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In all the examples I've seen so far, we always make the assumption that when we take a random sample from some population, each member of the sample comes from the same distribution as the other members i.e. they're identically distributed

Why is it okay to make this assumption?

Is it not possible to have members of the sample that follow some other distribution?

Is this a question worth pondering ?

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...we take a random sample from some population...

Here, population represents the distribution. If you take samples from different populations, your samples would be from different distributions, and therefore not identically distributed.

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  • $\begingroup$ How do we know that a population follows a particular distribution? Would we deduce what the population distribution isfrom the empirical distribution of a large sample from that distribution? $\endgroup$ – stochasticmrfox Feb 12 at 16:26
  • $\begingroup$ population doesn't follow a distribution, it represents it. individual samples follow distribution. I couldn't understand your second sentence. $\endgroup$ – gunes Feb 12 at 16:58
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Often it is simply because it is a reasonable assumption. If you sample from a population, say for instance, the height of men in Europe, it is reasonable to assume they follow the same distribution.

If you are in a setting where you are uncomfortable with the assumption of identically distributed, you could do the bayesian approach and and construct a hierarchical model. Then you can get different posterior distributions, but then these are based on some "deeper" assumption. But then you need to consider how to construct this model.

In the end I think it boils down to if you are going to make a model, you need a place to start. And if you take measurements of individuals in a population, the assumption is reasonable.

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