Independent random variable

The random variables $X$ and $Y$ are said to be independent if, for any two sets of real numbers $A$ and $B$ , $P X \in A, Y \in B = P X \in A P Y \in B$
Denote with $x ⊥ ⊥ y$
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 $a, b$
Event A and B are independent if any of the 3 conditions hold:
1. $P (A ∣ B) = A$
2. $P (B ∣ A) = B$
3. $P (A \cap B) = P (A) . P (B)$ : product rule for independent event
  - Use this to find independence: $P (A \cap B) = P (A) + P (B) - P (A \cap B)$
Events A, B and C are independent if all following equations hold:
1. $P (A, B) = P (A) P (B)$
2. $P (B, C) = P (B) P (C)$
3. $P (C, A) = P (C) P (A)$
4. $P (A, B, C) = P (A) P (B) P (C)$

StrixTheKiet Notes