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°ÝÃDªº±Ô­zÁö«Ü²³æ¡A¦ý²Ó«ä¤§¤U¡A«oµo²{¨ä¨Ã¤£«Ü²³æ¡C³o¹D²z¤£Ãø©ú¥Õ¡A¦]¬°¥i¤Uª`ªº¤èªk¹ê¦b¤Ó¦h¤F¡A­n¤@¤@¤ñ¸û¬O¤£¥i¯àªº¡C

¬°¤F­n§JªA¤W­±©Ò»¡ªº§xÃø¡A¼Æ¾Ç®a­º¥ý¦Ò¼{´XºØ¤ñ¸û¥i¯à¬°¤H­Ì±Ä¥Îªº¤èªk¡A ³o¨Ç¤èªk©Ò¥H¸û±`±Ä¥Î¡A®õ¥b¬O¥Ñ©óª½Ä±¤W»{¬°¥¦­Ì¥i³Q±Ä¦æ¡C·íµM¡Aª½Ä±ªº»{©w©¹©¹¬O¤£¥i¾aªº¡A©Ò¥H³Ì¦n¯à¦³²z½×¤ä«ù¡C¤U­±´N¤¶²Ð¤TºØ¥i¯àªº¤èªk¡A¨Ã¤ñ¸û¨äÀu¦H¡C

¤èªk¤@¡B¨C¦¸¥Ò§¡¤U½äª` 1 ¤¸¡C¡]ÅãµM¡A³o¼Ëªº¤Uª`ªk³Ì«O¦u¡A§Ú­ÌºÙ¤§¬°«O¦u«¬¤Uª`ªk¡C¡^
¤èªk¤G¡B­º¥ý¥Ò¤U 1 ¤¸½äª`¡C­Y¥LŤF¡A«h¤U¦¸¤´¤U 1 ¤¸¡F­Y¿é¤F¡A«h±N½äª`¥[­¿¡A¨Ì¦¹Ãþ±À¡C´«¨¥¤§¡A©¹«á¥u­n¤@Ĺ¡A¥L´N¤U 1 ¤¸¡A§_«h´N§â¤Uª`ª÷ÃB¥[­¿¡C·íµM¡A§Ú­Ì°²³]©Ò¤Uª÷ÃB¬O¦X²zªº¡C¡]ÅãµM«ù³oºØ¤Uªkªº²z¥Ñ¬O¦]¬°¥u­n¤@Ĺ¡A¨º»ò«D¦ý©Ò¦³¿éªºª÷ÃB§Y¥þ¼´¦^¨Ó¡A¨Ã¥B¤Ï¦hĹ 1 ¤¸¡A§Ú­Ì©h¥BºÙ¤§¬°¿é¤£°_«¬¤Uª`ªk¡C¡^
¤èªk¤T¡B¥u­n³\¥i¡A¥Ò´N±N©Ò¦³½ä¥»¤Uª`¡A¦]¦¹¥u­n¤@½ü¡A¬Y¥Ò´N¦å¥»µLÂk¡C¡]ÅãµM³oºØ¤èªk¬O³Ì¤jÁxªº¡A§Ú­Ì´NºÙ¤§¬°·¥ºÝ«¬¤Uª`ªk¡C¡^

§A·|±Ä¥Î­þºØ¤èªk©O¡H¯à»¡­Ó¹D²z¥X¨Ó¶Ü¡H¨Æ¹ê¤W¡Aµª®×¨Ã¤£Â²³æ¡A¥¦¸ò p ¨s³º¤j©ó¡Bµ¥©ó©Î¤p©ó 1/2 ¦³Ãö¡A¤]§Y¸ò§A¬O§_¤ñ²ø®a±j¦³Ãö¡C§Ú­Ì´NÁ| c=2 ªº¨Ò¤l¨Ó»¡©ú¡C¬°¤è«K­p¡A§Ú­Ì¥H¡u¡Ï¡vªí¥ÒĹ¡A¥H¡u¡Ð¡vªí¥Ò¿é¡A¨Ã¥H¡Ï¡B¡Ð©Ò§Î¦¨¤§¤¤¦Cªí¥Ü¥Ò¦b¾ãÁɧ½¿éĹªº¶¶§Ç¡C

­º¥ý§Ú­Ì¦Ò¼{«O¦u«¬¤Uª`ªk¡A¦¹®É¥u¦³¦b¤U¦C½Ñ³õ¦X¡A¥Ò¤~·|Ĺ¡]§Y²ø®a½ä¥»¿é¥ú¡^¡C

++,
+-++,-+++,
+-+-++,+-+++,-++-++,-+-+++,
$\vdots$                        $\vdots$                        $\vdots$                        $\vdots$                        ¡C
¦b²Ä¤@¦C ¡Ï¡Ï ¤¤¡A¥Ò³sŨ⦸¡A¦¹¦¸¾÷²v¬° $p\cdot p=p^2$¡C¦b²Ä¤G¦C¤¤¡A¥ÒŤF¤T¦¸¡A¿é¤F¤@¦¸¡A¨Ã¥B¦³¨âºØ¥i¯à©Ê¡A©Ò¥H¨ä¾÷²v¬° $2\cdot p^3\cdot q=2p^3q$¡]q ¬°¿éªº¾÷²v¡A¬G p+q=1¡^¡C¨Ì¦¹±À¾É¥i±o¦b²Ä n ¦C¤¤¡A¥ÒŤF n+1 ¦¸¡A¦Ó¿é¤F n-1 ¦¸¡A¨Ã¥B¦³ 2n-1 ºØ¥i¯à©Ê¡A©Ò¥H¨ä¾÷²v¬° 2n-1pn+1qn-1¡C¦]¦¹¥i±o¦b¾ã­ÓÁɧ½¤¤¡A¥ÒĹªº¾÷²v¬°


\begin{eqnarray*}
& & p^2+2p^3q+\cdots+2^{n-1}p^{n+1}q^{n-1}\\
&=& p^2(1+2pq+4p^2q^2+\cdots+2^{n-1}p^{n-1}q^{n-1})\\
&=& \frac{p^2}{1-2pq}
\end{eqnarray*}


²{¦bÅý§Ú­Ì¦Ò¼{¿é¤£°_«¬¤Uª`ªk¡C¦¹®É¥u¦³¦b¤U¦C½Ñ³õ¦X¡A¥Ò¤~·|Ĺ¡C

++,+-+,
-+++,-++-+,¡]ª`·N¡G¥Ò²Ä¤G¦¸¶È¯à¤Uª` 1 ¤¸¡^
-+-+++,-+-++-+,
                                
$-+-+\cdots-+++$, $-+-+\cdots-++-+$,
                                                                                ¡C

¥é¤W¤§­pºâ¡A¥i±o¦¹®É¥ÒĹªº¾÷²v¬°

\begin{eqnarray*}
&&(P^2+p^2q)+pq(p^2+p^2q)+p^2q^2(p^2+p^2q)+\cdots\\
&=&(p^2+p...
...\quad\mbox{{\fontfamily{cwM0}\fontseries{m}\selectfont \char 1}}
\end{eqnarray*}


³Ì«á³]¬Y¥Ò±Ä·¥ºÝªk¡A«h¥Ò²Ä¤@¦¸§Y¤Uª`2¤¸¡A¦]¦¹¤@¦¸´N¨M©w¤F¿éĹ¡A©Ò¥H¥ÒĹªº¾÷²v¬° p ¡C

²{¦b§Ú­Ì¦A¦^¨ì­ì°ÝÃD¡G¨s³º¦b³o¤TºØ¤èªk¤¤¡A¥H¨ººØ¤èªk³Ì¦n¡H¥Ñ©ó¬Û¹ïÀ³Ä¹ªº¾÷²v¤½¦¡¤w¨D±o¡A©Ò¥H§Ú­Ì¥u»Ý±N p ­È¥N¤J¡A¶i¦Ó¤ñ¸û¨ä¤j¤p§Y¥i¡AÁ|¨Ò¨Ó»¡¡A·í $p=\frac{1}{2}$ ®É¡A¤TªÌ¤§­È¬Ò¬° $\frac{1}{2}$¡F¦Ó·í $p=\frac{2}{3}$ ®É¡A¤TªÌ¤§­È¨Ì§Ç¬° $\frac{4}{5}$¡B$\frac{16}{21}$¡B$\frac{2}{3}$¡F¦Ü©ó·í $p=\frac{1}{3}$ ®É¡A«h¨ä­È¨Ì§Ç¬° $\frac{1}{5}$¡B$\frac{5}{21}$¡B$\frac{1}{3}$¡C ³o¨Ç¼Æ­È§i¶D§Ú­Ì¡A·í $p=\frac{1}{2}$ ®É¡A¤TºØ¤Uª`ªk¨S¼vÅT¥ÒĹªº¾÷·|¡F·í $p=\frac{2}{3}$ ®É¡A«h¥H«O¦uªk¸û¦n¡F·í $p=\frac{1}{3}$ ®É¡A«o¥H·¥ºÝªk³Ì¨Î¡A«O¦uªk³Ì®t¡C

³o¨Çµ²½×¡A¬O¤£¬O¦³¨Ç¥X§A·N®Æ©O¡H¨ä¹ê°ÝÃDÁÙ¨S¥þ³¡¸Ñ¨M¡A¨´¤µ§Ú­Ì¶È´N«O¦u¡B¿é¤£°_¡B·¥ºÝ¤T«¬¨Ó§@¤ñ¸û¡C¬O§_©|¦³¨ä¥L«¬ªº¤Uª`ªk·|¨Ï±oµª®×§ó¦n¡HÁÙ¦³¡A§Ú­Ì¶È´N¯S¨Ò¨Ó¦Ò¼{¡A¦b¤@¯ëªº±¡§Î¤U¡Aµª®×¤S¬O«ç¼Ë©O¡H

²{¦b¡A¥ý§â³Ì¤@¯ë©Êªºµ²ªG¼g¦b¤U­±¡A¨ä¤¤ $\nu(i)$ ¥Nªí·í¥Ò¦³ i ¤¸®É·|Ĺªº¾÷²v¡C

±¡ªp¤@¡G $p=\frac{1}{2}$

¦¹®É¤£½×¥Ò¦p¦ó¤Uª`¡A$\nu(c)$ ùÚµ¥©ó c/(m+c)¡C

±¡ªp¤G¡G $p>\frac{1}{2}$

¦¹®É¤£½×¥Ò¦p¦ó¤Uª`¡A $\nu(c)\leq[1-(\frac{q}{p})^c]/[1-(\frac{q}{p})^{m+c}]$¡A¦Ó¥kºÝ¬°«O¦u«¬¤Uª`ªkĹªº¾÷²v¡C¦]¦¹¡A¦b¦¹±¡ªp¥H«O¦u«¬ªº¤Uª`ªk¬°³Ìí·í¡C¥t¤@¤è­±¡A·¥ºÝ¤Uª`ªkªºÄ¹­±³Ì§C¡C

±¡ªp¤T¡G$p<\frac{1}{2}$

¦¹®É¥H·¥ºÝªk³Ì¨Î¡A«O¦uªk³Ì®t¡C¦P¼Ë¦a¡A«O¦u«¬¤Uª`ªkĹªº¾÷²v¬° $[(\frac{q}{p})^c-1]/[(\frac{q}{p})^{m+c}-1]$¡C

²{¦b§Ú­Ì´N¨Ó¬ã¨s¡A¬°¤°»ò·|¦³³o­Óµ²½×¡I³o¥Î¨ì¤F¤@¨Ç¼Æ¾Ç¤u¨ã¡A¤£¹L¹ï¨ä¤¤¸û½ÆÂøªº³¡¤À¡A¦]ÅU¤Î¥»¤åªº¥iŪ©Ê¡Aµ§ªÌ¥u«Ü§ã­nªº±Ô­z¤@¤U¡C

¥Ñ©ó¦b¤W­±ªºµ²½×ùØ¡A«O¦uªk³B©ó¤@­Ó©~¤¤ªº¦a¦ì¡A©Ò¥H§Ú­Ì¥ý´N¦¹ªk¶i¦æ°Q½×¡A µM«á¦A¶i¤@¨B¬ã¨s¾ã­Ó°ÝÃD¡C

¦p¦P¥H«e¡A$\nu(i)$ ¥Nªí·í¥Ò©Ò¾Ö¦³ªº¸ê¥»¹F i ¤¸®É¡A¥L·|Ĺªº¾÷²v¡C¥Ñ©ó¥Ò¤Î²ø®aªºÁ`¸ê¥»ÃB¬° m+c ¤¸¡A©Ò¥H i ¤§¥i¯à­È¬° i = 0, 1, ¡K, m + c¡CÅãµM¦a¡A$\nu(0)=0$¡A$\nu(c+m)=1$¡A¦Ó $\nu(c)$ ¬°§Ú­Ì³Ì¦­©Ò·Q¨D±o¤§¾÷²v¡C

±¡ªp¤@¡G $p=\frac{1}{2}$

°²©w¬Y¥Ò²{¦³ i ¤¸¡A¨º»ò¦³ $\frac{1}{2}$ ªº¾÷·|¡A¥Lªº¸ê¥»·|¦¨¬° i+1 ©Î i-1 ¤¸¡C¦]¦¹

\begin{eqnarray*}
\nu(i) &=& \frac{1}{2}\nu(i+1)+\frac{1}{2}\nu(i-1) \\
&& \quad i=1,2,\cdots,m+c-1
\end{eqnarray*}


³o¼Ëªº¨ç¼Æ £h¡A¦b¼Æ¾Ç¤W¬O¤@­Ó½u©Ê¨ç¼Æ¡A¦]¦¹¸Ñªº³q¦¡¬° $\nu(i)=a+bi$¡C¥Ñ©ó¡A$\nu(0)=0$¡B$\nu(c+m)=1$¡A±o a=0¡B $b=\frac{1}{c+m}$¡C¦]¦¹ $\nu(c)=\frac{c}{m+c}$¡A¥ç§Y¥ÒªºÄ¹­±¬° c/(m+c)¡C

±¡ªp¤G¡G $p>\frac{1}{2}$

¥O q=1-p¡C¦¹®É¹ï £h §Ú­Ì¦³¤èµ{¦¡

\begin{eqnarray*}
\nu(i) &=& p\nu(i+1)+q\nu(i-1) \\
&& \quad i=1,2,\cdots,m+c-1
\end{eqnarray*}


³o¼Ëªº¤@²Õ¤èµ{¦¡¡A¦b¼Æ¾Ç¤WºÙ§@¬O®t¤À¤èµ{¦¡¡C¥¦¤]¦³¤@­Ó¨D¸Ñªº¤@¯ë¤èªk¡A¦ý¨ä¹D²z¸û²`¡C¬°¦¹¤§¬G¡A§Ú­Ì¯S±Ä¥Î¤U­±ªº¤èªk¡C

§Q¥Îp+q=1¡A¤W²Õ¤èµ{¦¡¥i§ï¼g¬°

\begin{eqnarray*}
&& \nu(1)-\nu(0)=\nu(1)-\nu(0) \\
&& \nu(2)-\nu(1)=\frac{q}{p...
...\
&& \nu(m+c)-\nu(m+c-1) = (\frac{q}{p})^{m+c-1}[\nu(1)-\nu(0)]
\end{eqnarray*}


¨âÃä¬Û¥[¡A¨Ã§Q¥Î $\nu(m+c)=1$¡B$\nu(0)=0$¡A±o

\begin{displaymath}
1=\frac{1-(\frac{q}{p})^{m+c}}{1-\frac{q}{p}}\nu(1)\quad\mbo...
...uad\mbox{{\fontfamily{cwM0}\fontseries{m}\selectfont \char 1}}
\end{displaymath}

­Y¨ú«e c ¶µ¬Û¥[¡A«h±o

\begin{displaymath}
\nu(c)=\frac{1-(\frac{q}{p})^c}{1-\frac{q}{p}}\nu(1)=\frac{1...
...uad\mbox{{\fontfamily{cwM0}\fontseries{m}\selectfont \char 1}}
\end{displaymath}

±¡ªp¤T¡G $p<\frac{1}{2}$

¥é¤G¤§¸Ñªk¡A¥i¨D±o

\begin{displaymath}
\nu(c)=\frac{(\frac{q}{p})^c-1}{(\frac{q}{p})^{m+c}-1}\quad\mbox{{\fontfamily{cwM0}\fontseries{m}\selectfont \char 1}}
\end{displaymath}

«O¦uªkªº $\nu(c)$ ¤w¨D±o¡A²{¦b§Ú­Ì¨Ó¬ã¨s¬°¤°»ò¦b±¡ªp¤G®É¡A¥H«O¦u¤Uª`ªkªº $\nu(c)$ ¬°³Ì¤j¡F ¦Ó¦b±¡ªp¤T®É¡A¤Ï¥H«O¦u¤Uª`ªkªº $\nu(c)$ ¬°³Ì¤p¡F¦P®É¥t¤@¤è­±¡A¦b±¡ªp¤G®É¡A«hµL½×¦óºØ¤Uª`ªk¡A$\nu(c)$ ¬Ò¤@¼Ë¡C

­º¥ý§Ú­Ì¤Þ¶i¤@­Ó©w²z¡C¥O Sn ¥Nªí¦b²Ä n ¦¸Áɧ½®É¡A¥Ò©Ò¾Ö¦³¤§¸ê¥»ÃB¡A¦]¦¹ Sn ¬O¤@­ÓÀH¾÷ÅܼơC§Ú­Ì¨Ã³] S0=c¡A§Y­ì¸ê¥»¡C¥O N ªíµ²§ôÁɧ½©Ò»Ý¤§®É¶¡¡A¦]¦¹ SN=0 ©Î c+m¡C§Ú­Ì¨Ã¥H E ªí´Á±æ­È¡C

©w²z¡G
³] f ¬°¤@©w¸q©ó Sn ¤W¤§¦³¬É¨ç¼Æ¡C­Y¦b Sn ¤§±ø¥ó¤U¡Af(Sn+1) ¤§´Á±æ­È E[f(Sn+1)] = f(Sn)¡A«h E[f(SN)] = f(S0) = f(c)¡C ­Y±N¡u=¡v§ï¬°¡u$\geq$¡v¡A«hµ²½×¥ç¯u¡C

¦¹©w²z¦b¾÷²v¾Ç¤W¡A§YµÛ¦Wªº¿ï¾Ü¼Ë¥»©w²z (optional sampling theorem)¡A¥¦ªºÃÒ©ú¤w¶W¹L¥»¥Zµ{«×¡A©Ò¥H²¤¥h¤£ÃÒ¡A¦ý¥¦ªºª½Æ[·N¸q«o¤£Ãø¤F¸Ñ¡C´N®³¡u=¡vªº±¡§Î¨Ó»¡¡A¨ä¹ê¬O»¡­Y§Aªº²Ä n+1 ¦¸Áɧ½¡A¥­§¡¦Ó¨¥¨Ã¤£¯à§ïÅܦb²Ä n ¦¸Áɧ½®É f ¤§­È¡A«h·í¾ã­ÓÁɧ½µ²§ô®É¡Af ªº¥­§¡­È¤]»P­ì¥ý­È¤@¼Ë¡C¥t¤@¤è­±¡A­Y¦b¡u$\geq$¡vªº±¡ªp¡A¥ç§Y§Aªº²Ä n+1 ¦¸Áɧ½¥­§¡¦Ó¨¥·|§ï¶i f ¥ý«e¤§­È¡A«h·íÁɧ½µ²§ô®É¡Af ªº¥­§¡­È¤]´¿¤ñ­ì¥ý­È¬°¨Î¡C

²{¦b§Ú­Ì´N®³³o©w²z¨ÓÃÒ©ú¥ý«e§Ú­Ì©Ò¤U¤§µ²½×¡C

­º¥ý¡A§Ú­Ì¦Ò¼{±¡ªp¤@¡C¦¹®É¨ú f(Sn)=Sn¡A«h¤£½×¹ï¦óºØ¤Uª`ªk¡A¦]³Ó­t¾÷·|§¡µ¥¡A $p=q=\frac{1}{2}$¡A ©Ò¥H­Yµ¹©w Sn¡A«h ESn+1 = Sn¡C¦]¦¹¥Ñ¤W©w²zª¾ ESN = c¡C¦ý $ES_N = (c+m)\nu(c)+0[1-\nu(c)]$ = $(c+m)\nu(c)$¡A ©Ò¥Hª¾¤£½×¥H¦óºØ¤èªk¡A $\nu(c)=\frac{c}{c+m}$¡C

¦Ü©ó¦b±¡ªp¤G©Î¤T®É¡A§Ú­Ì¨ú $f(S_n)=(\frac{q}{p})^{S_n}$¡C¦¹®É­Yµ¹©w Sn¡A«h

\begin{eqnarray*}
E[f(S_{n+1})]
&=& E[(\frac{q}{p})^{S_{n+1}}] \\
&=& (\frac{...
...{-a}q] \\
&=& f(S_n)[(\frac{q}{p})^ap+(\frac{p}{q})^aq]\quad ,
\end{eqnarray*}


¨ä¤¤ $a\geq1$ ¬°©Ò¤Uª`¤§ª÷ÃB¡C§Q¥Î

\begin{displaymath}
(\frac{q}{p})^a p + (\frac{q}{p})^a q
\geq (\frac{q}{p}p + \frac{p}{q}q)^a = 1 \quad ,
\end{displaymath}

¥i±o¤£½×¥H¦óºØ¤Uª`ªk¤Uª`¡A­Yµ¹©w Sn¡A«h $E[f(S_{n+1})]\geq f(S_n)$¡C©Ò¥H¥Ñ©w²zª¾ $E[f(S_N)]\geq(\frac{q}{p})^c$¡C ¦ý

\begin{eqnarray*}
E[f(S_N)]
&=& E[(\frac{q}{p})^{S_N}]\\
&=& (\frac{q}{p})^{m...
...{p})^0[1-\nu(c)]\\
&=& 1-[1-(\frac{q}{p})^{m+c}]\nu(c) \quad ,
\end{eqnarray*}


¦]¦¹¥i±o¦b±¡ªp¤G¡A$p>\frac{1}{2}$ ®É¡A

\begin{displaymath}
\nu(c)\leq\frac{1-(\frac{q}{p})^c}{1-(\frac{q}{p})^{m+c}} \quad ;
\end{displaymath}

¦Ó¦b±¡ªp¤T¡A$p>\frac{1}{2}$ ®É¡A

\begin{displaymath}
\nu(c)\geq\frac{(\frac{q}{p})^c-1}{(\frac{q}{p})^{m+c}-1}\quad\mbox{{\fontfamily{cwM0}\fontseries{m}\selectfont \char 1}}
\end{displaymath}

¦ý $[1-(\frac{q}{p})^c]/[1-(\frac{q}{p})^{m+c}]$ ¬°±Ä¥Î«O¦u¤Uª`ªk®ÉĹªº¾÷²v¡A©Ò¥Hª¾¦b±¡ªp¤G®É¡A¥H«O¦uªkªº $\nu(c)$ ¬°³Ì¤j¡F¦ý¦b±¡ªp¤T®É¡A«o¥H«O¦uªkªº $\nu(c)$ ¬°³Ì¤p¡C

¦Ü©ó¬°¤°»ò¦b±¡ªp¤G®É¡A¥H·¥ºÝªkªºÄ¹­±¬°³Ì§C¡F¦ý¦b±¡ªp¤T®É¡A«o¥H·¥ºÝªkªºÄ¹­±¬°³Ì¤j¡C³o¨ä¤¤¤S²o¯A¨ì§ó²`ªº²z½×¡A¥u¦n±q²¤¤F¡C

   

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