Editing Joint probability distribution (section)

==Joint cumulative distribution function==
For a pair of random variables <math>X,Y</math>, the joint cumulative distribution function (CDF) <math>F_{X,Y}</math> is given by<ref name="KunIlPark">{{cite book | author=Park,Kun Il| title=Fundamentals of Probability and Stochastic Processes with Applications to Communications| publisher=Springer | year=2018 | isbn=978-3-319-68074-3}}</ref>{{rp|p. 89}}

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>F_{X,Y}(x,y) = \operatorname{P}(X\leq x,Y\leq y)</math>|{{EquationRef|Eq.1}}}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where the right-hand side represents the [[probability]] that the random variable <math>X</math> takes on a value less than or equal to <math>x</math> '''and''' that <math>Y</math> takes on a value less than or equal to <math>y</math>.

For <math>N</math> random variables <math>X_1,\ldots,X_N</math>, the joint CDF <math>F_{X_1,\ldots,X_N}</math> is given by

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>F_{X_1,\ldots,X_N}(x_1,\ldots,x_N) = \operatorname{P}(X_1 \leq x_1,\ldots,X_N \leq x_N)</math>|{{EquationRef|Eq.2}}}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

Interpreting the <math>N</math> random variables as a [[random vector]] <math>\mathbf{X} = (X_1,\ldots,X_N)^T</math> yields a shorter notation:

:<math>F_{\mathbf{X}}(\mathbf{x}) = \operatorname{P}(X_1 \leq x_1,\ldots,X_N \leq x_N)</math>