next up previous
Next: $B$3$NJ8=q$K$D$$$F(B... (PDF ե: hg.pdf)

$BJ?@.(B 13 $BG/(B 6 $B7n(B 8 $BF|(B
$BD64v2?J,I[$NJ?6Q!"J,;6!"6K8B(B
$B?73c9)2JBg3X(B $B>pJsEE;R9)3X2J(B $BC]LnLP<#(B

$BD64v2?J,I[(B $HG(N_1,N_0;n)$ $B$O!"3NN(JQ?t(B $x$$B!"3NN(4X?t(B $p(x)$ $B$,(B

\begin{displaymath}
x\in \{0,1,2,\ldots,n\},\hspace{1zw}
p(x)=\frac{\left(\begin...
...}\right)}{\left(\begin{array}{c}N_1+N_0\\ n\end{array}\right)}
\end{displaymath}

$B$GM?$($i$l$k3NN(J,I[$G$"$k!#(B


$BJdBj(B 1

\begin{eqnarray*}\sum_{k=0}^n\left(\begin{array}{c}N\\ k\end{array}\right)\left(...
...ight)\\
& = & \left(\begin{array}{c}N+M\\ n\end{array}\right)
\end{eqnarray*}




$B>ZL@(B

``$BJl4X?t$NJ}K!(B'' $B$rMQ$$$k!#Fs9`DjM}$K$h$j(B

\begin{displaymath}
(1+x)^N=\sum_{j=0}^N\left(\begin{array}{c}N\\ j\end{array}\...
...=\sum_{k=0}^M\left(\begin{array}{c}M\\ k\end{array}\right)x^k
\end{displaymath}

$B$G$"$k$N$G!"(B

\begin{eqnarray*}(1+x)^{N+M} & = & (1+x)^N(1+x)^M
= \left\{\sum_{j=0}^N\left(\...
...\end{array}\right)\left(\begin{array}{c}M\\ k\end{array}\right)
\end{eqnarray*}



$B$H$J$k$,!"Fs9`DjM}$h$j(B

\begin{displaymath}
(1+x)^{N+M} = \sum_{\ell=0}^{N+M}\left(\begin{array}{c}N+M\\ \ell\end{array}\right)x^\ell
\end{displaymath}

$B$G$"$k$N$G!"$f$($K(B

\begin{displaymath}
\sum_{k+j=\ell}\left(\begin{array}{c}N\\ j\end{array}\right...
...ay}\right)=\left(\begin{array}{c}N+M\\ \ell\end{array}\right)
\end{displaymath}



$BL?Bj(B 2

$BD64v2?J,I[(B $HG(N_1,N_0;n)$ $B$KBP$7$F!"(B $E[x]=np$ ( $p=N_1/(N_1+N_0)$)


$B>ZL@(B

\begin{eqnarray*}E[x]
& = & \sum_{x=0}^n xp(x)
= \sum_{k=0}^n k
\frac{\left(...
...{array}\right)\left(\begin{array}{c}N_0\\ n-k\end{array}\right)
\end{eqnarray*}



$B$H$J$k$,!"$3$3$G!"(B

\begin{eqnarray*}k\left(\begin{array}{c}N_1\\ k\end{array}\right)
& = & k\frac...
...
& = & N_1\left(\begin{array}{c}N_1-1\\ k-1\end{array}\right)
\end{eqnarray*}



$B$J$N$G!"(B

\begin{eqnarray*}
% latex2html id marker 675
E[x]
& = & \frac{1}{\left(\begin{...
...zw}(\mbox{$BJdBj(B \ref{lemm:1} $B$N(B $N,M,n$\ $B$,(B $N_1-1,N_0,n-1$})\\
\end{eqnarray*}



$B$H$J$k!#$H$3$m$G!"(B

\begin{eqnarray*}\left(\begin{array}{c}N_1+N_0\\ n\end{array}\right)
& = & \fr...
...+N_0}{n}\left(\begin{array}{c}N_1+N_0-1\\ n-1\end{array}\right)
\end{eqnarray*}



$B$h$j!":G8e$N<0$OLsJ,$5$l$F(B

\begin{displaymath}
E[x] = \frac{N_1}{\displaystyle \frac{N_1+N_0}{n}} = n\frac{N_1}{N_1+N_0}=np
\end{displaymath}

$B$H$J$k!#(B



$BL?Bj(B 3

$BD64v2?J,I[(B $HG(N_1,N_0;n)$ $B$KBP$7$F!"(B $V[x]=npq\displaystyle \frac{N_1+N_0-n}{N_1+N_0-1}$ ( $q=1-p=N_0/(N_1+N_0)$)


$B>ZL@(B

\begin{eqnarray*}E[x(x-1)]
& = & \sum_{x=0}^n x(x-1)p(x) = \sum_{x=2}^n x(x-1)...
...ray}\right)\left(\begin{array}{c}N_0\\ n-k\end{array}\right)\\
\end{eqnarray*}



$B$G$"$j!"L?Bj(B 2 $B$N>ZL@$HF1MM$K$7$F(B

\begin{displaymath}
k(k-1)\left(\begin{array}{c}N_1\\ k\end{array}\right)=N_1(N_1-1)\left(\begin{array}{c}N_1-2\\ k-2\end{array}\right)
\end{displaymath}

$B$,$$$($k$N$G!"JdBj(B 1 $B$r;H$$!"L?Bj(B 2 $B$N>ZL@$HF1MM$N(B $B7W;;$r9T$&$H(B

\begin{eqnarray*}E[x(x-1)] & = & \frac{N_1(N_1-1)}{\left(\begin{array}{c}N_1+N_0...
...\right)\\
& = & N_1(N_1-1)\frac{n(n-1)}{(N_1+N_0)(N_1+N_0-1)}
\end{eqnarray*}



$B$H$J$k!#0lJ}!"(B

\begin{displaymath}
E[x(x-1)]=E[x^2-x]=E[x^2]-E[x]
\mbox{ $B$h$j(B }
E[x^2]=E[x(x-1)]+E[x]
\end{displaymath}

$B$G$"$j!"$h$C$FL?Bj(B 2 $B$h$j(B

\begin{eqnarray*}V[x] & = & E[x^2]-E[x]^2=E[x(x-1)]+E[x]-E[x]^2\\
& = & N_1(N_...
..._1+N_0}\right\}
\hspace{1zw}\left(p=\frac{N_1}{N_1+N_0}\right)
\end{eqnarray*}



$B$3$3$G(B $A=N_1+N_0$ $B$H$*$/$H(B

\begin{eqnarray*}V[x] & = & np\left\{\frac{(n-1)(N_1-1)}{A-1}+1-n\frac{N_1}{A}\r...
...0-n}{N_1+N_0-1}
\hspace{1zw}\left(q=\frac{N_0}{N_1+N_0}\right)
\end{eqnarray*}




$B0J8e!"(B$N_1=N$, $N_0=M$ $B$H=q$/$3$H$K$9$k!#(B


$BL?Bj(B 4

$BD64v2?J,I[(B $HG(N,M;n)$ $B$KBP$7$F!"(B$p=N/(N+M)$ $B$r8GDj$7$F(B $N,M\rightarrow\infty$ $B$H$9$k$H(B $HG(N,M;n)\rightarrow B(n,p)$


$B>ZL@(B


\begin{displaymath}
p=\frac{N}{N+M}
\end{displaymath}

$B$r8GDj$9$k$H$$$&$3$H$O!"(B$p:q=N:M$ $B$h$j(B

\begin{displaymath}
M=\frac{q}{p}N
\end{displaymath}

$B$H$7$F(B $N\rightarrow\infty$ $B$9$k$H$$$&$3$H!#$3$N$H$-!"(B

\begin{displaymath}
p(x)=\frac{\left(\begin{array}{c}N\\ x\end{array}\right)\le...
...rrow
\left(\begin{array}{c}n\\ x\end{array}\right)p^xq^{n-x}
\end{displaymath}

$B$H$J$k$3$H$r<($;$P$h$$(B ($0\leq x\leq n$)$B!#0J8e!"(B$n-x=y$ $B$H$9$k!#(B

\begin{eqnarray*}p(x) & = &
\frac{N(N-1)\cdots(N-x+1)}{x!}\times
\frac{M(M-1)...
...0}^{x-1}\frac{N-j}{N+M-j}
\prod_{k=0}^{y-1}\frac{M-k}{N+M-x-k}
\end{eqnarray*}



$B$3$3$G!"(B $\displaystyle M=\frac{q}{p}N$ $B$h$j(B

\begin{eqnarray*}\lefteqn{\frac{N-j}{N+M-j}
=\frac{\displaystyle 1-\frac{j}{N}}...
...le \frac{q}{p}}{\displaystyle 1+\frac{q}{p}} = \frac{q}{p+q}=q}
\end{eqnarray*}



$B$H$J$k$N$G!"7k6I(B

\begin{displaymath}
p(x)\stackrel{N\rightarrow\infty}{\longrightarrow}
\left(\...
...^xq^y=\left(\begin{array}{c}n\\ x\end{array}\right)p^xq^{n-x}
\end{displaymath}

$B$H$J$k!#(B





next up previous
Next: $B$3$NJ8=q$K$D$$$F(B...
Shigeharu TAKENO
2001$BG/(B 8$B7n(B 9$BF|(B