Basic concept

Probability Space

• Sample space(표본공간) $\Omega$: a (fixed) set of all possible outcomes
• event(사건) $\mathcal F$: A set of events
• probability measure $\mathbb P:\mathcal F \rightarrow[0,1]$

<aside> 📐 공리

$\{A_i\}\subseteq\mathcal F$에 대해

1. $0 \le P(A_i)\le1$
2. $\mathbb P(\Omega)=1$
3. $if\ i\ne j\ and\ A_i\cap A_j=\emptyset,\ \mathbb P(\cup_iA_i)=\sum_i\mathbb P(A_i)$ </aside>

$(\Omega, \mathcal F, \mathbb P)$를 probability space라 한다.

Basic properties of probability

$A$를 사건이라 하자.

1. $\mathbb P(A^c) = 1-\mathbb P(A)$
2. If $B$가 사건이고$B\subseteq A,\ \mathbb P(B)\le \mathbb P(A)$
3. $0=\mathbb P(\emptyset)\le\mathbb P(A)\le\mathbb P(\Omega)=1$

Other concepts

Boole’s inequality (Union bound)

$$ ⁍ $$

Conditional probability

$$ \mathbb P(A|B)={\mathbb P(A\cap B)\over \mathbb P(B)} $$

Bayes’ rule

$$ \mathbb P(A|B)={\mathbb P(B|A)\mathbb P(A)\over\mathbb P(B)}\\ posterior={likelihood\times prior\over evidence} $$