 Quote by musicgold
Why do I have to use the qualifier 'identically distributed'
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Because if the distributions are not identical the steps that follow would not be valid.
Quote by musicgold
how does a non-identically distributed sample of a random variable look
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The face value of a playing card drawn from a pack without replacement is a simple example.
(Crossposted with HallsofIvy, but I think my example is better ;) so I will let it stand)