structural:summarization
More than two random variables

A finite set of n {\displaystyle n} random variables { X 1, …, X n } {\displaystyle \{X_{1},\ldots,X_{n}\}} is pairwise independent if and only if every pair of random variables is independent. Even if the set of random variables is pairwise independent, it is not necessarily mutually independent as defined next.

A finite set of n {\displaystyle n} random variables { X 1, …, X n } {\displaystyle \{X_{1},\ldots,X_{n}\}} is mutually independent if and only if for any sequence of numbers { x 1, …, x n } {\displaystyle \{x_{1},\ldots,x_{n}\}}, the events { X 1 ≤ x 1 }, …, { X n ≤ x n } {\displaystyle \{X_{1}\leq x_{1}\},\ldots,\{X_{n}\leq x_{n}\}} are mutually independent events (as defined above in Eq.3). This is equivalent to the following condition on the joint cumulative distribution function F X 1, …, X n (x 1, …, x n) {\displaystyle F_{X_{1},\ldots,X_{n}}(x_{1},\ldots,x_{n})}. A finite set of n {\displaystyle n} random variables { X 1, …, X n } {\displaystyle \{X_{1},\ldots,X_{n}\}} is mutually independent if and only if

It is not necessary here to require that the probability distribution factorizes for all possible k {\displaystyle k} -element subsets as in the case for n {\displaystyle n} events. This is not required because e.g. F X 1, X 2, X 3 (x 1, x 2, x 3) = F X 1 (x 1) ⋅ F X 2 (x 2) ⋅ F X 3 (x 3) {\displaystyle F_{X_{1},X_{2},X_{3}}(x_{1},x_{2},x_{3})=F_{X_{1}}(x_{1})\cdot F_{X_{2}}(x_{2})\cdot F_{X_{3}}(x_{3})} implies F X 1, X 3 (x 1, x 3) = F X 1 (x 1) ⋅ F X 3 (x 3) {\displaystyle F_{X_{1},X_{3}}(x_{1},x_{3})=F_{X_{1}}(x_{1})\cdot F_{X_{3}}(x_{3})}.

The measure-theoretically inclined may prefer to substitute events { X ∈ A } {\displaystyle \{X\in A\}} for events { X ≤ x } {\displaystyle \{X\leq x\}} in the above definition, where A {\displaystyle A} is any Borel set. That definition is exactly equivalent to the one above when the values of the random variables are real numbers. It has the advantage of working also for complex-valued random variables or for random variables taking values in any measurable space (which includes topological spaces endowed by appropriate σ-algebras).