
%\title{�W�{�f�[�^�̒��o�A�W�{����}

\section{�L����W�c�Ɩ�����W�c}
��W�c�̍\���v�f�A���ꂩ�璲�ׂ悤�Ƃ���Ώۂ̌����L���������ɂ���āA�L����W�c�A
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�ł���ꍇ��������W�c�̋c�_�ŁA�J��Ԃ��ɂ���āA�����̑Ώۂ̏󋵂��ς���Ă���ꍇ��
�L����W�c�ł���B���������_�I�ł͂Ȃ��A�ߎ��I�ɑ傫�Ȑ��A���Ƃ��΁@$N$���傫��
 $1/N$���[���Ƌ߂��Ƃ݂Ȃ���ꍇ�ɂ́A�W�{�̒��o�̑O��łɂ���ĕω����Ȃ��ƍl�����邩��A
�L����W�c�Ɩ�����W�c�͓����ƍl������B
����ɑ΂��āA���o�̑O��ŕ�W�c���ς���Ă���Ƃ��A�������o�Ɣ񕜌����o�̋�ʂ�����K�v��
�łĂ���B

���ܕ�W�c�ɂ�2��ނ̑���($\heartsuit ��\clubsuit $)�����ꂼ��M,N������̂Ƃ���
���ꂩ��W�{�����o���B�ԍ������ċ�ʂ���Ƃ��āA�N���肦�邠����ꍇ��
�\�ɂ܂Ƃ߂Ă݂�B�\�̍s�͍ŏ��̎��o���ŁA���̗��2��ڂ̎��o���̌��ʂ�
�\���Ƃ��悤�B
�܂���W�c��
$\overbrace{\clubsuit 1 \ \clubsuit 2 \ \cdots \clubsuit M}^{M}$
$\overbrace{\heartsuit 1 \ \heartsuit 2 \ \cdots \heartsuit N}^{N}$
\hspace{2em}
$\clubsuit k, k=1,2,\cdots M:$ $\heartsuit j, j=1,2,\cdots N$
�ł���A2�̕W�{���Ƃ肾�����Ƃ�����l����B


[�������o]

\begin{tabular}{c|cccc|ccc|}
\backslashbox{�s\\�i�P��ځj}{��\\(�Q��ځj}
  & $\clubsuit 1$ & $\clubsuit 2$ & $\cdots$  &  $\clubsuit M$ & 
$\heartsuit 1$  &  $\cdots$ & $\heartsuit N$ \\
     \hline
%
  $\clubsuit 1$ & $\clubsuit 1$ $\clubsuit 1$ & $\clubsuit 1$ $\clubsuit 2$  &         & 
$\clubsuit 1$ $\clubsuit M$ & $\clubsuit 1$ $\heartsuit 1$ &         & $\clubsuit 1$ $\heartsuit N$                      \\
%
  $\clubsuit 2$ & $\clubsuit 2$ $\clubsuit 1$ & $\clubsuit 2$ $\clubsuit 2$ &          & 
$\clubsuit 2$ $\clubsuit M$  & $\clubsuit 2$ $\heartsuit 1$ &        & $\clubsuit 2$ $\heartsuit N$                      \\
%
  $\vdots$  &         &        &             &          &            &          &                        \\
%
  $\clubsuit M$ & $\clubsuit M$ $\clubsuit 1$  & $\clubsuit M$ $\clubsuit 2$  &         & 
$\clubsuit M$ $\clubsuit M$ & $\clubsuit M$ $\heartsuit 1$ &        & $\clubsuit M$ $\heartsuit N$                      \\
\hline
%
 $\heartsuit 1$ & $\heartsuit 1$ $\clubsuit 1$ & $\heartsuit 1$  $\clubsuit 2$  &           
 & $\heartsuit 1$ $\clubsuit M$  & $\heartsuit 1$ $\heartsuit 1$ &         & $\heartsuit 1$ $\heartsuit N$                      \\
%
  $\vdots$ &         &        &             &          &            &          &                        \\
%
 $\heartsuit N$ & $\heartsuit N$ $\clubsuit 1$  & $\heartsuit N$  $\clubsuit 2$  &         & 
$\heartsuit N$  $\clubsuit M$  & $\heartsuit N$  $\heartsuit 1$ &          & $\heartsuit N$  $\heartsuit N$  \\
\hline
\end{tabular}

\bigskip
��̕\�ɂ�����v�f�̌�
\begin{center}
\begin{tabular}{c|cccc|ccc|}
\backslashbox{1���}{2���}
 & $\clubsuit$ 1 & $\clubsuit$ 2 & $\cdots$  &  $\clubsuit$ M & $\heartsuit$ 1  &  $\cdots$ & $\heartsuit$ N \\
  \hline
%
  $\clubsuit$ 1 &   &   &   &  &  &    &     \\
%
  $\clubsuit$ 2 &   & $M^2$  &   &   &    & MN  &     \\
%
  $\vdots$  &         &        &             &          &            &          &                        \\
%
  $\clubsuit$ M &  &   &   &   &   &   &   \\
\hline
%
 $\heartsuit$ 1 &  &  &   &  &  &   &   \\
%
  $\vdots$ &         &   $MN$     &             &          &            &   $N^2$       &                        \\
%
 $\heartsuit$ N &  &  &   &  &   &     &   \\
\hline
\end{tabular}
\end{center} 


2�̌��ʂ𑮐��ł܂Ƃ߂č��v���Ă݂��
$$
\left.
\begin{array}{ll}
   \clubsuit \clubsuit  & : M^2   \\
   \clubsuit \heartsuit + \heartsuit \clubsuit& : 2MN  \\
   \heartsuit \heartsuit & : N^2 \\
\end{array}
 \right\} = (M+N)^2 = (M+N)(M+N)
$$


���ɔ񕜌��ł�1��ڂ�2��ڂɓ������̂͂Ȃ�����A�܂�񕜌��̏ꍇ�ɂ�
�Ίp���̕����������B
[�񕜌����o] �\����$\times$ �͋N����Ȃ����Ƃ��Ӗ����Ă���
$$
\begin{array}{c|cccc|ccc|}
                  & \clubsuit 1 & \clubsuit 2 & \cdots  &  \clubsuit M & \heartsuit 1  &  \cdots & \heartsuit N \\
     \hline
%
  \clubsuit 1 & \times & \clubsuit 1 \clubsuit 2  &         & 
\clubsuit 1 \clubsuit M & \clubsuit 1 \heartsuit 1 &         & \clubsuit 1 \heartsuit N                      \\
%
  \clubsuit 2 & \clubsuit 2 \clubsuit 1 & \times &          & 
\clubsuit 2 \clubsuit M  & \clubsuit 2 \heartsuit 1 &        & \clubsuit 2 \heartsuit N                      \\
%
  \vdots  &         &        &             &          &            &          &                        \\
%
  \clubsuit M & \clubsuit M \clubsuit 1  & \clubsuit M \clubsuit 2  &         & 
\times & \clubsuit M \heartsuit 1 &        & \clubsuit M \heartsuit N                      \\
\hline
%
 \heartsuit 1 & \heartsuit 1 \clubsuit 1 & \heartsuit 1  \clubsuit 2  &           
 & \heartsuit 1 \clubsuit M  & \times &         & \heartsuit 1 \heartsuit N                      \\
%
  \vdots &         &        &             &          &            &          &                        \\
%
 \heartsuit N & \heartsuit N \clubsuit 1  & \heartsuit N  \clubsuit 2  &         & 
\heartsuit N  \clubsuit M  & \heartsuit N  \heartsuit 1 &          & \times  \\
\hline
\end{array}
$$

���������ėv�f�̌���
$$
\begin{array}{c|cccc|ccc|}
                  & \clubsuit 1 & \clubsuit 2 & \cdots  &  \clubsuit M & \heartsuit 1  &  \cdots & \heartsuit N \\
     \hline
%
  \clubsuit 1 &  &   &   &  &  &    &     \\
%
  \clubsuit 2 &   & (M)_2  &   &  &    & MN  &     \\
%
  \vdots  &         &        &             &          &            &          &                        \\
%
  \clubsuit M &  &   &   &   &   &   &   \\
\hline
%
 \heartsuit 1 &  &  &   &  &  &   &   \\
%
  \vdots &         &   MN     &             &          &            &   (N)_2       &                        \\
%
 \heartsuit N &  &  &   &  &   &     &   \\
\hline
\end{array}
$$
���v��
$$
\left.
\begin{array}{ll}
   \clubsuit \clubsuit  & : (M)_2   \\
   \clubsuit \heartsuit + \heartsuit \clubsuit& : 2MN  \\
   \heartsuit \heartsuit & : (N)_2 \\
\end{array}
 \right\} = (M+N)_2 =(M+N)(M+N-1)
$$


�����3�񂭂�Ԃ����l����ƁA���̕\�̍s�ɂ�2��ڂ܂ł̑����ɂ�鍇�v���ʂ�
�\���A���̗��3��ڂ̌��ʂŕ����Ă݂�B

\vspace{2ex}

[�������o]�ł̗v�f�̐��Ƃ��̍��v��


\begin{tabular}{c|cc|}
\backslashbox{2���\\�܂ł̌���}{3���}%
                              &  $\clubsuit$ & $\heartsuit$ \\ 
\hline
   $\clubsuit$ $\clubsuit$ : $M^2$  &     $M^3$    &    $M^2 N$      \\
   $\clubsuit$ $\heartsuit$  : $2MN$  &   $2M^2 N$   &    $2MN^2$    \\
   $\heartsuit$ $\heartsuit$ : $N^2$  &   $MN^2$ &   $N^3$ \\
\hline
\end{tabular}
$
\left.
\begin{array}{ll}
   \clubsuit + \clubsuit + \clubsuit & : M^3   \\
   \clubsuit + \clubsuit + \heartsuit & : 3M^2 N  \\
   \clubsuit + \heartsuit + \heartsuit & : 3MN^2  \\
   \heartsuit + \heartsuit + \heartsuit& : N^3 \\
\end{array}
 \right\} = (M+N)^3 
$


\vspace{2ex}

[�񕜌����o]�ł̗v�f�̐��Ƃ��̍��v��


\begin{tabular}{c|cc|}
\backslashbox{2���\\�܂ł̌���}{3���}%
               &   $\clubsuit$ & $\heartsuit$ \\ 
\hline
   $\clubsuit$ $\clubsuit$ : $(M)_2$  &     $(M)_3$    &    $(M)_2 N$           \\
   $\clubsuit$ $\heartsuit$  : $2MN$  &   $2(M)_2 N$   &    $2M(N)_2$    \\
   $\heartsuit$ $\heartsuit$ : $(N)_2$  &   $M(N)_2$ &   $(N)_3$ \\
\hline 
\end{tabular}
$
\left.
\begin{array}{ll}
   \clubsuit + \clubsuit + \clubsuit & : (M)_3   \\
   \clubsuit + \clubsuit + \heartsuit & : 3(M)_2 N  \\
   \clubsuit + \heartsuit + \heartsuit & : 3M(N)_2  \\
   \heartsuit + \heartsuit + \heartsuit& : (N)_3 \\
\end{array}
 \right\} = (M+N)_3 
$

\vspace{3ex}
�ȏ�̌v�Z����A��ʂ�$n$�񂭂�Ԃ����ꍇ�ɂ�

�u�������o�v

$$
\left.
\begin{array}{ll}
   \overbrace{\clubsuit + \clubsuit + \cdots +\clubsuit}^{n} & : 
      \displaystyle\binom{n}{0}M^n   \\
   \overbrace{\clubsuit +  \cdots + \clubsuit}^{n-1} + \heartsuit & :
      \displaystyle\binom{n}{1}M^{n-1} N  \\
   \overbrace{\clubsuit + \cdots + \clubsuit}^{n-k} + 
      \overbrace{\heartsuit + \cdots + \heartsuit}^{k}  & : 
      \displaystyle\binom{n}{r} M^{n-k}N^k  \\
   \overbrace{\heartsuit + \heartsuit + \cdots + \heartsuit}^{n}& : 
      \displaystyle\binom{n}{n}N^n \\
\end{array}
 \right\} = (M+N)^n 
$$


�u�񕜌����o�v

$$
\left.
\begin{array}{ll}
   \overbrace{\clubsuit + \clubsuit + \cdots +\clubsuit}^{n} & : 
      \displaystyle\binom{n}{0}(M)_n   \\
   \overbrace{\clubsuit +  \cdots + \clubsuit}^{n-1} + \heartsuit & :
      \displaystyle\binom{n}{1}(M)_{n-1} N  \\
   \overbrace{\clubsuit + \cdots + \clubsuit}^{n-k} + 
      \overbrace{\heartsuit + \cdots + \heartsuit}^{k}  & : 
      \displaystyle\binom{n}{k} (M)_{n-k}(N)_k  \\
   \overbrace{\heartsuit + \heartsuit + \cdots + \heartsuit}^{n}& : 
      \displaystyle\binom{n}{n}(N)_n \\
\end{array}
 \right\} = (M+N)_n 
$$


\vspace{3ex}
�m���ϐ�$X$�𒊏o�ɂ�����W�{�ł�$n$�̂���$\clubsuit$�̌���\���Ƃ����
$k=0,1,2\cdots,n$�ɑ΂���
$$
    \begin{array}{ll}
      P(X=k)  
  & =   P(\overbrace{\clubsuit + \cdots + \clubsuit}^{k} + 
    \overbrace{\heartsuit + \cdots + \heartsuit}^{n-k}  ) \\ 
  & =  
   \left\{
       \begin{array}{ll}
          \displaystyle\binom{n}{k}\dfrac{M^{k}N^{n-k}}{(M+N)^n} & 
            \quad \mbox{:�������o} \\[1em]
          \displaystyle\binom{n}{k}\dfrac{(M)_{k}(N)_{n-k}}{(M+N)_n} & 
            \quad \mbox{:�񕜌����o} 
       \end{array}
    \right.
  \end{array}
$$



�������o�̓p�����[�^ $n, p=\dfrac{M}{M+N}$��2�����z$Binom\left(n,\dfrac{M}{M+N}\right)$�ł���A
$$
  f(k) = \binom{n}{k} p^k (1-p)^{n-k}, k=0,1,2,\cdots,n
$$
�܂��񕜌����o�̏ꍇ�ɂ͒��􉽕��z$Hyper(M+N,M,n)$�ƌĂ΂�A
$$
  f(k) = \binom{n}{k}
         \dfrac{(M)_{k} (N)_{n-k}}{(M+N)_n}
       = \dfrac{\binom{M}{k} \binom{N}{n-k}}{\binom{M+N}{n}}, 
         k=0,1,2,\cdots,n
$$
�ƕ\����邱�Ƃ������B

������r����΂킩��悤�A�ׂ���̐��Ƒg�����̐����Ή����Ă���B
�������o�ł�$M^n = \overbrace{M \cdot M \cdots M}^{n}$
�ɑ΂��āA�񕜌����o�ł�$(M)_n = \overbrace{M (M-1)  \cdots (M-n+1)}^{n}$
�ƂȂ��Ă��邱�Ƃɒ��ӂ���B


\section{�W�{���ς̕��ςƕ��U}
�W�{�f�[�^���狁�߂����{�I�ȓ��v�ʂƂ��ẮA�W�{���ς�����B�P���ɎZ�p���ς�
���߂邱�Ƃł���B
���ܕ�W�c�ɂ�2��ނ̑���������Ƃ��ɁA�W�{���ςɑ΂��镽�ρi���Ғl�j�ƕ��U��
���߂�B������2��ނ����Ȃ�����A�W�{���ςƂ������A�W�{�䗦���邢�͕W�{������
�������Ƃ������B
�܂��W�c�̑�����$\clubsuit, \heartsuit $�Ƃ��āA$i$�Ԗ�$(i=1,2,\cdots ,n)$��
���o���ꂽ�W�{�ɑ΂���
$
   X_i = \left\{
     \begin{array}{l}
        1 \quad \mbox{if} \quad \clubsuit \mbox{�����o���ꂽ�Ƃ�} \\ 
        0 \quad \mbox{if} \quad  \heartsuit \mbox{�����o���ꂽ�Ƃ�} \\
     \end{array}
     \right. 
$
�Ƃ����ƁA�����̘a$X= X_1 + X_2 +\cdots + X_n$�̕��z��
�������o�Ȃ�΁A2�����z$Binom\left(n,\dfrac{M}{M+N}\right)$�ł����āA
�����񕜌����o�Ȃ�΁A���􉽕��z$Hyper(M+N,M,n)$�ɂ��������B
���������Ă���𗘗p����΁A�W�{���ρi�W�{�䗦�j
$$\overline{X}_n = \dfrac{X}{n}= \dfrac{1}{n}\left( X_1 + X_2 +\cdots + X_n\right)$$
�̕��ςƕ��U�����߂���B
$$ 
\begin{array}{ll}
\mbox{�������o:}\quad 
  E(\overline{X}_n) = \dfrac{M}{M+N}, & 
  V(\overline{X}_n ) = \dfrac{1}{n}\dfrac{M}{M+N}\dfrac{N}{M+N} \\[2ex]
\mbox{�񕜌����o:}\quad 
  E(\overline{X}_n) = \dfrac{M}{M+N}, & 
  V(\overline{X}_n ) = \dfrac{1}{n}\dfrac{M}{M+N}\dfrac{N}{M+N}
                        \dfrac{M+N-n}{M+N-1} 
\end{array}
$$
�񕜌����o�̕��U��$\dfrac{M+N-n}{M+N-1}$�͗L���␳���Ƃ�΂�邪�A
�W�{��$n$�ɔ�ׂĕ�W�c�T�C�Y���傫��$M+N$����Ƃ���ƁA���̒l��
$1$�ɋ߂��Ƃ݂Ȃ��邩��A�����Ɣ񕜌��̈Ⴂ�͂Ȃ��Ȃ��Ă��邱�Ƃ�������B

\begin{Toi}
�񕜌����o��
����$E(X_i)= p$, ���U$V(X_j)= p(1-p)$������$p=M/(M+N)$�������A�܂�
�����U$Cov(X_i,X_j)=P(X_j=1)P(X_j=1 \vert X_i =1) - p^2$���v�Z����B
\end{Toi}

