Editing Independence (probability theory) (section)

===For real valued random variables===
====Two random variables====
Two random variables <math>X</math> and <math>Y</math> are independent [[if and only if]] (iff) the elements of the [[Pi system|{{pi}}-system]] generated by them are independent; that is to say, for every <math>x</math> and <math>y</math>, the events <math>\{ X \le x\}</math> and <math>\{ Y \le y\}</math> are independent events (as defined above in {{EquationNote|Eq.1}}). That is, <math>X</math> and <math>Y</math> with [[cumulative distribution function]]s <math>F_X(x)</math> and <math>F_Y(y)</math>, are independent [[if and only if|iff]] the combined random variable <math>(X,Y)</math> has a [[joint distribution|joint]] cumulative distribution function<ref name=Gallager>{{cite book | first=Robert G. | last=Gallager| title=Stochastic Processes Theory for Applications| publisher=Cambridge University Press| year=2013 | isbn=978-1-107-03975-9}}</ref>{{rp|p. 15}}

{{Equation box 1
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|equation = {{NumBlk||<math>F_{X,Y}(x,y) = F_X(x) F_Y(y) \quad \text{for all } x,y</math>|{{EquationRef|Eq.4}}}}
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|border colour = #0073CF
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or equivalently, if the [[probability density function|probability densities]] <math>f_X(x)</math> and <math>f_Y(y)</math> and the joint probability density <math>f_{X,Y}(x,y)</math> exist,

:<math>f_{X,Y}(x,y) = f_X(x) f_Y(y) \quad \text{for all } x,y.</math>

====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}}}}
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|border colour = #0073CF
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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).