확률변수,RV (rev. 1.10)
Definition of RV ¶
A RV $\displaystyle X$ is a real-valued function of the experimental outcome.
$\displaystyle X:\mathbb{\Omega}\to\mathbb{R}$
여기서
Discrete RV (DRV) ¶
A RV $\displaystyle X$ is discrete if its range is finite or countably infinite.
여기서, range $\displaystyle r(X)=\left{x\middle|\exists\omega\in\mathbb{\Omega}\textrm{ such that }X(\omega)=x\right}$
X = # of heads
Ω = {HH, HT, TH, TT}
X(HH) = 2
X(TT) = 0
r(X) = {0, 1, 2}
여기서, range $\displaystyle r(X)=\left{x\middle|\exists\omega\in\mathbb{\Omega}\textrm{ such that }X(\omega)=x\right}$
즉 sample $\displaystyle \omega$ 를 $\displaystyle x$ 라는 값으로 대응시키는?
ex. two fair coin tosses그런 함수가 $\displaystyle X$ 이고 그것을 독립변수(or 정의역)로 하는 대응관계가 range r? 조건제시법 이해가 잘...
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］
$\displaystyle X(\omega)=\begin{cases}1,&\textrm{ if }\omega>0\\0,&\textrm{ if }\omega=0\\-1,&\textrm{ if }\omega<0\end{cases}$
$\displaystyle X(\omega)=\begin{cases}1,&\textrm{ if }\omega>0\\0,&\textrm{ if }\omega=0\\-1,&\textrm{ if }\omega<0\end{cases}$
페이지: 이산확률변수,discrete_RV
Probability mass function (PMF) ¶
The PMF $\displaystyle p_X(x)$ of a DRV $\displaystyle X$ is defined as:
$\displaystyle p_X(x)=$
$\displaystyle p_X(x)=P(X=x)=P\left(\left{\omega\in\mathbb{\Omega}\middle|X(\omega)=x\right}\right)$
위의 동전던지기를 예로 들면$\displaystyle p_X(x)=$
¼ if x=0 TT
½ if x=1 HT TH
¼ if x=2 HH
½ if x=1 HT TH
¼ if x=2 HH
Mean or expectation ¶
평균 or 기대값
$\displaystyle E[X]=\sum_x x\cdot P_X(x)$
위 동전던지기를 예로 들면 앞면이 나오는 횟수의 기대값은E［Ｘ］= 0·¼ + 1·½ + 2·¼ = 1
Variance ¶
분산
$\displaystyle V[X]=E[(X-E[X])^2]=\sum_x(x-E[X])^2P_X(x)$
Properties 특징i) $\displaystyle E[aX+b]=aE[X]+b$
ii) $\displaystyle {\rm Var}[aX+b]=a^2{\rm Var}(X)$
ii) $\displaystyle {\rm Var}[aX+b]=a^2{\rm Var}(X)$
Conditioning RV on event ¶
Given an event $\displaystyle A$ with $\displaystyle P(A)>0,$ the conditional PMF $\displaystyle P_{X|A}$ of a DRV $\displaystyle X$ is defined as
$\displaystyle P_{X|A}(x)=P(X=x|A)=\frac{P(\left{X=x\right}\cap A)}{P(A)}$
Conditioning X on Y:$\displaystyle P_{X|Y}(x|y)=P(X=x|Y=y)$
$\displaystyle =\frac{P(X=x,Y=y)}{P(Y=y)}$
$\displaystyle =\frac{P_{X,Y}(x,y)}{P_Y(y)}$
$\displaystyle =P(\{X=x\}\cap\{Y=y\})$
$\displaystyle =P(\{\omega\in\mathbb{\Omega}|X(\omega)=x\textrm{ and }Y(\omega)=y\})$
$\displaystyle =\frac{P(X=x,Y=y)}{P(Y=y)}$
$\displaystyle =\frac{P_{X,Y}(x,y)}{P_Y(y)}$
$\displaystyle =P(\{X=x\}\cap\{Y=y\})$
$\displaystyle =P(\{\omega\in\mathbb{\Omega}|X(\omega)=x\textrm{ and }Y(\omega)=y\})$
Independence ¶
Two DRVs $\displaystyle X$ and $\displaystyle Y$ are independent if
$\displaystyle P_{X,Y}(x,y)=P_X(x)P_Y(y)\;\;\;\forall x,y$
Continuous RV (CRV) ¶
A RV $\displaystyle X$ is continuous if there exists a non-negative function $\displaystyle f_X()$ such that
$\displaystyle P(X\in B)=\int_B f_X(x)dx,\;\;\forall B\subset\mathbb{R}$
interval B가 $\displaystyle B=[a,b]$ 라면$\displaystyle P(a\le X\le b)=\int_a^b f_X(x)dx$
페이지: 연속확률변수,continuous_RVDRV and CRV ¶
| DRV | CRV | |
| PF | PMF $\displaystyle P_X(x)=P(X=x)$ | PDF $\displaystyle f_X(x),P(X\in B)=\int_B f_X(x)dx$ |
| DF | CDF $\displaystyle F_X(k)=P(X\le k)=\sum_{x\le k} P_X(x)$ | CDF $\displaystyle F_X(x)=P(X\le x)=\int_{-\infty}^x f_X(t)dt$ |
| Mean | $\displaystyle E[X]=\sum_x xP_X(x)$ | $\displaystyle E[X]=\int_{-\infty}^{\infty} xf_X(x)dx$ |
| Var | $\displaystyle V(X)=\sum(x-E[X])^2P_X(x)$ | $\displaystyle V(X)=\int_{-\infty}^{\infty}(x-E[X])^2dx$ |
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