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In a probability space, we say that the random events
 are independent if
for all
 such that
 .
An arbitrary family of random events is independent if every finite subfamily is independent.
The random variables
 are independent if, given any Borel sets
 , the random events
![$ [X_1\in B_1],\dots,[X_n\in B_n]$](http://images.planetmath.org:8080/cache/objects/1053/l2h/img7.png "$ [X_1\in B_1],\dots,[X_n\in B_n]$") are independent. This is equivalent to saying that
where
 are the distribution functions of
 , respectively, and
 is the joint distribution function. When the density functions
 and
 exist, an equivalent condition for independence is that
An arbitrary family of random variables is independent if every finite subfamily is independent.
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