Independent Random Variable

Two variables $X,Y$ are independent if: $\forall A,B\subset \mathbb{R}$ $P(X\in A,Y\in B)=P(X\in A)P(Y\in B)$

CDF/PMF/PDF Factorization

$$X\perp Y⟺\{\begin{array}{ll}{F}_{X,Y}(x,y)=F_X(x)F_Y(y)& (any)\\ f_{X,Y}(x,y)=f_X(x)f_Y(y)& (cont)\\ p_{X,Y}(x,y)=p_X(x)p_Y(y)& (disc)\end{array}$$