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Basic terminology question.

I hear “let the sample be i.i.d.“ and “let these random variables be i.i.d.” being used interchangeably.

Even Wikipedia uses both:

A collection of random variables is independent and identically distributed if...

vs.

It is commonly assumed that observations in a sample are effectively i.i.d.

Are there different nuances or are they equivalent?

Also, are the points in a sample regarded as repeated measurements of one random variable, or one measurement each of multiple i.i.d. random variables?

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From Wikipedia, two Random Variables (RVs) (remark: you can generalize this to any number of RVs) are independent and identically distributed (i.i.d.) if their CDF is the same for any element of the domain $I$ and if their joint CDF factorizes in the product of the marginal CDFs: $${\begin{aligned}&F_{X}(x)=F_{Y}(x)\,&\forall x\in I\\&F_{X,Y}(x,y)=F_{X}(x)\cdot F_{Y}(y)\,&\forall x,y\in I\end{aligned}}$$

(Note that this also imply that their pdfs are the same (almost everywhere, i.e. on the whole domain except for sets of measure zero, but this is a technical condition so don't worry about it)

Realizations of a RV are usually referred to as samples, i.e. roughly speaking their outcome. The assumption that samples generated by a RV are i.i.d. simply refers to the fact that underlying RVs, whose realizations you observe in the samples, are i.i.d..

So replying to your questions:

  1. they are essentially the same thing.

  2. You can regard as repeated measurements of one random variable since the CDFs of two i.i.d. RV are the same.

I hope I clarified your doubts!

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