In a probability space, we say that the random events $A_{1},\dots,A_{n}$ are independent if

for all $i_{1},\dots,i_{k}$ such that $1\leq i_{1}<i_{2}<\cdots<i_{k}\leq n$.

An arbitrary family of random events is independent if every finite subfamily is independent.

The random variables $X_{1},\dots,X_{n}$ are independent if, given any Borel sets $B_{1},\dots,B_{n}$, the random events $[X_{1}\in B_{1}],\dots,[X_{n}\in B_{n}]$ are independent. This is equivalent to saying that

where $F_{{X_{1}}},\dots,F_{{X_{n}}}$ are the distribution functions of $X_{1},\dots,X_{n}$, respectively, and $F_{{X_{1},\dots,X_{n}}}$ is the joint distribution function. When the density functions $f_{{X_{1}}},\dots,f_{{X_{n}}}$ and $f_{{X_{1},\dots,X_{n}}}$ exist, an equivalent condition for independence is that

An arbitrary family of random variables is independent if every finite subfamily is independent.