Joint Cumulative Distribution Function

Describes the probability two Random Variable being less than two given values. A CDF used to describe the joint distribution of RV

Definition

$F_{X,Y}(x,y)=P(X\le x,Y\le y)=P(\{X\le x\}\cap \{Y\le y\})$

Expressing Cartesian Product of intervals

With $B=(x_1,x_2]\times [y_1,y_2)=\{(x,y):x\in (x_1,x_2]\wedge y\in (y_1,y_2]\}\subset {\mathbb{R}}^2$ Using Joint CDF directly, $P((X,Y)\in B)=P(x_1<X\le x_2,y_1<Y\le y_2)=F_{X,Y}(x_2,y_2)-F_{X,Y}(x_2,y_1)-F_{X,Y}(x_1,y_2)-F_{X,Y}(x_1,y_1)$