Bayes theorem:

Descriptions:

• Can only be used on CDF not PDF

Proof:

• Let $E$ and $F$ be events, $E=EF\cup E{F}^{\prime }$
  • Then $P(E)=P(EF)+P(E{F}^{\prime })$ $=P(E|F).P(F)+P(E|F^{\prime }).P(F^{\prime })$ $⟹P(E)=P(E|F).P(F)+P(E|F^{\prime }).P(F^{\prime })$

Special case:

• $P(F|E)=\frac{P(F).P(E|F)}{P(F).P(E|F)+P(F^{\prime }).P(E|F^{\prime })}=\frac{P(F\cap E)}{P(E)}$

$P(F_i|E)=\frac{P(F_i).P(E|F_i)}{P(F_1).P(E|F_1)+...+P(F_n).P(E|F_n)}$

• $F_1,F_2,...,F_n$ where all events F are mutually exclusive events (that is disjoint events) whose union is the sample space where $E$ is an event that has occurred