With finite sample sizes, of course the distributions are different (cumulative empirical distribution function has jumps, unlike the true underlying distribution).

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