Editing Independence (probability theory) (section)

====More than two random variables====
A finite set of <math>n</math> random variables <math>\{X_1,\ldots,X_n\}</math> 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 <math>n</math> random variables <math>\{X_1,\ldots,X_n\}</math> is '''mutually independent''' if and only if for any sequence of numbers <math>\{x_1, \ldots, x_n\}</math>, the events <math>\{X_1 \le x_1\}, \ldots, \{X_n \le x_n \}</math> are mutually independent events (as defined above in {{EquationNote|Eq.3}}). This is equivalent to the following condition on the joint cumulative distribution function {{nowrap|<math>F_{X_1,\ldots,X_n}(x_1,\ldots,x_n)</math>.}} A finite set of <math>n</math> random variables <math>\{X_1,\ldots,X_n\}</math> is mutually independent if and only if<ref name=Gallager/>{{rp|p. 16}}

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>F_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = F_{X_1}(x_1) \cdot \ldots \cdot F_{X_n}(x_n) \quad \text{for all } x_1,\ldots,x_n</math>|{{EquationRef|Eq.5}}}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

It is not necessary here to require that the probability distribution factorizes for all possible {{nowrap|<math>k</math>-element}} subsets as in the case for <math>n</math> events. This is not required because e.g. <math>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)</math> implies <math>F_{X_1,X_3}(x_1,x_3) = F_{X_1}(x_1) \cdot F_{X_3}(x_3)</math>.

The measure-theoretically inclined reader may prefer to substitute events <math>\{ X \in A \}</math> for events <math>\{ X \leq x \}</math> in the above definition, where <math>A</math> is any [[Borel algebra|Borel set]]. That definition is exactly equivalent to the one above when the values of the random variables are [[real number]]s. 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 space]]s endowed by appropriate σ-algebras).