AKA [[확률과정,stochastic_process]]



=random walk? 
Related example: [[브라운_운동,Brownian_motion]]

현실을 모델링

관련: [[확률변수,random_variable]] (RV)
[[마르코프_연쇄,Markov_chain]]과 관계가?

See [[확률및랜덤프로세스]]

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Def. 
For each outcome $\zeta\in S,$ (ζ: [[결과,outcome]], S: [[표본공간,sample_space]])
consider function $X:\zeta\mapsto X(t,\zeta),\quad t\in I$

먼저 결과를(outcome을) 픽스.
then, for fixed $\zeta,$ the graph[* 이 그래프는 이름이 여러가지임. sample path(표본경로), sample function(표본함수), realization of random process(확률 과정의 실현)] of function $X(t,\zeta):$

다음은 시간을 픽스.
and for fixed $t=t_k,\;X(t_k,\forall\zeta)=X_k$ : random variable

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
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[[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

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Up: [[과정,process]]