Schervish's (1995) Theory of Statistics defines exchangeability like this (p. 7):

A finite set $X_1, …, X_n$ of random quantities is said to be exchangeable if every permutation of $(X_1, …, X_n)$ has the same joint distribution as every other permutation. An infinite collection is exchangeable if every finite subcollection is exchangeable.

Likewise, Wikipedia currently says:

Formally, an exchangeable sequence of random variables is a finite or infinite sequence $X_1, X_2, X_3, …$ of random variables such that for any finite permutation of the indices, the joint probability distribution of the permuted sequence is the same as the joint probability distribution of the original sequence.

and also:

In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed (i.i.d.) if each random variable has the same probability distribution as the others and all are mutually independent.

But what exactly is meant in these cases to say that, e.g., a random variable $X$ has "the same distribution" as another random variable $Y$? Does it mean that the CDFs of $X$ and $Y$ are equal? Or does it mean the CDFs are equal almost everywhere? Or is it something else?

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    $\begingroup$ Two CDFs that are equal almost everywhere are equal. ("Almost everywhere" has to be understood with respect to Lebesgue measure on $\mathbb{R}$.) $\endgroup$
    – whuber
    Jun 10, 2016 at 13:36
  • $\begingroup$ @whuber Ah, I suspected that the right-continuity of CDFs might be enough for that, but I didn't check. $\endgroup$ Jun 10, 2016 at 13:45
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    $\begingroup$ Strictly speaking, $L^2$ does not consist of functions - it consists of equivalence classes of functions, where equivalence is given by equality almost everywhere. $\endgroup$ Jun 11, 2016 at 19:27

1 Answer 1


Schervish seems to use "distribution" to mean "induced probability measure", so I guess what is meant is that the probability measures of $X$ and $Y$ are equal, or equivalently, their CDFs are equal.


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