Independent variable

The random variables $X$ and $Y$ are said to be independent if, for any two sets of real numbers $A$ and $B$ , $P X \in A, Y \in B = P X \in A P Y \in B$
In other words, $X$ and $Y$ are independent if, for all $A$ and $B$ ,
the events $X \in A$ and $Y \in B$ are independent.
It will follow if and only if, for all $a$ and $b$ $P X \leq a, Y \leq b = P X \leq a P Y \leq b$
In terms of the joint distribution function, $X$ and $Y$ are independent if $F (a, b) = F_{X} (a) F_{Y} (b)$ for all $a, b$

Synaptic Vault 🔒