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Subsections

$BDj>o%]%"%=%s2aDx(B

$B;X?tJ,I[(B

$B3NN(JQ?t(B$ X$ $B$,3NN(L)EY4X?t(B (probability density function ) $ f\left( x\right) =\lambda e^{-\lambda x}$ $B!J$?$@$7(B$ x$ $B$OHsIi$N$&$H$-$K!$(B$ X$ $B$O;X?tJ,I[(B$B$K=>$&$H$$$&!%;X?tJ,I[$O5-21$N$J$$J,I[$H8@$o$l$k(B ($BL55-21@-(B: Memorylessness, memoryless property)$B!%$3$3$G5-21$,$J$$$H$O(B$ s,t\geq0$ $B$KBP$7$F(B

$\displaystyle P\left( X>s+t\vert X>t\right) =P\left( X>s\right)%
$ (1.1)

$B$,@.$jN)$D$3$H$r8@$&!%Nc$H$7$F?tG/A0$KGc$C$?NdB"8K$r9M$($F$_$h$&!%9XF~Ev=i$O=gD4$K2TF0$7$F$$$?NdB"8K$b$d$,$F8E$/$J$j$$$D$+$O2u$l$F$7$^$&$@$m$&!%$3$3$GNdB"8K$,2u$l$k$^$G$NBQ5WG/?t$NJ,I[$,L55-21@-$r;}$D$H$7$h$&!%9XF~$7$?NdB"8K$O(B$ t$ $BG/7P$C$?8=:_$b=gD4$KF0$$$F$$$k!%$5$F!$$3$NNdB"8K$O$"$H2?G/$b$D$@$m$&$+!%NdB"8K$N $B$H$9$k$H(B$ t$ $BG/4V2TF/ $BG/2TF/$9$k3NN($O>r7oIU$-3NN((B $ P\left(X>s+t\vert X>t\right)$ $B$GI=$9$3$H$,$G$-$k!%$H$3$m$,L55-21@-(B1.1$B$r;H$($P!$$3$N>r7oIU$-3NN($O(B $ P\left(X>s\right)$ $B$GM?$($i$l$k!%$D$^$j2aDx$,L55-21@-$rM-$9$k$H$O!$(B$ t$ $BG/4VNdB"8K$,2TF0$7$F$$$?$3$H$,$3$l$+$i@h$K5/$3$k8N>c$NM=B,$K$J$s$N>pJs$b$b$?$i$5$J$$$3$H$r0UL#$9$k(B.

$B$=$l$G$O$I$N$h$&$JJ,I[$,5-21$J$7$NFCD'$r;}$D$N$+$r9M$($F$_$h$&!%>r7oIUJ,I[$O

$\displaystyle P\left( X>s+t\vert X>t\right)$ $\displaystyle =\frac{P\left( X>s+t,X>t\right) }{P\left(
 X>t\right) }$    
  $\displaystyle =\frac{P\left( X>s+t\right) }{P\left( X>t\right) }$    

$B$3$l$r(B1.1$B$KBeF~$7$F(B

$\displaystyle P\left( X>s+t\right) =P\left( X>s\right) P\left( X>t\right)$

$B$J$k4X78<0$rF@$k!%;X?tJ,I[4X?t(B

$\displaystyle P\left( X>x\right) =e^{-\lambda x}$

$B$,$3$N@-

$\displaystyle f\left( x\right) =\lambda e^{-\lambda x}$ (1.2)

$B$O5-21$J$7$NFCD'$r;}$DJ,I[$G$"$k$3$H$,$o$+$k!%5-21$J$7$NFCD'$r;}$DO"B33NN(J,I[$O;X?tJ,I[$N$_$G$"$k!%(B

$B%$%Y%s%H$N4V3V$,;X?tJ,I[$K=>$&2aDx$rDj>o%]%"%=%s2aDx$H$$$&!J0lMM%]%"%=%s2aDx!$(Bhomogeneous Poisson process$B$H$b8F$V!K!%%]%"%=%s2aDx$H$O5-21$N$J$$;v>]$N@8@.2aDx$G$"$j!$CO?L3X!&1V3X!&6bM;9)3X!&J]81?t3X$J$I!$%$%Y%s%H!JCO?L!&=P@8;`K4!&:DL3ITMz9T!&;v8NEy!K$r2r@O$9$kI}9-$$J,Ln$G4pK\$H$J$k3NN(2aDx$G$"$k!%?@7P2J3X$K$*$1$k?@7P:YK&$N%9%Q%$%/H/2P3hF0$rI=8=$9$k>e$G$b4pK\E*$JLr3d$r2L$?$9!%0J2<$G$O%$%Y%s%H$N$3$H$r%9%Q%$%/$HI=8=$9$k!%(B


$B0lDj;~4VFb$N%9%Q%$%/$N8D?t!J%]%"%=%sJ,I[!K(B

$B%9%Q%$%/4V3V$NJ,I[$,;X?tJ,I[(B1.2$B$K=>$&>l9g$K!$(B$ T$ $BIC4V$N4V$K@8$8$k%9%Q%$%/$N8D?t(B$ N_{T}$ $B$NJ,I[$r5a$a$h$&!%;O$a$K$R$H$D$b%9%Q%$%/$,@8$8$J$$3NN($r5a$a$k!%$3$l$O0l$DL\$N%9%Q%$%/4V3V$,(B$ T$ $B$h$jBg$-$$$3$H$r0UL#$9$k$+$i@8B84X?t$GM?$($i$l$k(B.

$\displaystyle P\left( N_{T}=0\right) =\bar{F}\left( T\right) =e^{-\lambda T}%
$

$B $B!K$,@8$8$k3NN($r5a$a$k!%0l$DL\$N%9%Q%$%/4V3V$r(B$ s_{1}$ $B$H$9$k$H!$Fs$DL\$N%9%Q%$%/4V3V$O(B$ T-s_{1}$ $B$h$jBg$-$$I,MW$,$"$k!%$3$N3NN($O(B

$\displaystyle = P\left( s_1 < X_1 < s_1 + ds_1 \right) P\left( X_2 > T \right)$    
  $\displaystyle = f(s_1) ds_1 \bar{F}\left(T-s_{1}\right)$    

$ s_{1}$ $B$O(B$ [0,T)$ $B$NHO0O$r

$\displaystyle P\left( N_{T}=1\right)$ $\displaystyle =\int_{0}^{T}f\left( s_{1}\right) \bar{F}\left(
 T-s_{1}\right) ds_{1}$    
  $\displaystyle =\int_{0}^{T}\lambda e^{-\lambda s_{1}}e^{-\lambda\left( T-s_{1}\right)
 }ds_{1}$    
  $\displaystyle =\lambda Te^{-\lambda T}%
$    

$BF1MM$K$7$F!$Fs$D$N%9%Q%$%/!J(B$ N_{T}=2$ $B!K$,@8$8$k>l9g$O#22sL\$N%9%Q%$%/;~9o$r(B$ s_2$ $B$H$7$F(B

$\displaystyle P\left( N_{T}=2\right)$ $\displaystyle =\int_{0}^{T}ds_{1}\int_{s_{1}}^{T}ds_{2}f\left(
 s_{1}\right) f\left( s_{2}-s_{1}\right) \bar{F}\left( T-s_{2}\right)$    
  $\displaystyle =\int_{0}^{T}ds_{1}\int_{s_{1}}^{T}ds_{2}\lambda e^{-\lambda s_{1...
...ambda
 e^{-\lambda\left( s_{2}-s_{1}\right) }e^{-\lambda\left( T-s_{2}\right) }$    
  $\displaystyle =\int_{0}^{T}ds_{1}\lambda^{2}e^{-\lambda T}\left( T-s_{1}\right)$    
  $\displaystyle =\frac{1}{2}\left( \lambda T\right) ^{2}e^{-\lambda T}%
$    

$B$3$l$r7+$jJV$9$H0lHL$K%9%Q%$%/?t$NJ,I[$O(B

$\displaystyle P\left( N_{T}=n\right) =\frac{1}{n!}\left( \lambda T\right) ^{n}%
e^{-\lambda T}%
$ (1.3)

$B$H$J$k!%$3$NJ,I[$r%]%"%=%sJ,I[(B$B$H8F$V!%$h$j@53N$JF3=P$O8e$G=R$Y$k!%(B
$B%>(B 1.1: $BecZe&1eb%9ec%&ih,g%/(B $ \lambda =1,3,5$
\includegraphics[width=1\columnwidth]{graphics/book__1.eps}

$BLdBj(B 1   $B%]%"%=%sJ,I[$NJ?6Q$*$h$SJ,;6$,(B$ \lambda T$ $B$K$J$k$3$H$r3N$+$a$h(B.

$BLdBj(B 2   $B%]%"%=%sJ,I[$NFC@-4X?t$,

$\displaystyle f^{\ast}\left( u\right) =\exp\left[ \lambda\left( e^{iu}-1\right) \right]$ (1.4)

$B2rK!(B 3   $B;X?t4X?t$NDj5A(B $ \exp\left[ x\right]=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}$ $B$rMQ$$$k!%(B

$\displaystyle f^{\ast}\left( u\right)$ $\displaystyle =E\left[ e^{-iun}\right] 
 =\sum_{n=0}^{\infty}\frac{e^{-\lambda}\lambda^{n}}{n!}e^{-iun}$    
  $\displaystyle =e^{-\lambda}\sum_{n=0}^{\infty}\frac{\left[ e^{-iu+\ln\lambda}\right]
 ^{n}}{n!}
 =e^{-\lambda}\exp\left[ e^{-iu+\ln \lambda}\right]$    
  $\displaystyle =\exp\left[ \lambda\left( e^{iu}-1\right) \right]$    

$BLdBj(B 4   $BFC@-4X?t$rMQ$$$F%]%"%=%sJ,I[$NJ?6Q$HJ,;6$r5a$a$h!%(B


$B0lDj%9%Q%$%/?t$NBT$A;~4V!J%"!<%i%sJ,I[!K(B

$B5-21$J$7$NFCD'$r;}$D%9%Q%$%/$,(B$ n$ $B2s5/$3$k$^$G$N;~4V$,(B$ T$ $BIC$H$J$k3NN($r7W;;$7$h$&!%(B$ n=1$ $B$N>l9g$K;X?tJ,I[$K$J$k$3$H$O8+$?!%$=$3$G $B$N>l9g!$#22sL\$N%9%Q%$%/$N;~9o$,(B$ T$ $B$H$J$k3NN($r9M$($k!%#12sL\$N%9%Q%$%/$,5/$3$k;~9o$r(B$ s_{1}$ $B$H$9$k$H!$$=$N3NN(L)EY$O(B $ f\left(s_{1}\right)$ $B$GM?$($i$l$k!%#22sL\$N%9%Q%$%/$,(B$ T$ $B$G@8$8$k3NN(L)EY$O(B$ s_{1}$ $B$O(B $ f\left(T-s_{1}\right)$ $B$G$"$k!%Fs$D$N;v>]$OFHN)$@$+$i!$#2$D$N%9%Q%$%/$,5/$3$k$^$G$N;~4V$,(B$ T$ $B$G$"$k3NN((B $ f\left(s_{1}\right) f\left(T-s_{1}\right) ds_1$ $B$K$J$k!%(B $ s_{1}$ $B$O(B$ (0,T]$ $B$N$I$l$r

$\displaystyle f_2\left( T\right)$ $\displaystyle =\int_{0}^{T}ds_{1}f\left( s_{1}\right) f\left(
 T-s_{1}\right)$    
  $\displaystyle =\int_{0}^{T}ds_{1}\lambda e^{-\lambda s_{1}}\lambda e^{-\lambda\left(
 T-s_{1}\right) }$    
  $\displaystyle =\lambda^{2}Te^{-\lambda T}$    

$B$H$J$k!%F1MM$K$7$F(B$ n=3$ $B$N>l9g$O#22sL\$N%9%Q%$%/;~9o$r(B$ s_2$ $B$H$7$F(B

$\displaystyle f_3(T)$ $\displaystyle =\int_{0}^{T}ds_1 \int_{s_1}^{T}ds_{2}f\left(s_1\right) f\left(s_2 - s_1\right) f\left(T-s_2\right)$    
  $\displaystyle =\int_{0}^{T}ds_1 \int_{s_1}^{T}ds_2\lambda e^{-\lambda s_1}\lambda e^{-\lambda (s_2-s_1)}\lambda e^{-\lambda\left(T-s_2\right)
 }$    
  $\displaystyle =\lambda^{3}e^{-\lambda T}\int_{0}^{T}ds_1 \left(T-s_1 \right)$    
  $\displaystyle =\frac{1}{2}\lambda^{3}T^{2}e^{-\lambda T}$    

$B$G$"$k!%$3$l$r7+$jJV$;$P0lHL$K(B$ n$ $B2s$N%9%Q%$%/$,@8$8$k$^$G$N;~4V$N3NN(L)EY$G$"$k%"!<%i%sJ,I[(B

$\displaystyle f_n\left( T\right) =\frac{1}{\left( n-1\right) !}\lambda^{n}T^{n-1}e^{-\lambda T}$ (1.5)

$B$,F3$+$l$k!%$3$NJ,I[$O5"GZL@$9$k$3$H$,$G$-$k$+$i<+J,$G$d$C$F$_$k$H$h$$!%%"!<%i%sJ,I[$NJ?6Q$O(B$ n/\lambda$ $B!$J,;6$O(B $ n/\lambda^{2}$ $B$G$"$k!%:#$N>l9g!$(B$ n$ $B$,@0?t$J$N$G%"!<%i%sJ,I[!J(BErlang distribution$B!K$H8F$P$l$k!%(B$ n$ $B$r $B$K3HD%$7$?J,I[$O%,%s%^J,I[(B(Gamma distribution)$B$H$$$&!%(B

$BLdBj(B 5   $B5"G$&$3$H$r>ZL@$;$h!%(B

$B2rK!(B 6   $ f_{n}$ $B$,<0(B1.5$B$GM?$($i$l$k$H2>Dj$7$F!$(B$ n+1$ $B2s$N%9%Q%$%/$,@8$8$k3NN(L)EY4X?t$O(B

$\displaystyle \int_0^T f_n \left(s \right) f_1 \left(T-s \right) ds$ $\displaystyle = \int_0^T \frac{1}{\left( n-1\right) !}\lambda^{n}s^{n-1}e^{-\lambda s} \lambda e^{-\lambda \left(T -s \right)} ds$    
  $\displaystyle = \frac{1}{\left( n-1\right) !}\lambda^{n} \lambda e^{-\lambda T } \int_0^T s^{n-1} ds$    
  $\displaystyle = \frac{1}{\left( n-1\right) !}\lambda^{n} \lambda e^{-\lambda T } \frac{1}{n} T^{n}$    
  $\displaystyle = \frac{1}{n!}\lambda^{n+1} T^{n} e^{-\lambda T }$    

$B$3$l$O(B $ f_{n+1} \left( T \right)$ $B$KB>$J$i$J$$!%(B

$BLdBj(B 7   $B%"!<%i%sJ,I[$r;X?tJ,I[$N%i%W%i%9JQ49$rMQ$$$F5a$a$h!%(B

$B2rK!(B 8   $B$^$:%"!<%i%sJ,I[$N%i%W%i%9JQ49(B $ G_{n}^{\ast}\left(
s\right) $ $B$r5a$a$F$*$/$H(B

$\displaystyle G_{n}^{\ast}\left( s\right)$ $\displaystyle =\int_{0}^{\infty}\frac{\lambda^{n}}%
{\Gamma\left( n\right) }t^{n-1}e^{-\lambda t}e^{-st}dt$    
  $\displaystyle =\frac{\lambda^{n}}{\Gamma\left( \alpha\right) }\int_{0}^{\infty}%
t^{n-1}e^{-\left( \lambda+s\right) t}dt$    

$ u=\left( \lambda+s\right) t$ $B$H$*$$$F!$(B

$\displaystyle G_{\alpha}^{\ast}\left( s\right)$ $\displaystyle =\frac{\lambda^{n}}{\Gamma\left(
 n\right) }\int_{0}^{\infty}\left( \frac{u}{\lambda+s}\right) ^{n-1}e^{-u}\frac{du}{\lambda+s}$    
  $\displaystyle =\left( \frac{\lambda}{\lambda+s}\right) ^{n}\frac{1}{\Gamma\left(
 \alpha\right) }\int_{0}^{\infty}u^{n-1}e^{-u}du$    
  $\displaystyle =\left( \frac{\lambda}{\lambda+s}\right) ^{n}$    

$B $ X_{1}+X_{2}+\cdots+X_{n}$ $B$NFC@-4X?t(B $ f_{n}^{\ast}\left( s\right) $ $B$O(B

$\displaystyle f_{n}^{\ast}\left( s\right)$ $\displaystyle =E\left[ \exp\left\{ -s\left( X_{1}+X_{2}+\cdots+X_{n}\right) \right\} \right]$    
  $\displaystyle =E\left[ e^{-sX_{1}}\right] E\left[ e^{-sX_{2}}\right] \cdots E\left[
 e^{-sX_{1}}\right]$    
  $\displaystyle =f^{\ast}\left( x_{1}\right) f^{\ast}\left( x_{2}\right) \cdots
 f^{\ast}\left( x_{n}\right)$    

$BJ,I[$,F10l$N>l9g(B(i.i.d.)$B$O(B

$\displaystyle f_{n}^{\ast}\left( s\right) =\left\{ f_{1}^{\ast}\left( s\right)
\right\} ^{n}
$

$BNc$($P!$(B $ f_{1}^{\ast}\left( s\right) $ $B$,;X?tJ,I[$N>l9g$O(B

$\displaystyle f_{n}^{\ast}\left( s\right) =\left( \frac{\lambda}{\lambda+s}\right) ^{n}%
$

$B5U%i%W%i%9JQ49$9$k$H%"!<%i%sJ,I[(B

$\displaystyle \mathcal{L}^{-1}\left[ f_{n}^{\ast}\left( s\right) \right] =\frac{\lambda^{n}}%
{\Gamma\left( n\right) }t^{n-1}e^{-\lambda t}%
$

$B$K$J$k!%(B


$B%"!<%i%sJ,I[$H%]%"%=%sJ,I[$N4X78(B

$B%>(B 1.2: $Beb%1ecBe$&eb%Cku!
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$Bd;$O5U$K(B$ T$ $BIC4V$K@8$8$k%9%Q%$%/$N8D?t(B$ n$ $B$NJ,I[$rD4$Y$F$_$h$&!%$3$N$?$a$K$O%9%Q%$%/4V3V$H%9%Q%$%/8D?t$N4X78$rDj5A$9$kI,MW$,$"$k!%(B$ N_{T}$ $B$r;~9o(B0 $B$+$i(B$ T$ $B$^$G$N%9%Q%$%/$N8D?t!$(B$ S_{n}$ $B$rBh(B$ n$ $BHVL\$N%9%Q%$%/$^$G$N;~4V$H$9$k!%:#!$(B$ T$ $B$,(B$ S_{n}$ $B$h$jD9$$;~4V$@$C$?$H$7$h$&!J(B$ S_{n}>T$ $B!K!%$3$N$H$-;~9o(B$ T$ $B$^$G$K4^$^$l$k%9%Q%$%/$N?t$O9b!9(B$ n-1$ $B8D$G$"$k!J?^;2>H!K!%$3$N$3$H$+$i

$\displaystyle P\left( N_{T}<n\right) =P\left( S_{n}>T\right)%
$ (1.6)

$B1&JU$O(B$ n$ $B2s$N%9%Q%$%/$,@8$8$k$^$G$NBT$A;~4V$,(B$ T$ $BIC0J>e$G$"$k3NN(!$$9$J$o$A%"!<%i%sJ,I[$N@8B84X?t$G(B

$\displaystyle P\left( S_{n}>T\right)$ $\displaystyle =\int_{T}^{\infty}\frac{1}{\left( n-1\right)
 !}\lambda^{n}t^{n-1}e^{-\lambda t}dt$    
  $\displaystyle =\sum_{k=0}^{n-1}\frac{\left( \lambda T\right) ^{k}}{k!}e^{-\lambda T}%
$    

$B$3$l$OItJ,@QJ,$rMQ$$$F5a$a$k$3$H$,$G$-$k$+$i<+J,$G5a$a$F$_$k$H$h$$!%=>$C$F%9%Q%$%/?t$NN_@QJ,I[4X?t$O(B1.6$B$KBeF~$7$F(B

$\displaystyle P\left( N_{T}<n\right) =\sum_{k=0}^{n-1}\frac{\left( \lambda T\right)
^{k}}{k!}e^{-\lambda T}
$

$B;~9o(Bt$B$^$G$N%9%Q%$%/$N8D?t$,(B$ n$ $B$G$"$k3NN($O(B $ P\left( N_{T}=n\right) =P\left( N_{T}<n+1\right) -P\left( N_{T}<n\right)$ $B$G$"$k!%$3$l$rMQ$$$l$P(B

$\displaystyle P\left( N_{T}=n\right) =\frac{\left( \lambda T\right) ^{n}}{n!}e^{-\lambda
 T}$ (1.7)

$B$H$J$j!$:F$S%]%"%=%sJ,I[$,F@$i$l$k!%(B

$BLdBj(B 9   $B%"!<%i%sJ,I[(B$ f_{n}$ $B$rMQ$$$F(B$ T$ $BIC4V$N4V$K@8$8$k%9%Q%$%/$N8D?t(B$ N_{T}$ $B$,%]%"%=%sJ,I[$K=>$&$3$H$r<($;!%(B

$B2rK!(B 10   $ n$ $B2s$N%9%Q%$%/$,(B$ T$ $BIC0JFb$K@8$8$k3NN($O(B$ f_n(s) ds$ ($ 0<s<T$ )$B!%$5$i$K(B$ n+1$ $B2sL\$N%9%Q%$%/$,(B$ T$ $BIC8e$K@8$8$k3NN($O(B $ \overline{F} \left(T-s \right)$ $B$@$+$i(B

$\displaystyle \int_0^T f_n \left(s \right) \overline{F} \left(T-s \right) ds$ $\displaystyle = \int_0^T \frac{1}{\left( n-1\right) !}\lambda^{n}s^{n-1}e^{-\lambda s} e^{-\lambda \left(T -s \right)} ds$    
  $\displaystyle = \frac{1}{\left( n-1\right) !}\lambda^{n} e^{-\lambda T } \int_0^T s^{n-1} ds$    
  $\displaystyle = \frac{1}{n!} \left(\lambda T\right)^{n} e^{-\lambda T }.$    

$B=V4V%9%Q%$%/@8@.N($K$h$k%]%"%=%s2aDx$NDj5A(B

$B$3$3$^$GL55-21@-$+$iF3$+$l$k;X?tJ,I[$r$b$H$KDj>o%]%"%=%s2aDx$N@-1.1.2$B$G$O;X?tJ,I[$+$i%]%"%=%sJ,I[$r!$(B1.1.3$B5Z$S(B1.1.4$B$G$O;X?tJ,I[$+$i%"!<%i%sJ,I[$r7P$F%]%"%=%sJ,I[$r5a$a$?!%$3$N@a$G$O%9%Q%$%/$N=V4VH/@8N((B $ \lambda\left[\text{spike}/s\right]$ $B$r=PH/E@$H$7$F%]%"%=%s2aDx$rDj5A$9$k!%(B

$BDj5A(B 11   $BHy>.6h4VFb$N%$%Y%s%H@8@.3NN($,

  $\displaystyle P\left(\textrm{one spike in}\left[t,t+\Delta\right)\right)=\lambda\Delta+o\left(\Delta\right)$    
  $\displaystyle P\left(\textrm{more than one spike in}\left[t,t+\Delta\right)\right)=o\left(\Delta\right)$    
  $\displaystyle P\left(\textrm{no spike in}\left[t,t+\Delta\right)\right)=1-\lambda\Delta+o\left(\Delta\right)$ (1.8)

$B$3$3$G(B $ o\left(\Delta\right)$ $B$O(B$ \Delta$ $B$r>.$5$/$9$k$H@~7A9`(B $ \lambda \Delta$ $B$KHf$Y$FL5;k$G$-$k$[$I>.$5$/$J$k4X?t$G$"$k!J(B $ \lim_{\Delta\rightarrow 0}o\left(\Delta\right)/\Delta=0$ $B!K!%F1$8(B $ o\left(\Delta\right)$ $B$,0[$J$k<0$G;H$o$l$F$$$k$,!$$3$N@- $B$C$FHy>.6h4V(B$ \Delta$ $B$KF~$k%9%Q%$%/$N?t$r9M$($F$_$h$&!%;~9o(B$ T$ $B$r==J,$K>.$5$JI}(B$ \Delta$ $B$N6h4V$K$h$C$F(B $ N=\Delta/T$ $B8D$KJ,$1$k!%%]%"%=%s2aDx$G$O6h4V(B$ \Delta$ $BFb$N@8@.%9%Q%$%/8D?t$N3NN($OJ?6Q(B $ \lambda \Delta$ $B$N%]%"%=%sJ,I[$GM?$($i$l$?!%$3$N3NN($rHy>.NL(B $ \lambda \Delta$ $B$GE83+$7$h$&!%(B

$\displaystyle p\left(N_{\Delta}=n\right)=\frac{\left(\lambda\Delta\right)^{n}}{...
...!}\left[1-\lambda\Delta+\frac{1}{2}\left(\lambda\Delta\right)^{2}+\cdots\right]$ (1.9)

$B$3$3$G%9%Q%$%/$,@8$8$J$$3NN($H>/?t%9%Q%$%/$N@8@.3NN($r9M$($k$H(B

$\displaystyle p\left(N_{\Delta}=0\right)$ $\displaystyle =1\left[1-\lambda\Delta+\frac{1}{2}\left(\lambda\Delta\right)^{2}+\cdots\right]=1-\lambda\Delta+o\left(\Delta\right)$    
$\displaystyle p\left(N_{\Delta}=1\right)$ $\displaystyle =\left(\lambda\Delta\right)\left[1-\lambda\Delta+\frac{1}{2}\left(\lambda\Delta\right)^{2}+\cdots\right]=\lambda\Delta+o\left(\Delta\right)$    
$\displaystyle p\left(N_{\Delta}=2\right)$ $\displaystyle =\left(\lambda\Delta\right)^{2}\left[1-\lambda\Delta+\frac{1}{2}\left(\lambda\Delta\right)^{2}+\cdots\right]=o\left(\Delta\right)$ (1.10)

$B=>$C$F%]%"%=%s2aDx$G$"$l$P<0(B[*]$B$rK~$?$9!%$^$?[*]$B$K$h$kDj5A$+$i%]%"%=%sJ,I[!&;X?tJ,I[$,F3$+$l!$%]%"%=%s2aDx$N@- $B$3$3$G%]%"%=%s2aDx$H%Y%k%L!<%$2aDx$H$N0c$$$r=R$Y$F$*$/!%%Y%k%L!<%$2aDx$H$OFsCM$rl9g!$N%;6Hy>.6h4VFb$K%9%Q%$%/$,@8@.$5$l$k$+H]$+$NFs$l(B $ P\left(\textrm{one spike in}\left[t,t+\Delta\right)\right)=\lambda\Delta$ , $ P\left(\textrm{no spike in}\left[t,t+\Delta\right)\right)=1-\lambda\Delta$ $B$GM?$($i$l$k!%$3$N$H$-(B$ N$ $B2s$N;n9T$K(B$ n$ $B2s$N%9%Q%$%/$,4^$^$l$k3NN($OFs9`J,I[(B(Binomial distribution)$B$GM?$($i$l$k!%$^$?:G=i$N(B$ N-1$ $B2s$K%9%Q%$%/$,@8@.$;$:!$(B$ N$ $B2sL\$N;n9T$G%9%Q%$%/$,@8@.$9$k3NN($O4v2?J,I[!J(BGeometric distribution$B!K$GM?$($i$l$k!%$D$^$jO"B3;~4V$G$N%+%&%s%HJ,I[$G$"$k%]%"%=%sJ,I[$HBT$A;~4VJ,I[$G$"$k;X?tJ,I[$N$=$l$>$l$KBP1~$7$FFs9`J,I[$H4v2?J,I[$,F@$i$l$k!%O"B3J,I[$GL55-21@-$r;}$DJ,I[$,M#0l;X?tJ,I[$G$"$C$?$N$HF1$8$h$&$K!$N%;6J,I[$GL55-21@-$r;}$DJ,I[$O4v2?J,I[$N$_$G$"$k!%(B

$B;X?tJ,I[$NF3=P(B

$B;~9o(B$ T$ $B!J(B$ N$ $BHVL\$N%S%s!K$^$G%9%Q%$%/$,5/$3$i$:!$;~9o(B$ T$ $B!J(B$ N$ $BHVL\$N%S%s!K$K$*$$$F%9%Q%$%/$,@8$8$k3NN($O!$Dj5A(B[*]$B$K=>$($P!$(B

$\displaystyle P(x<T<x+\Delta) = \left( 1-\lambda\Delta\right)^{N-1}\lambda \Delta + o\left(\Delta\right).$ (1.11)

$B$GM?$($i$l$k!%$3$l$O4v2?J,I[$K$h$k6a;w<0$G$"$k!%$3$N<0$r

$\displaystyle P(x<T<x+\Delta) = \left( 1-\lambda\Delta\right) ^{N}\frac{\lambda\Delta}{1-\lambda \Delta} + o\left(\Delta\right).$ (1.12)

$B$3$3$G(B $ \left\vert x\right\vert <1$ $B$KBP$9$k8x<0(B

$\displaystyle \log\left( 1-x\right) =-x-\frac{1}{2}x^{2}-\frac{1}{3}x^{3}\cdots$ 

$B5Z$S(B

$\displaystyle \frac{1}{1-x}=1+x+x^{2}+x^{3}\cdots$

$B$r;W$$=P$;$P!$Bh0l@.J,$O==J,$K>.$5$J(B$ \Delta$ $B$G$KBP$7$F(B

$\displaystyle \left( 1-\lambda\Delta\right) ^{N}$ $\displaystyle =\exp\left[ N\log\left\{
 1-\lambda\Delta\right\} \right]$    
  $\displaystyle =\exp\left[ N\left\{ -\lambda\Delta-\frac{1}{2}\left( \lambda
 \Delta\right) ^{2}+\cdots\right\} \right]$    
  $\displaystyle \simeq \exp\left[ -\lambda T\right]$    

$B$h$j4JC1$J5a$aJ}$H$7$F(B

$\displaystyle \lim_{\Delta\rightarrow0}\left( 1-\lambda\Delta\right) ^{N}=\lim
_{N\rightarrow\infty}\left( 1-\lambda T\frac{1}{N}\right) ^{N}=e^{-\lambda T}%
$

$B$H$7$F$b$h$$!%BhFs@.J,$O(B

$\displaystyle \frac{\lambda\Delta}{1-\lambda\Delta}$ $\displaystyle =\lambda\Delta\left\{
 1+\lambda\Delta+\left( \lambda\Delta\right) ^{2}+\cdots\right\}$    
  $\displaystyle \simeq \lambda\Delta$    

$B$H$J$k!%(B

$B0J>e$h$j(B

$\displaystyle P(x<T<x+\Delta) \sim \exp\left[ -\lambda T\right] \lambda \Delta + o\left(\Delta\right)
$

$B$h$C$F;X?tL)EY4X?t$,F@$i$l$k!%(B

$\displaystyle f\left(T\right) = \lim_{\Delta \rightarrow 0} \frac{P(x<T<x+\Delta)}{\Delta} = \lambda e^{-\lambda T}%
$

$B$3$NF3=PK!$O0lHLE@2aDx$N9M;!$G$b=EMW$K$J$C$F$/$k!%=V4V%9%Q%$%/@8@.N($r(B$ \lambda$ $B$H$7$F=PH/$7!$=V4V%9%Q%$%/H/2PN((B$ \lambda$ $B$K=>$&%9%Q%$%/;~7ONs$N%9%Q%$%/L)EYJ,I[$OJ?6Q(B$ 1/\lambda$ $B$N;X?tJ,I[$K$J$k$3$H$,$o$+$C$?!%(B

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$\displaystyle p\left( N_{T}=n\right)$ $\displaystyle \simeq \left(
 \begin{array}[c]{c}
 N\\ 
 n
 \end{array}
 \right) \left( \lambda\Delta\right) ^{n}\left( 1-\lambda\Delta\right)
 ^{N-n}$    
  $\displaystyle =\frac{N!}{n!\left( N-n\right) !}\left( \frac{\lambda\Delta}%
{1-\lambda\Delta}\right) ^{n}\left( 1-\lambda\Delta\right) ^{N}$    
  $\displaystyle \simeq \frac{N!}{\left( N-n\right) !}\frac{1}{n!}\left( \lambda
 \Delta\right) ^{n}\exp\left( -\lambda T\right)$    
  $\displaystyle =\frac{N!}{N^{n}\left( N-n\right) !}\frac{1}{n!}\left( \lambda T\right)
 ^{n}\exp\left( -\lambda T\right)$    
  $\displaystyle \rightarrow\frac{\left( \lambda T\right) ^{n}}{n!}\exp\left( -\lambda
 T\right)$    

$B$?$@$7:G8e$N%9%F%C%W$G$O!$%9%?!<%j%s$N8x<0(B $ \ln N!\sim N\ln N-N$ $B$r;H$C$F(B

$\displaystyle \ln\frac{N!}{N^{n}\left( N-n\right) !}$ $\displaystyle =\ln N!-n\ln N-\ln\left(
 N-n\right) !$    
  $\displaystyle \sim N\ln N-N-n\ln N-\left( N-n\right) \ln\left( N-n\right) +\left(
 N-n\right)$    
  $\displaystyle = - \left( 1-\frac{n}{N}\right) \ln\left( 1-\frac{n}{N}\right) ^{N}-n$    
  $\displaystyle \rightarrow -1\cdot\ln e^{-n}-n$    
  $\displaystyle =0$    

$B$rMQ$$$?!%$J$*%9%?!<%j%s$N8x<0$O%i%W%i%96a;w(B$B!J0HE@K!!K$rMQ$$$F5a$a$k$3$H$,$G$-$k!%(B

$B%]%"%=%s2aDx$NL`EY4X?t(B

$B%]%"%=%s2aDx$G$O%$%Y%s%H4V3V$,FHN)$K;X?tJ,I[$K=>$&!%=>$C$F;~9o(B $ t_{1},\ldots,t_{n}$ $B$G%9%Q%$%/$,4QB,$5$l$k3NN($O(B

$\displaystyle p\left(t_{1},\ldots,t_{n}\cap N_{T}=n\right)\Delta^{n}$ $\displaystyle =f\left(t_{1}\right)\Delta\prod_{i=2}^{n}f\left(t_{i}-t_{i-1}\right)\Delta\overline{F}\left(T-t_{n}\right)$    
  $\displaystyle =\lambda e^{-\lambda t_{1}}\Delta\prod_{i=2}^{n}\lambda e^{-\lambda\left(t_{i}-t_{i-1}\right)}e^{-\lambda\left(T-t_{i}\right)}$    
  $\displaystyle =\lambda^{n}e^{-\lambda T}\Delta^{n}$ (1.13)

$B$3$l$h$j%]%"%=%s2aDx$NL`EY4X?t!J(Blikelihood function$B!K$O

$\displaystyle p\left(t_{1},\ldots,t_{n}\cap N_{T}=n\right)=\lambda^{n}e^{-\lambda T}$ (1.14)


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