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°²©w¹ï¥ô·N¤@­Ó t¡AX(t) ¬O¦b®É¶¡ t ®Éªº¤@­ÓÀH¾÷ÅܼơA §Ú­Ì´N¥s³o¤@²ÕÀH¾÷ÅÜ¼Æ $\{X(t) , 0 \leq t < \infty\}$¬°¤@­ÓÀH¾÷¹Lµ{ (random process©Îstochastic process)¡C

©w¸q¤@ °²©w¤@­ÓÀH¾÷¹Lµ{ $\{X(t) , 0 \leq t < \infty\}$ º¡¨¬¤U¦Cªº±ø¥ó®É¡A §Ú­Ì´NºÙ¥¦¬°¤@­Ó¤RªQ¹Lµ{ (Poisson process)¡A

(i) X(0)=0

(ii)¹ï©ó¥ô¦ó¤@¦C®É¶¡ $0 \leq s_1 \leq t_1$ $\leq s_2 \leq t_2$ $\leq$ ¡K $\leq s_n \leq t_n$¡An ¬°¤@¥ô·N¾ã¼Æ

\begin{displaymath}
X(t_1) - X(s_1),X(t_2)-X(s_2), \cdots, X(t_n) - X(s_n)
\end{displaymath}

¬O n ­Ó¿W¥ßÀH¾÷ÅܼơC

(iii)¹ï¥ô¦ó $t \geq 0$,$s \geq 0$,X(t)-X(0)¡]=X(t)¡^©M X(t+s)-X(s) ¦³¬Û¦Pªº¾÷²v¤À§G¡C

(iv)¹ï¥ô¦ó¤@­Ó t ©M¤@­Ó k

\begin{displaymath}
P(X(t)=k)=\frac{\lambda^k}{k!}e^{-\lambda t} \quad k=0,1,2\cdots
\end{displaymath}

¨ä¤¤ £f ¬O¤@µ¹©wªº°Ñ¼Æ¡C

§Ú­Ì¥ý¥Î¤@­Ó¹ê¨Ò¨Ó¸ÑÄÀ³o¨Ç±ø¥óªº·N¸q¡C °²·Q§Ú­Ì¦b¤@±ø¤½¸ô¤W¬YÂIÆ[¹î³q¹L³oÂIªº¥æ³q¶q¡C ¥O X(t) ¬°±q®É¶¡ 0 ¨ì®É¶¡ t ³q¹L³oÂIªº¨T¨®Á`¼Æ¡C «h±ø¥ó (i)ªí¥Ü¦b®É¶¡ 0 ®É¡A³q¹Lªº¨®¼Æ¬O 0¡C ±ø¥ó(ii)»¡¦b¤¬¤£¬Û¥æªº®É¬q (time intervals) ¤º³q¹Lªº¨®¼Æ¬O¬Û¤¬¿W¥ßªº¡C ±ø¥ó(iii)»¡¥u­n®É¬qªø«×¬Û¦P¡A ¤£¦P®É¬q¤º³q¹Lªº¨®¼Æ¦³¬Û¦Pªº¾÷²v¤À§G¡C ±ø¥ó (iv) »¡ X(t) ¦³¤@­Ó¨ã¦³°Ñ¼Æ $\lambda t$ ªº¤RªQ¤À§G¡C ±ø¥ó (i) ¬O¦X²zªº¡C ¦pªGÆ[¹îªº¥þ³¡®É¶¡¤£¤Óªøªº¸Ü¡A ±ø¥ó(ii)©M (iii)¤]¬O«Ü¦X²zªº¡C ±ø¥ó(iv)¦ü¥G¦³ÂI¬ðµM¡A¦ý¬O¦b²Ä¤T¸`ùاڭ̱N·|½Í¨ì¥¦¤]¬O«Ü¦X²zªº¡A ¦Ó¥B¤@¯ë¹ê´úªº¥æ³q¶qªº¬ö¿ý¤]¤ä«ù³o¤@°²©wªº¡C

¨ä¥LÃþ¦üªº¨Ò¤l¡A¦p X(t) ¥Nªí¦b®É¬q [0,t] ¤º¬Y®ÈÀ]¤ºÁ`¦@¥´¶i¨Óªº¹q¸Ü¦¸¼Æ¡A©Î X(t) ¥Nªí¬YÂå°|¦b®É¬q [0,t] ¤º¨Ó±¾¸¹ªº¯f¤H­Ó¼Æµ¥µ¥¡A °²©w¬°¤@¤RªQ¹Lµ{¡A³£»P¨Æ¹ê«Ü¬Ûªñ¡C ©Ò¥H¤RªQ¹Lµ{ªºÀ³¥Î«D±`¤§¼s¡C

¹ï©ó¤RªQ¹Lµ{¤@¯ë±Ð¬ì®Ñùؤ]±`±Ä¥Î¤U­±ªº©w¸q¡C

©w¸q¤G °²¦p¤@­ÓÀH¾÷¹Lµ{ $\{X(t) , 0 \leq t < \infty\}$ º¡¨¬¥H¤Wªº±ø¥ó (i),(ii),(iii) ¤Î

(iv') $P(X(h)=0) = 1 - \lambda h + o(h)$, $P(X(h)=1) = \lambda h + o(h)$, $P(X(h)\geq2)=o(h)$¡C ·í h «Ü¤p®É¡A¦¹¦a £f ¬°¤@±`¼Æ¡C
«h§Ú­ÌºÙ $\{X(t) , 0 \leq t < \infty\}$ ¬°¤@¤RªQ¹Lµ{¡C

³oùاڥΨì²Å¸¹ f(t)=o(t) °²¦p $f(t)/t \longrightarrow 0$¡A ·í t $\longrightarrow$ 0 ¡C

¥H¤U§Ú­nÃÒ§Y¦b±ø¥ó (i),(ii) ©M (iii) ¤§¤U¡A ±ø¥ó (iv) ©M (iv') ¬O¬Ûµ¥ªº¡A©Ò¥H©w¸q¤@©M©w¸q¤G¬Ûµ¥ªº¡C

¥ýÃÒ (iv) $\Rightarrow$ (iv')¡G
±ø¥ó(iv)»¡

\begin{displaymath}
P(X(h)=k)=\frac{(\lambda h)^k}{k!}e^{-\lambda k}, \quad k=0,1,2\cdots
\end{displaymath}

©Ò¥H·í k=0 ¤Î 1 ®É

\begin{eqnarray*}
P(X(h)=0)=e^{-\lambda h} &=& 1 -\lambda h + \frac{ (\lambda h)...
... &=& \lambda h (1-\lambda h + o(h)) \\
& = & \lambda h + o(h) ,
\end{eqnarray*}


©ó¬O

\begin{displaymath}
P(X(h) \geq 2) = 1 - P(X(h)=0) - P(X(h)=1)=o(h)
\end{displaymath}

©Ò¥H§Ú­Ì±o¨ì (iv')¡C

¦AÃÒ (iv') $\Rightarrow$ (iv):
¥O

Pk(t) = P(X(t)) =k) (3)

©ó¬O

\begin{eqnarray*}
P_k(t+h) &=& P(X(t+h)=k) \\
&=& P( X(t) =k,X(t+h)-X(t)=0)\\ ...
...\\
&=& P_k(t) - \lambda h P_k(t) + \lambda h P_{k-1}(t) + o(h)
\end{eqnarray*}


©Ò¥H

\begin{displaymath}
\frac{P_k(t+h)-P_k(t)}{h} = - \lambda P_k(t) + \lambda P_{k-1}(t) + o(1)
\end{displaymath}

¥H¤W¬O°²©w $k \geq 1$¡C¦p k=0 ¤W¦¡¤]¦¨¥ß¡A±©¥kÃä²Ä¤G¶µ¤£¦s¦b¡C

¥O $h \longrightarrow 0$ ±o¨ì¤èµ{¦¡

\begin{displaymath}
\begin{cases}
P_k'(t) = -\lambda P_k(t) + \lambda P_{k-1}(t)...
...231} $k=1,2,\cdots$\ } \\
P_0'(t) =-\lambda P_0(t)
\end{cases}\end{displaymath} (4)

¤èµ{¦¡ (4) ³q±`¥s°µ¬ìº¸²öªGº¸¤Ò¤èµ{¦¡ (Kolmogorov equation)¡C

¥ý¸Ñ $ P_0'(t) =-\lambda P_0(t) $ ±o

\begin{displaymath}
\log P_0(t) = -\lambda t+ \mbox{const. \quad {\fontfamily{cw...
...ries{m}\selectfont \char 67} \quad} P_0(t)
= c e^{-\lambda t}
\end{displaymath}

¥Îªì­È±ø¥ó P0(0) = P(X(0)=0)=1¡A±o¨ì c=1¡A©Ò¥H $P_0(t) = e^{-\lambda t}$¡C ¦A¥Î¼Æ¾ÇÂk¯Çªk¡A¸Ñ $P_k'(t) = -\lambda P_k(t) + \lambda
P_{k-1}(t)$¡Aª`·Nªì­È±ø¥ó¬O $P_k(0) = P(X(0)=k)=0,k \neq 0$¡A«h¥i±o¨ì

\begin{displaymath}
P_k(t)=P(X(t)=k)=\frac{(\lambda t)^k}{k!} e^{-\lambda t} , \quad
k=1,2,\cdots
\end{displaymath}

©ó¬O§Ú­Ì±o¨ì±ø¥ó(iv)¡C

   

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