Difference between r1.3 and the current
@@ -8,6 +8,7 @@
현실을 모델링관련: [[확률변수,random_variable]] (RV)
[[마르코프_연쇄,Markov_chain]]과 관계가?
See [[확률및랜덤프로세스]]
@@ -25,8 +26,47 @@
then, $\lbrace X(t,\forall\zeta\in S),t\in I\rbrace\equiv X(t),\quad t\in I$⇔ '''random process, r.p., 확률과정'''
여기서 만약 $I$ 가
countable set이면, $X(t)$ 는 discrete-time r.p. (이산시간확률과정)
continuous set이면, $X(t)$ 는 continuous-time r.p. (연속시간확률과정)
from http://www.kocw.net/home/search/kemView.do?kemId=1279832 22. Random process 1:11
----
[[자기상관,autocorrelation]] R
[[자기공분산,autocovariance]] C
Def.
For continuous-time random process $X(t),$
(i) the mean function $m_X(t)$ of random process $X(t):$
$m_X(t)=E(X(t))=\int_{-\infty}^{\infty}xf_{X(t)}(x)dx$
(ii) the variance function $VAR(X(t))$ of random process $X(t):$
$VAR(X(t))=\int_{-\infty}^{\infty}\left{x-m_X(t)\right}^2 f_{X(t)}(x)dx$
$=\int_{-\infty}^{\infty}x^2 f_{X(t)}(x)dx-m_X(t)^2$
(iii) the autocorrelation $R_X(t_1,t_2)$ of random process $X(t):$
$X(t_1)=X_1,X(t_2)=X_2$ 일 때
$R_X(t_1,t_2)=E\left(X(t_1)X(t_2)\right)=E(X_1X_2)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}xyf_{X_1,X_2}(x,y)dxdy$
여기서 $f_{X_1,X_2}(x,y)$ 는 joint pdf of X,,1,, and X,,2,,
$\to R_X(t,t)=E\left(X(t)^2\right)$
(iv) the autocovariance $C_X(t_1,t_2)$ of random process $X(t):$
$X(t_1)=X_1,X(t_2)=X_2$ 일 때
$C_X(t_1,t_2)=E\left(\left(X(t_1)-m_X(t_1)\right)\left(X(t_2)-m_X(t_2)\right)\right)$
$=R_X(t_1,t_2)-m_X(t_1)m_X(t_2)$
$=E(X(t_1)X(t_2))-m_X(t_1)m_X(t_2)$
$\to C_X(t,t)=VAR(X(t))$
(v) the correlation coefficient $\rho_X$ of random process
$\rho_X(t_1,t_2)=\frac{C_X(t_1,t_2)}{\sqrt{C_X(t_1,t_1)}\times\sqrt{C_X(t_2,t_2)}}$
see also [[상관계수,correlation_coefficient]]
from 23. Specifying a Random Process
----
Up: [[과정,process]]
현실을 모델링
See 확률및랜덤프로세스
다음은 시간을 픽스.
and for fixed : random variable
and for fixed : random variable
then,
⇔ random process, r.p., 확률과정
여기서 만약 가countable set이면, 는 discrete-time r.p. (이산시간확률과정)
continuous set이면, 는 continuous-time r.p. (연속시간확률과정)
from http://www.kocw.net/home/search/kemView.do?kemId=1279832 22. Random process 1:11continuous set이면, 는 continuous-time r.p. (연속시간확률과정)
Def.
For continuous-time random process
(i) the mean function of random process
(ii) the variance function of random process
(iii) the autocorrelation of random process
see also 상관계수,correlation_coefficient
일 때
(iv) the autocovariance of random process 여기서 는 joint pdf of X1 and X2
일 때
(v) the correlation coefficient of random processfrom 23. Specifying a Random Process
Up: 과정,process
----
- [1] 이 그래프는 이름이 여러가지임. sample path(표본경로), sample function(표본함수), realization of random process(확률 과정의 실현)