Editing Independence (probability theory) (section)

====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
|indent =
|title=
|equation = {{NumBlk||<math>F_{X,Y}(x,y) = F_X(x) F_Y(y) \quad \text{for all } x,y</math>|{{EquationRef|Eq.4}}}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

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>