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These equations show the joint distribution or density characterised as a mixture distribution based on the underlying limiting empirical distribution (or a parameter indexing this distribution).
Note that not all finite exchangeable sequences are mixtures of i.i.d. To see this, consider sampling without replacement from a [[finite set]] until no elements are left. The resulting sequence is exchangeable, but not a mixture of i.i.d. Indeed, conditioned on all other elements in the sequence, the remaining element is known.
== Covariance and correlation ==
Exchangeable sequences have some basic [[covariance and correlation]] properties which mean that they are generally positively correlated. For infinite sequences of exchangeable random variables, the covariance between the random variables is equal to the variance of the mean of the underlying distribution function.<ref name="O'Neill"/> For finite exchangeable sequences the covariance is also a fixed value which does not depend on the particular random variables in the sequence. There is a weaker lower bound than for infinite exchangeability and it is possible for negative correlation to exist.
'''Covariance for exchangeable sequences (infinite):''' If the sequence <math>X_1,X_2,X_3,\ldots</math> is exchangeable, then
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* {{sectionlink|Resampling (statistics)|Permutation tests}}, statistical tests based on exchanging between groups
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