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Mr¡G©ñ±ó«e­± k-1 ¦ì¡C¦Û²Ä r ¦ì°_¡A­º¦¸¹J¨ì¸û«e§¡¨ÎªÌ§Y°±¡C

¤]´N¬O»¡¡A¦Û²Ä r ¦ì°_¡A¿ï¾Ü­º¦¸¥X²{ªºÁ{®É²Ä¤@¦W¡C½Ðª`·N¡A©Ò¿×©ñ±ó«e­± r-1 ¦ì¡A·íµM¬O§A¨M¤£¿ï¦o­Ì¡A¡]¥i¼¦ªº¥ý¾W¡I¡^¡A¦ý¬O§A¤´À³¹ï¦o­Ì¨C¦ì¥´¶q¤@µf¡A¥H«K»P«á¨Óªº§@­Ó¤ñ¸û¡A¤]´N¬O»¡¡A¥H«K¨M©w«á¨ÓªÌªºÁ{®É¦W¦¸¡C¤S M1¡A¨S¦³¤°»ò·N«ä¡G§A¥²¶·¿ï¨ú²Ä¤@¦ì¡A¦]¬°²Ä¤@¦ì¤@©w¬O¤@­ÓÁ{®É²Ä¤@¦W¡C

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P(Mr) ¬°®Ú¾Ú¤W­z¿ìªk¡A¿ï¤¤²Ä¤@¦Wªº¾÷²v
Sk ¬°®Ú¾Ú¤W­z¿ìªk¡A²Ä k ¦ì»Ó¨q³Q¿ïªº¨Æ¥ó

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\begin{displaymath}
P(M_r)=\sum^n_{k=r} P(S_k,x_k=1)=\sum^n_{k=r}\,P(S_k\vert x_k=1)P(x_k=1)
\end{displaymath} (1)

³oùØ P(A,B) ¥Nªí P(A ¥B B)¡CÅãµM¡A $P(x_k=1) = \frac{1}{n}$¡C¡]n ­Ó¤£¦P¼Æ¦r¶Ã±Æ¡A³Ì¤pªº­n¦b²Ä k ­Ó¦ì¸m¡^¡C¦Ü©ó P(Sk|xk=1) ¬O¤°»ò¡H°²¦p²Ä¤@¦W¦b²Ä k ­Ó¦ì¸m¡A¨º»ò¡A«ö·Ó Mr¡A¥u¦³¤U­±ªº±¡§Î§Ú­Ì¤~¯à¿ï¨ì¦o¡G³o´N¬O¡A«e­± k-1 ­Ó¼Æ¦r¡]»Ó¨q¡^¤¤¡A³Ì¤pªº¡]³Ì¦nªº¡^­nµo¥Í¦b³Ì«e­± r-1 ­Ó¦ì¸mùØ¡C¡]Á|¨Ò»¡¡Ar=3, k=6, ¦Ó¦U¦ì»Ó¨qªº¦W¦¸¡A¨Ì¥X³õ¦¸§Ç±Æ¡A¤À§O¬° 4,8,5,6,3,1,¡K «h«ö·Ó M3¡A§Ú­Ì¥²¶·¿ï¾Ü²Ä¤­¦ì¡A¦]¬°¦o¬OÁ{®É²Ä¤@¦W¡A©Ò¥H¯u¥¿ªº²Ä¤@¦W´N¨S¦³³Q¿ï¨ì¡^¡C ÅãµM¡A¤W­z¨Æ¥óªº¾÷·|¬O $\frac{(r-1)}{(k-1)}$¡A¦]¦¹

\begin{displaymath}
P(M_r)=\frac{r-1}{n}\,\sum^{n}_{k=r}\,\frac{1}{k-1}\qquad r\geq 2
\end{displaymath} (2)

·íµM¡A¥H«e¤w¸g»¡¹L¡AP(M1)=1/n¡C

§Ú­Ì²{¦b­nÀˬd¤@¤U¡A¦b¤W­z¿ìªkùØ¡A­þ¤@­Ór¯à¨ÏP(Mr)³Ì¤j¡C §â³o­Ór¥s°µr*¡A¨º»òMr*´N¬O³Ì¨Îªº¿ï¾Üªk¡C¬Ý¨ì³oùØ¡A ¤ñ¸û²Ó¤ß¤@ÂIªºÅªªÌ¤]³\·|°Ý¡G§A«ç»ò¥i¥H«OÃÒ¨S¦³¦A¦nªº¿ìªk©O¡H´«¥y¸Ü»¡¡G³Ì¦nªº¿ìªk¡A¤@©w¬O¦b¤W­z³oÃþ $(M_r:r=2,3\cdots)$ ùØ­±¶Ü¡H µª®×¬OªÖ©wªº¡AÃÒ©ú¤]¤£¬O«ÜÃø¡F¦ý¬O¤S­n¦A¶O¤@µf¤f¦Þ¡A§Ú·Q´Nºâ¤F½}¡C ŪªÌ¦p­n²`¨s¡A¥i°Ñ·Ó¸ê®Æ[4]¡A[5]¡A¦n¦n·Q¤@·Q¡C 4 §Ú²{¦b±N P(Mr) ²¼g§@ f(r)¡A¥H $\triangle f(r)=f(r+1)-f(r)$ ¥Nªí f(r) ªº¡u¼W²v¡v¡C±q(2)¦¡§Ú­Ì¬Ý¨ì

\begin{displaymath}
\begin{eqalign}
\triangle f(r)= & \frac{r}{n}(\frac{1}{r}+\c...
...\
=&\frac{1}{n}(\sum^{n-1}_{k=r}\,\frac{1}{k}-1)
\end{eqalign}\end{displaymath} (3)



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¥Ñ³oùاA¥i¥Hª`·N¨ì¡A $\triangle f(r)$ ÀH r ¼W¥[¦Ó»¼´î¡C§A¤S¥i¥HÀËÅç¥X¡A $\triangle f(2)>0$¡A¡]°²³] n>4¡^¡A $\triangle f(n-1)<0$¡C³oªí¥Ü·í r ¼W¥[®É¡Af(r) ¥ý¼W¦Ó«á´î¡A¦p¹Ï1¡C ¦]¦¹ r* ¤@©w¬O­º¦¸¨Ï±o $\triangle f(r)\leq 0$ ªº¨º­Ó r¡A¤]´N¬O»¡¡A $\sum^{n-1}_{r^*}\,(\frac{1}{k})-1\leq 0$ ¦ý¬O $\sum^{n-1}_{r^*-1}(\frac{1}{k})-1>0$¡C©Ò¥H r* º¡¨¬¤U­±ªº¦¡¤l

\begin{displaymath}
\frac{1}{r^*}+\cdots+\frac{1}{n-1}\leq 1< \frac{1}{r^*-1}+
\frac{1}{r^*}+\cdots+\frac{1}{n-1}
\end{displaymath} (4)

¥ô·N©w¤F n «á¡Ar* ´N¥i¦Û(4)¦¡¤¤­pºâ¥X¨Ó¡C

³oùئ³½ìªº¬O¡AÁöµM r* ÀH n ¦ÓÅÜ¡A¦ý·í n «Ü¤j®É¡A$\frac{r^*}{n}$ ´X¥G¨S¦³¤°»òÅܤơC¡]¨Æ¹ê¤W¡A¹ê»Úªº¼Æ¾ÚÅã¥Ü¡An=20 ¤w¸g¥iºâ¡u«Ü¤j¡v¤F¡^¡C«ç¼Ë¬Ý¥X³oÂI¨Ó¡H³oùاڭ̭n´£¨ì¡u¦ÛµM¹ï¼Æ¡v$\log$¡C³o¬O¥H $e\approx2.718$ ¬°©³ªº¹ï¼Æ¨ç¼Æ¡C ¡]¦³¤H±N³o $\log$ ¼g§@ $\log_e$ ©ÎªÌ $\ln$¡^¡C¥¦¬O«ç¼Ë¨Óªº¡H³Ì¦nªº©w¸q¤èªk¬O¡G $\log{t}$ ¬O¦±½u $y=\frac{1}{x}$ »P x ¶b¦b x=1 ©M x=t ¤§¶¡ªº­±¿n¡A¦p¹Ï2¡C³o¼Ë»¡ªº ¸Ü¡Ae´N¬O$\log {t}=1$ªº°ß¤@ªº¹ê¼Æ¸Ñ¡C



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\begin{displaymath}
1+\frac{1}{2}+\cdots+\frac{1}{n-1}>\log{n}>\frac{1}{2}+\frac{1}{3}+\cdots+
\frac{1}{n}
\end{displaymath}



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\begin{displaymath}
0<1+\frac{1}{2}+\cdots+\frac{1}{n-1}-\log{n}<1-\frac{1}{n}
\end{displaymath}

©ÎªÌ»¡
\begin{displaymath}
1+\frac{1}{2}+\cdots+\frac{1}{n-1}\approx\log{n}
\end{displaymath} (5)

¨âªÌ¬Û®t³Ì¦h¤£¶W¹L1¡C(¨Æ¹ê¤W·ín¼W¥[®É¡A $1+\frac{1}{2}+\cdots+\frac{1}{(n-1)}-\log{n}$·|¦³¤@­Ó·¥­­,³o­Ó·¥­­¥s°µ¤×©ÔEuler±`¼Æ,¨ä­È¬ù¬°0.5772)¡C

®Ú¾Ú(5), $\frac{1}{r^*}+\cdots+\frac{1}{(n-1)}\approx\log{n}-\log{r^*}=\log{(\frac{n}{r^*})}$¡C ·ín«Ü¤j®É,(4)¦¡¤¤¥ª¥k¨âÃä¬Û®tµL´X,(¨âÃä¬Û®t¥u¤@¶µ¡C½Ðª`·N(4)¦¡ªº¥ªÃäÅã¥Ü,·ín«Ü¤j®É¡Ar*¤]¥²«Ü¤j),¦]¦¹(4)¦¡¥i¥H¤j¬ù¼g¬°

\begin{displaymath}
\log{\frac{n}{r^*}}\approx 1
\end{displaymath}

¦]¬°$\log{e}=1$¡A©Ò¥H

\begin{displaymath}
r^*\approx \frac{n}{e} \approx 0.36n
\end{displaymath}

³o®É­Ôªºf(r*)¤j¬ù¬O

\begin{displaymath}
P(M_{r^*})=\frac{r^*-1}{n}\,\sum^n_{r^*}\,\frac{1}{k-1}\appr...
...c{n}{r^*-1}}\approx\frac{1}{e}\log{e}=\frac{1}{e}
\approx 0.36
\end{displaymath}

¥H§d¾Ç¤Hªº¤T¤Q¦ì»Ó¨q¨Ó»¡¡Ar*¤j¬ù¬O11¡A³Ì¦nªº¿ìªk¬O¡u©ñ±ó«e10¦ì $\cdots\cdots$¡v¡A³o¼Ë¿ï¤¤²Ä¤@¦Wªº¾÷·|¦Ü¤Ö·|¦³0.36¡C´«­Ó¨Ò¤l»¡¡A ¦pªG§A¥´ºâ±q¤G¤Q·³¨ì¥|¤Q·³¡A¨C¦~³£¥i¦Ò¼{µ²±B¡C°²¦p§A¨C¦~¥u¦Ò¼{¤@¦ì¹ï ¶H¡A¨º»ò³Ì¦nªº¤èªk¤j¬ù¬OM8¡A¤]´N¬O»¡¡A¦Û§A¤G¤Q¤K·³°_¡A¤~¶}©l­n ¡u»{¯u¡v°_¨Ó¡C(¦~«Cªº®É«J¦hª±ª±µL§«)¡C

   

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