(PDF ¥Õ¥¡¥¤¥ë: 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

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

$B$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

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

$B$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

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

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

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

$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

$$E[x] = \frac{N_1}{\displaystyle \frac{N_1+N_0}{n}} = n\frac{N_1}{N_1+N_0}=np$$

$B$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

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

$B$,$$$($k$N$G!"JdBj(B 1 $B$r;H$$!"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

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

$B$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

$$p=\frac{N}{N+M}$$

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

$$M=\frac{q}{p}N$$

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

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

$B$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 $$M=\frac{q}{p}N$$ $B$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

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

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