Conditioning a Random Variable by an Event

Definition

Event B가 주어졌을 떄 랜덤변수 X에 대하여

Conditional CDF

$$ F_{X|B}(x) = P[X\le x\mid B] $$

Conditional PMF

$$ P_{X|B}(x) = P[X=x\mid B] $$

Conditional PDF

$$ f_{X|B}(x) = \frac{dF_{X|B}(x)}{dx} $$

Theorem : Conditional PDF, PMF

Discrete

$$ P_{X|B}=\begin{cases}\displaystyle{\frac{P_X(x)}{P[B]}}&&x\in B\\\\0&& e/w\end{cases} $$

Continuous

$$ f_{X|B}=\begin{cases}\displaystyle{\frac{f_X(x)}{P[B]}}&&x\in B\\\\0&& e/w\end{cases} $$

Theorem : Partition B1, B2, ..., Bm

$X$의 Partition $B_1, B_2, \dots, B_m$이 주어졌을 떄

Discrete

$$ P_X(x) = \sum_{i = 1}^{m}P_{X|B_i}(x)P[B_i] $$