Independence

Overview

Events $A$ and $B$ are independent if $P(A\cap B)=P(A)P(B)$. If $P(A)>0$ and $P(B)>0$, then this is equivalent to saying

Let $\mathcal{A}$ be a collection of more than two events. We say the events in $\mathcal{A}$ are pairwise independent if any two events in $\mathcal{A}$ are independent. We say the events in $\mathcal{A}$ are mutually independent if every event in $\mathcal{A}$ is independent of any combination of other events in $\mathcal{A}$.

Dependence

We say two events $A$ and $B$ are dependent if

• If $P(A\cap B)>P(A)P(B)$, then $A$ and $B$ are positively dependent.
• If $P(A\cap B)<P(A)P(B)$, then $A$ and $B$ are negatively dependent.

Conditional

Let $C$ be an event such that $P(C)>0$. Events $A$ and $B$ are conditionally independent given $C$ if either of the following equivalent identities hold:

1. Factorization: $P(A,B|C)=P(A|C)P(B|C)$
2. Irrelevance: $P(A|B,C)=P(A|C)$

This is typically denoted as $(A\perp \perp B\mid C)$.