Sub:
 [[이산확률변수,discrete_RV]]
 [[연속확률변수,continuous_RV]]

= Definition of RV =
A RV $X$ is a real-valued function of the experimental outcome.
 $X:\mathbb{\Omega}\to\mathbb{R}$
여기서
 $\mathbb{\Omega}$ = [[VG:표본공간,sample_space]]

= Discrete RV (DRV) =
A RV $X$ is discrete if its range is finite or countably infinite.
여기서, range $r(X)=\left{x\middle|\exists\omega\in\mathbb{\Omega}\textrm{ such that }X(\omega)=x\right}$
 즉 sample $\omega$ 를 $x$ 라는 값으로 대응시키는?
  그런 함수가 $X$ 이고 그것을 독립변수(or 정의역)로 하는 대응관계가 range r? 조건제시법 이해가 잘...

ex. two fair coin tosses
X = # of heads
Ω = {HH, HT, TH, TT}
X(HH) = 2
X(TT) = 0
r(X) = {0, 1, 2}

ex. sampling a number ω from ［-1,1］
$X(\omega)=\begin{cases}1,&\textrm{ if }\omega>0\\0,&\textrm{ if }\omega=0\\-1,&\textrm{ if }\omega<0\end{cases}$

페이지: [[이산확률변수,discrete_RV]]

= Probability mass function (PMF) =
The PMF $p_X(x)$ of a DRV $X$ is defined as:
 $p_X(x)=P(X=x)=P\left(\left{\omega\in\mathbb{\Omega}\middle|X(\omega)=x\right}\right)$

위의 동전던지기를 예로 들면
$p_X(x)=$
 ¼ if x=0 TT
 ½ if x=1 HT TH
 ¼ if x=2 HH

= Mean or expectation =
평균 or 기대값
 $E[X]=\sum_x x\cdot P_X(x)$

위 동전던지기를 예로 들면 앞면이 나오는 횟수의 기대값은
 E［Ｘ］= 0·¼ + 1·½ + 2·¼ = 1

= Variance =
분산
 $V[X]=E[(X-E[X])^2]=\sum_x(x-E[X])^2P_X(x)$

Properties 특징
 i) $E[aX+b]=aE[X]+b$
 ii) ${\rm Var}[aX+b]=a^2{\rm Var}(X)$

= Conditioning RV on event =
Given an event $A$ with $P(A)>0,$ the conditional PMF $P_{X|A}$ of a DRV $X$ is defined as
 $P_{X|A}(x)=P(X=x|A)=\frac{P(\left{X=x\right}\cap A)}{P(A)}$

Conditioning X on Y:
 $P_{X|Y}(x|y)=P(X=x|Y=y)$
 $=\frac{P(X=x,Y=y)}{P(Y=y)}$
 $=\frac{P_{X,Y}(x,y)}{P_Y(y)}$

 $=P(\{X=x\}\cap\{Y=y\})$
 $=P(\{\omega\in\mathbb{\Omega}|X(\omega)=x\textrm{ and }Y(\omega)=y\})$

= Conditional expectation =
 $E[X|A]=\sum_x x\cdot P_{X|A}(x)$

= Joint PMF of two DRVs =
X, Y: DRVs
 $P_{X,Y}(x,y)=P(X=x,Y=y)$

= Independence =
Two DRVs $X$ and $Y$ are independent if
 $P_{X,Y}(x,y)=P_X(x)P_Y(y)\;\;\;\forall x,y$

= Continuous RV (CRV) =
A RV $X$ is continuous if there exists a non-negative function $f_X()$ such that
 $P(X\in B)=\int_B f_X(x)dx,\;\;\forall B\subset\mathbb{R}$
interval B가 $B=[a,b]$ 라면
 $P(a\le X\le b)=\int_a^b f_X(x)dx$

페이지: [[연속확률변수,continuous_RV]]

= DRV and CRV =
||     || DRV || CRV ||
||PF   || PMF [[br]] $P_X(x)=P(X=x)$ || PDF [[br]] $f_X(x),P(X\in B)=\int_B f_X(x)dx$ ||
||DF   || CDF [[br]] $F_X(k)=P(X\le k)=\sum_{x\le k} P_X(x)$ || CDF [[br]] $F_X(x)=P(X\le x)=\int_{-\infty}^x f_X(t)dt$ ||
||Mean || $E[X]=\sum_x xP_X(x)$ || $E[X]=\int_{-\infty}^{\infty} xf_X(x)dx$ ||
||Var  || $V(X)=\sum(x-E[X])^2P_X(x)$ || $V(X)=\int_{-\infty}^{\infty}(x-E[X])^2dx$ ||

= 독립 and Joint =
discrete에 independence가 있다면
continuous에 joint가 있다?

raised comma=⸴

||Independence ||P,,X⸴Y,,(x,y)=P,,X,,(x)·P,,Y,,(y) ||f,,X⸴Y,,(x,y)=f,,X,,(x)·f,,Y,,(y), ∀x,y ||
||Conditioning ||P,,X|Y,,(x|y)=P,,X⸴Y,,(x,y) / P,,Y,,(y) ||f,,X|Y,,(x|y)=f,,X⸴Y,,(x,y) / f,,Y,,(y) ||

(from [[http://kocw.net/home/search/kemView.do?kemId=991018 건대강의]])

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Up: [[확률및랜덤프로세스,probability_and_random_process]]