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\begin{document}
%\maketitle

\hfill
{\bf\LARGE 
\begin{center}
  �m�����z�̃v���b�g\\
�\�\���������\�t�g�̗��p
\end{center}
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\section{��l���z�̘a}


�m���ϐ�$X_1,X_2,\cdots,$�͓Ɨ��œ�����l���z$U(0,1)$�ɏ]����
����B
$S_n = X_1 + X_2 + \cdots+ X_n,\, (n \geq 1)$�Ƃ��A
���z�֐�$U_n(x) = P(S_n \leq x)$, ���x�֐�$u_n(x) = \dfrac{d U_n}{dx}(x)$
���l����B�m���ϐ��̘a�����߂鎮������̊֌W����������B

\begin{theorem}
\begin{enumerate}
\item[(i)] �P�ʋ��$(0,1)$��̈�l���z�̖��x�֐��F
\[
  u_1(x) = 1, \, \mbox{if} \quad 0 < x <1 
\]
\item[(ii)] �a�̌����ɂ�閧�x�֐��̍ċA�֌W���ƕ��z�֐�����̌v�Z
\[
 u_{n+1}(x) = \int_{0}^1 u_n(x-y) dy = U_n(x) - U_{n+1}(x-1),
 \,n =1,2,\cdots
\]
\item[(iii)] ��l���z�̕��z�֐��F
\[
 U_1(x) = x_+ -(x-1)_+, \quad -\infty < x < \infty \, 
{������} x_+ =\max\{0,x\} = \dfrac{x+|x|}{2}
\]
\item[(iv)] ��ʂ̂΂����̕��z�֐��F
\[
 U_n(x) = \dfrac{1}{n!} \sum_{\nu=0}^{n}
  (-1)^{\nu} \binom{n}{\nu} (x - \nu)_+^{n},
  \,n =1,2,\cdots
\]
\item[(v)] ��ʂ̂΂����̖��x�֐��F
\[
 u_n(x) = \dfrac{1}{(n-1)!} \sum_{\nu=0}^{n}
  (-1)^{\nu} \binom{n}{\nu} (x - \nu)_+^{n-1},
  \,n =1,2,\cdots
\]
\end{enumerate}
\end{theorem}
�ؖ��͋A�[�@�ɂ��ƂÂ��B
�Ƃ���2���W���̉��@���F
$\binom{n}{\nu} + \binom{n}{\nu-1} = \binom{n+1}{\nu}$
��p����B���������x�͈̔͂���ʂ̂΂���$u_n(x)$��$0 < x < n$�ł��邪�A
�֌W������A$(x-n)_+$�ł͖��炩�ł��邪�A���̍�����̌v�Z������邩��
���������Ȃ��Ƃł͂Ȃ��B
���ςƕ��U�����Ƃ߂āA���S�Ɍ��藝�ɂ��A$n$���傫����ΐ��K���z�ŋߎ������B
�Ƃ���$n=12$�Ȃ�΁A�ȒP�Ȍv�Z�ɂ��A1�̕W�����K���z�ɂ�������
�m���ϐ���������B


\begin{theorem}
\[
 X_i \, \sim \, U(0,1) \quad \Rightarrow \quad
  "n=12",\,  S_{12} - 6 \sim N(0,1)
\]
\end{theorem}
�Ȃ��Ȃ�e$U(0,1)$�̕��ς�$1/2$,���U��$1/12$�ł��邩��A
�����������12��$S_{12}$�̕��ς�$6$,���U��$1$�ƂȂ�B
���������āA$S_{12} - 6 \sim N(0,1)$�ł���B

�ł͋�̓I�ɕ��z���v�Z���āA����ɐ}�ɕ\�����Ƃ������Ȃ��B
�Q�l�����F��l���z�̘a Feller Vol .2, p .26

\(xp[x] = \max\{0,x\}\) �̒�`�F
\(\text{xp}[\text{x$\_$}]=(x+\text{Abs}[x])/2\)


\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  (1) \, 2 �̘a�F �i�O�p�`���z�j 
  \begin{center}
   \begin{FRAME}
    \(\text{usma2}[\text{x$\_$}]=\)\\
    \(\text{Sum}[(-1)^{}\{k\}*\text{Binomial}[2,k]*\)\\
    \((\text{xp}[x-k]){}^{\{2-1\}},\{k,0,2\}]/(2-1)!\)\\
%\(\left\{1-x+\frac{1}{2} (-2+x+\text{Abs}[-2+x])-\text{Abs}[-1+x]+
% \frac{1}{2} (x+\text{Abs}[x])\right\}\)
   Plot[usma2[x],\{x,0,2\}]
   \end{FRAME}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[scale=0.8]{sum_of_Unif02.eps}
   \caption{�O�p�`���z}2�{�̈ꎟ���F$(0 < x < 1, 1 < x < 2)$
   \label{fig:tri}
  \end{center}
 \end{minipage}
\end{figure}


\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  (2) \, 3 �̘a�F%3�{�̂Q���Ȑ�
   %$( [0, 1], [1, 2], [2, 3])$
  \begin{center}
   \includegraphics[scale=0.8]{sum_of_Unif3.eps}
   \caption{3�{��2���Ȑ��̕��z$( [0, 1], [1, 2], [2, 3])$}
   \begin{comment}
    \begin{FRAME}
    \(\text{usma3}[\text{x$\_$}]=
    \text{Sum}[(-1){}^{\wedge}k*\text{Binomial}[3,k]*(\text{xp}[x-k])
    {}^{\wedge}\{3-1\},\{k,0,3\}]/(3-1)!\)
   %\(\left\{\frac{1}{2} \left(-\frac{1}{4}
   % (-3+x+\text{Abs}[-3+x])^2+\frac{3}{4} (
   % -2+x+\text{Abs}[-2+x])^2-\frac{3}{4} (-1+x+\text{Abs}[-1+x])^2+\frac{1}{4}    % (x+\text{Abs}[x])^2\right)\right\}\)
   % Plot[usma3[x],\{x,0,4\}]
   \end{FRAME}
   \end{comment}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
(3) \, 
5�̘a�F%5�{��4���Ȑ� \(�i[0, 1], [1, 2], \cdots, [4, 5]�j\)
  \begin{center}
   \includegraphics[scale=0.8]{sum_of_Unif5.eps}
   %\includegraphics[scale=0.8]{sum_of_Unif3.eps}
   \caption{5�{��4���Ȑ��̕��z\(�i[0, 1], [1, 2], \cdots, [4, 5]�j\)}
   \label{fig:5}
  \end{center}
 \end{minipage}
\end{figure}



\begin{comment}
(3) \, 
�T�̘a�F�T�{�̂S���Ȑ� \(�i[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]�j\)

\(\text{usma5}[\text{x$\_$}]=\text{Sum}[(-1){}^{\wedge}k*\text{Binomial}[5,k]*(\text{xp}[x-k]){}^{\wedge}\{5-1\},\{k,0,5\}]/(5-1)!\)
%\(\left\{\frac{1}{24} \left(-\frac{1}{16} (-5+x+\text{Abs}[-5+x])^4
% +\frac{5}{16} (-4+x+\text{Abs}[-4+x])^4-\frac{5}{8} (-3+x
% +\text{Abs}[-3+x])^4+\frac{5}{8} (-2+x+\text{Abs}[-2+x])^4-\frac{5}{16} (-1
% +x+\text{Abs}[-1+x])^4+\frac{1}{16} (x+\text{Abs}[x])^4\right)\right\}\)
Plot[usma5[x],\{x,0,5\}]
\includegraphics{sum_of_Unif5.eps}
\end{comment}


�P�Q�̘a�F�P�Q�{�̂P�P���Ȑ�
\(�i[0, 1], [1, 2], \cdots, [11, 12]�j\)

% \(\text{usma12}[\text{x$\_$}]
%  =\text{Sum}[(-1){}^{\wedge}
%  k*\text{Binomial}[12,k]*
% (\text{xp}[x-k]){}^{\wedge}\{12-1\},\{k,0,12\}]/(12-1)!\)

%\(\left\{\frac{1}{39916800}\left(\frac{(-12+x+\text{Abs}[-12+x])^{11}}{2048}
%-\frac{3}{512} (-11+x+\text{Abs}[-11+x])^{11}+\frac{33 (-10+x+\text{Abs}[-10
%+x])^{11}}{1024}-\frac{55}{512} (-9+x+\text{Abs}[-9+x])^{11}+\frac{495 (-8+x
%+\text{Abs}[-8+x])^{11}}{2048}-\frac{99}{256} (-7+x+\text{Abs}[-7+x])^{11}
%+\frac{231}{512} (-6+x+\text{Abs}[-6+x])^{11}-\frac{99}{256} (-5+x
%+\text{Abs}[-5+x])^{11}+\frac{495 (-4+x+\text{Abs}[-4+x])^{11}}{2048}
%-\frac{55}{512} (-3+x+\text{Abs}[-3+x])^{11}+\frac{33 (-2+x+\text{Abs}[-2
%+x])^{11}}{1024}-\frac{3}{512} (-1+x+\text{Abs}[-1+x])^{11}+\frac{(x
%+\text{Abs}[x])^{11}}{2048}\right)\right\}\)

%\text{Plot}[\text{usma12}[x],\{x,0,12\}]

%\includegraphics{sum_of_Unif121.eps}



�W�����K���z $N (0, 1)$ �̋ߎ���:
$\sum_{i = 1}^{12} X_i - 6$, 

$12$����������̂U�����������́i�a�̎��𕽍s�ړ��j�A����$0$�A���U$1$�ƂȂ�.
�e���$[0,1],[1,2],\cdots, [11.12]$�ł�11����������12�{�`�����O���t�ł���B
\begin{FRAME}
\noindent
    \(\text{usma12}[\text{x$\_$}]=
    \text{Sum}[(-1){\wedge}\{k\}*\text{Binomial}[12,k]*
    (\text{xp}[x-k]){\wedge}{\{12-1\}},\{k,0,12\}]/(12-1)!\\
\text{Plot}[\text{usma12}[x+6],\{x,-4,4\}]\)
\end{FRAME}


\begin{figure}[htbp]
 \begin{minipage}{0.45\hsize}
  \begin{center}
   \includegraphics[scale=0.8]{sum_of_Unif12.eps}
   \caption{12�̘a�ɂ��ߎ����z}
   \label{fig:normalappro}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.45\hsize}
  \begin{center}
   \includegraphics[scale=0.8]{normal01.eps}
   \caption{�W�����K���z$N(0,1)$}
   \label{fig:normal}
  \end{center}
 \end{minipage}
\end{figure}

\begin{comment}
\begin{tabular}{lll} \hline
 $x$	& Approx &	TRUE \\ \hline
0	&0.393926&	0.39894228\\
0.1	&0.392057&	0.396952547\\
0.2	&0.386503&	0.391042694\\
0.3	&0.377416&	0.381387815\\
0.4	&0.365039&	0.36827014\\
0.5	&0.349702&	0.352065327\\
0.6	&0.331802&	0.333224603\\
0.7	&0.311788&	0.312253933\\
0.8	&0.290145&	0.289691553\\
0.9	&0.26737&	0.26608525\\
1	&0.24396&	0.241970725\\
1.1	&0.220391&	0.217852177\\
1.2	&0.197104&	0.194186055\\
1.3	&0.17449&	0.171368592\\
1.4	&0.152885&	0.149727466\\
1.5	&0.132561&	0.129517596\\
1.6	&0.113726&	0.110920835\\
1.7	&0.0965207&	0.094049077\\
1.8	&0.0810246&	0.078950158\\
1.9	&0.0672606&	0.065615815\\
2	&0.055202&	0.053990967\\
2.1	&0.0447809&	0.043983596\\
2.2	&0.0358971&	0.035474593\\
2.3	&0.0284267&	0.028327038\\
2.4	&0.0222309&	0.02239453\\
2.5	&0.0171631&	0.0175283\\
2.6	&0.0130762&	0.013582969\\
2.7	&0.00982711&	0.010420935\\
2.8	&0.00728165&	0.007915452\\
2.9	&0.00531705&	0.005952532\\
3	&0.00382388&	0.004431848\\
\end{tabular}
\end{comment}

\vspace{2ex}

�Q�̐}\ref{fig:normal}�Ɛ}\ref{fig:normalappro}��
��ׂĂ݂�ƁA�������̃O���t�ł͂قƂ�Ǎ���
������Ȃ����A���Ό덷�̌v�Z
$\dfrac{�ߎ��l-�^�l}{�^�l}*100$\%�����߂Ă݂��
�ȉ��̂悤�ɂȂ�B����̃p�\�R����
�ȒP�ɐ��K���z�̃V�~�����[�V�������s�����Ƃ�
����������Ƃł͂Ȃ����A���ɊȒP�Ȉ�l������
12����1�̐��K���������邱�Ƃ��ł���B

\begin{figure}[htbp]
 \begin{minipage}{0.45\hsize}
\begin{tabular}{lll} \hline
 $x$	& Approx &	TRUE \\ \hline
0.0	&0.3939&	0.3989\\
0.2	&0.3865&	0.3910\\
0.4	&0.3650&	0.3683\\
0.6	&0.3318&	0.3332\\
0.8	&0.2901&	0.2897\\
1.0	&0.2440&	0.2420\\
1.2	&0.1971&	0.1942\\
1.4	&0.1529&	0.1497\\
1.6	&0.1137&	0.1109\\
1.8	&0.0810&	0.0790\\
2.0	&0.0552&	0.0540\\
2.2	&0.0359&	0.0355\\
2.4	&0.0222&	0.0224\\
2.6	&0.0131&	0.0136\\
2.8	&0.0073&	0.00792\\
3.0	&0.0038&	0.0044\\ \hline
\end{tabular}
 \end{minipage}
 \begin{minipage}{0.45\hsize}
\begin{center}
   \includegraphics[scale=0.8]{erro-size.eps}
   \caption{�^�l�Ƃ̑��Ό덷�i�ߎ��\�^�l�j�^�^�l��}
   \label{fig:erro-size}
\end{center}
 \end{minipage}
\end{figure}

\begin{Toi}
2�̂Ƃ�,3�̂Ƃ��̃O���t��
\[
 \dfrac{1}{2}\{ (x-2)_+ - 2 (x-1)_+ + x_+\}, \quad
 \dfrac{1}{2!}\{ - (x-3)_+^2 + 3 (x-2)_+^2 - 3 (x-1)_+^2 + x_+^2\}
\]
�ƂȂ邱�Ƃ������A3�ł͊��炩�ł��邱�Ƃ��m���߂�B
\end{Toi}

%\end{document}

\section{�v���b�g����}
%\noindent
���x�֐��̊֐��`��\�������閽�߁F
\begin{center}
{\bf PDF[\{���z��\}[�p�����[�^],�ϐ�],
\{�ϐ��A�ŏ��l�A�ő�l\}}
\end{center}
�O���t��`�����߁F
\begin{center}
{\bf Plot[PDF[\{���z��\}[�p�����[�^],�ϐ�],
\{�ϐ��A�ŏ��l�A�ő�l\}]}
\end{center}
�����̃O���t�������Ƃ��ɂ�PDF[...]�̕�����\{ \}�Ŗ��x�֐���
���ׂĊ���F
\begin{center}
{\bf \{PDF[\{���z��1\}[�p�����[�^],�ϐ�],
$\cdots$,
PDF[\{���z��n\}[�p�����[�^],�ϐ�]\}}
\end{center}
�Ƃ����`�ɂ���B

\begin{center}
\begin{tabular}{lp{25zw}}\hline
  ���x�֐��̊֐��`��\�������閽�߁F&
    PDF[���z��[�p�����[�^],�ϐ�],
    \{�ϐ��A�ŏ��l�A�ő�l\}\\ \hline
  �O���t��`�����߁F&
    Plot[PDF[���z��[�p�����[�^],�ϐ�],
    \{�ϐ��A�ŏ��l�A�ő�l\}]\\ \hline
\end{tabular}
\end{center}

\vspace{2ex}


\section{���z�̖��x�֐�}

\begin{enumerate}
\item 
\textbf{���U��l���z:} 
 DiscreteUniformDistribution[$n_{min}$,$n_{max}$] \\
��l�����𐶐����邽�߂�
�p������B�T�C�R���̃V�~�����[�V�����ł�$n_{min}= 1, n_{max}=6$�B
�����m���炵���������ۂ̊m���i�������m���j�ŋN�錋�ʂ�
$n_{min}$����$n_{max}$�܂ł̐����ŕ\�������s�ł���D
 \[
\text{U}\{n_{min}, \cdots,n_{max}\} \,\, \sim \,\, 
    \dfrac{1}{n_{max} - n_{min}+1}, \, x = n_{min}, \cdots, n_{max}�@(���l)
 \]



\item 
\textbf{�x���k�[�C���z:}BernoulliDistribution[$p$]\\
 �����i�l1�ɑΉ��j�� 
 $p$�̊m���ŋN��C���s�i�l0�ɑΉ��j
 ��$1-p$�̊m���ŋN��$1$��(�P��)
 ���s�̊m�����z�ł���D
$\displaystyle\binom{n}{k} = \dfrac{(n)_k}{k!} = \dfrac{(n)_k = n (n-1) \cdots (n-k+1)}{k!}$
�Ƃ���B�Q�����zBinom�� $n=1$�Ƃ����ꍇ�B
\[ 
\text{Binom}[1,p] \,\, \sim \,\,
\binom{1}{x}p^x (1-p)^{1-x}
=\left\{ \begin{array}{ll} p & x=1 \\ 1-p & x=0 \end{array} \right.
\]


\item
\textbf{ �񍀕��z:}
 BinomialDistribution[$n,p$] \\
 �e���s�ɂ����鐬���m����
 $p$ �Ƃ����Ƃ��ɁC$n$��̓Ɨ��������s�ŋN�鐬����
 �W�v�����񐔒l�̕��z�ł���D
\[ 
\text{Binom}[n,p] \,\, \sim \,\,
\binom{n}{x}p^x (1-p)^{n-x}, x=0,1,2,\cdots,n
\]


\begin{theorem}
\noindent
{\textrm (i)} \, $n$�̓Ɨ��ȃx���k�[�C���z�������a�̕��z�́A2�����z��
���������B 
\[X_i \, \sim \, \text{Binom}[1,p] \quad \Rightarrow \quad \sum_{i=1}^n X_i  \, \sim \, \text{Binom}[n,p]
\]

\noindent
{\textrm (ii)} \, $2$�̓Ɨ��ȓ����p�����[�^$p$������
2�����z�������a�̕��z�́A2�����z��
���������B 
\[X \, \sim \, \text{Binom}[n,p], \, 
Y \, \sim \, \text{Binom}[m,p]
\quad \Rightarrow \quad X + Y  \, \sim \, \text{Binom}[n+m,p]
\]
\end{theorem}



\item
\textbf{�􉽕��z:}
GeometricDistribution[$p$]\\
�e���s�̐�������$p$�Ƃ����Ƃ��ɁC�ŏ��ɐ������N��܂ł�
�K�v�Ƃ������s���̕��z�ł���D$q=1-p$�Ƃ���B�����^���s�Ƃ���
�Q�ʂ�̌��ʂ��J��Ԃ����Ƃ��A���߂Đ������\���܂ł̎��ԊԊu��
�\���B�҂����Ԃ̉�͂ɂ悭�p�����A�A���^�ł͎w�����z�ƑΔ䂳���B
\[
   \text{Geom}(p) \, \, \sim \, \, p q^x, \, x=0,1,2,\cdots
\]

\begin{theorem}
$n$�̓Ɨ��Ȋ􉽕��z�������a�̕��z�́A����2�����z��
���������B 
\[X_i \, \sim \, \text{Geom}[p] \quad \Rightarrow \quad \sum_{i=1}^n X_i  \, \sim \, \text{NegBinom}[n,p]
\]
\end{theorem}


\item
\textbf{ ���̓񍀕��z:}
  NegativeBinomialDistribution[$n,p$]\\
���̐���$n$�ɑ΂��镉�̓񍀕��z 
�Ƃ́C�e���s�ɂ����鐬���m����$p$�Ƃ����Ƃ��ɁC$n$
��̐�������܂łɕK�v�Ȃ��߂ɋN�������s�i$q=1-p$�̊m���j�񐔂�
���z�ł���D
�͂��߂Đ������N����܂ł̑҂����Ԃ��􉽕��z�ł��邩��A
�����$n$��܂Ő�������܂łɗv����Ԋu�̉񐔒l���z�ł���B
�܂�􉽕��z�̘a�ƍl����΂悢�B$n=1$���􉽕��z�ɑ��Ȃ�Ȃ��B
�ŏI��ȑO�͉񐔂��w�肵�Ȃ�����A2���W�����\���B�܂�
2���W���ɕ��̒l������A\(
\binom{-n}{x} = \frac{(-n)_x}{x!} = 
\frac{(-n)(-n-1)\cdots(-n-x+1)}{x!}
= (-1)^x \frac{(n+x-1)_x}{x!}
= (-1)^x \binom{n+x -1}{x}
\)
�ƂȂ�
���Ƃɒ��ӁB

\[ 
\text{NegBinom}[n,p] \,\, \sim \,\, 
\binom{-n}{x}p^n (-q)^{x}
=\binom{n +x -1}{x}p^n q^{x} , x=0,1,2,\cdots,
\]



\item
\textbf{�x�[�^�񍀕��z:}
BetaBinomialDistribution[$\alpha$,$\beta$,$n$]\\
�񍀕��z�ƃx�[�^���z��g�ݍ������������z�ł���D
���̊m���ϐ��́C�����m��$p$
���̂��m���ϐ��Ƃ��āA���̕��z��
�x�[�^���zBetaDistribution[$\alpha$,$\beta$]
�ɏ]���Ƃ��A���̍������z��BinomialDistribution[$n$,$p$]�ɏ]��
�Ƃ����D
$B(\alpha,\beta)
= \displaystyle\int_0^1 t^{\alpha - 1} (1-t)^{\beta-1} dt$
\, $(\alpha,\beta > 0)$�̓x�[�^�֐��B
�w������$ 0 < \alpha < 1, \alpha=1 , 1 < \alpha$�ɂ���đ傫��
�ω�����B
\[ 
\text{BetaBinom}[\alpha,\beta,n] \,\, \sim \,\, 
\binom{n}{x} \dfrac{B(x+\alpha, n + \beta-x)}{B(\alpha, \beta)}, x=0,1,2,\cdots,n
\]



\item
\textbf{�x�[�^���̓񍀕��z:}
BetaNegativeBinomialDistribution[$\alpha$,$\beta$,$n$]\\
�x�[�^���z�ƕ��̓񍀕��z��g�ݍ������������z�ł���D
$p$�̒l�����m�̂��̂Ƃ���̂ł͂Ȃ��A
�ϓ�������̂ƍl����B���̂悤�Ƀp�����[�^$p$���x�[�^���z�Ƃ��闝�R��
�x�[�^���z���_��ŁA��l���z���܂ނ��܂��܂Ȍ`���\������B
\[
\text{BetaNegBinom}(\alpha,\beta,n) \, \, \sim \, \,
 \binom{-1 + n + x}{ -1 + n}\dfrac{B[\alpha + n, \beta + x]}{B[\alpha, \beta]},
 \, x=0,1,2,\cdots,n
\]




\item
\textbf{���􉽕��z:}
HypergeometricDistribution[$n$,$n_{succ}$,$n_{tot}$]\\
$n$ �񒆂̂�\{������,�O��(�܂��͐����A���s)\}���s�ł̓������
�ł��邪�A���o���̌��ʂɂ���ē�����A�O��̊m�����ω�����B
��W�c�T�C�Y�i�����̌��j�� $n_{tot}$ �Ƃ��āA
�ŏ���$n_{tot}$�����̂���������A���̓���
�����肭����(�����Ƃ����)$n_{succ}$���
�O�ꂭ�����i���s�Ƃ���񐔁j$n_{fail}=n_{tot} - n_{succ}$��
�\������Ă���B��x���o���������͔j������
2�x�Ǝ��o����邱�Ƃ͂Ȃ����̂Ƃ���B
���Ȃ킿�A���ɖ߂����ɃT���v�����O����i�񕜌����o�j�B
�͂��߂ɂ́A����$n_{tot}$�̂�����
�����萔$n_{succ}$�ƊO�ꐔ$n_{fail}=n_{tot}-n_{succ}$,
"tot = total"�ł���A
���o���Ƃɉ��ꂩ�������Ă����񕜌����o�������s���B
���􉽕��z�Ƃ́A�������������݂���$n$��̂�����
�����肭�����������񐔂�\�����z�ł���B
���̂���Ԃ����s�͓Ɨ��ł͂Ȃ����Ƃɒ��ӁB
�񍀕��z�̓x���k�[�C���z�̂���Ԃ����s�œƗ��ł���ƂƂɈႢ������B
�����肭���������m���i�����肭���ƊO�ꂭ���̔䗦�j��
���i�������o�j�̏ꍇ�ł���B
\[ 
\text{HyperGeom}[n,n_{succ},n_{tot}] \, \, \sim \,\,
  \dfrac{\binom{n_{succ}}{x} \binom{n_{fail}}{n-x}}{\binom{n_{tot}}{n}}
= \binom{n}{x} 
   \dfrac{(n_{succ})_x \, (n_{fail})_{n-x}}{(n_{tot})_n}, \, x=0,1,2,\cdots,n
\]
\begin{theorem}
���􉽕��z�̃p�����[�^ $n_{tot}= n_{succ} + n_{fail}, (n_{tot} \to \infty)$ 
�ɂ�����
\[
 \dfrac{n_{succ}}{n_{tot}} \to p, \dfrac{n_{fail}}{n_{tot}} \to q=1-p 
  \, \Rightarrow \, 
 \text{HyperGeom}[n,n_{succ},n_{tot}] 
 \doteqdot \text{Binom}[n,p]
\]
����������ƁA��W�c�T�C�Y���傫���Ȃ�΁A�����Ɣ񕜌��̈Ⴂ��
���Ȃ��Ȃ�B�u�ΐ��l�̐^���͐s����Ƃ����ɓ��l�̎�͐s���܂��v
(�ΐ�܉E�q��)
\end{theorem}
���Ƃ��΁A�X�s�[�h�O�p�������Ђ��B��x�����΂����2�x�ƈ�����邱�Ƃ�
�Ȃ��B���̂Ƃ�10�{�̂���������C���̂���4�{��������Ŏc��6�{���O���
����D����3�{�����������āC���̂���2�{��������m���́H�Ƃ������͒��􉽕��z��
����B$n_{tot} =10, n_{succ} = 4, n_{fail} = 6, n=3, x= 2$
�Ƃ��Čv�Z����B


\item
\textbf{�|�A�\�����z:}
PoissonDistribution[$\lambda$]\\
�w�肳�ꂽ���ԓ��ɋN�鎖�ې����L�q����D������
$1/\lambda$�͈�莞�Ԃɑ΂��鎖�ۂ̕��ω�(�����Ƃ͌���Ȃ�)�ł���D

2�����z Binom($n$,$p_n$) �̋Ɍ��Ƃ��ē�����A�H�Ȍ��ۂ��L�q����
�Ƃ��ɗp�����A�d�v�ȕ��z�ł���B�H�Ƃ����Ӗ���2�����z�ł̎��ۂ�
�N����(��������)�m��$p_n$���[���ɋ߂Â��邪�A
���s�̌J��Ԃ���$n$�𑝂₷���Ƃɂ��B
�w�����o�Ă��闝�R�� $\lim_n(1 - p_n)^n = e^{-\lambda}$, 
�������A$n p_n \to \lambda (n \to \infty)$�ŁA�����̊m��
�i���ۂ��N����m���j$p_n \to 0$�Ƃ��Ȃ���΂Ȃ�Ȃ�����A���ꂪ
�H�Ƃ������Ƃ��\���Ă���B
\[{Po}[\lambda] \,\, \sim \,\,
  e^{- \lambda} \dfrac{\lambda^x}{x!}, \, x= 0,1,2,\cdots
\]
\begin{theorem}
2�����zBinom[$n$,$p_n$] �ɂ�����, ����萔$\lambda >0 $��
������ $(n \to \infty, p_n \to 0)$�̂Ƃ��A
\[
  n p_n \to \lambda \, \Rightarrow \, 
  {\textrm Binom}[n,p] \, \Rightarrow \, \text{Po}[\lambda]
\]
\end{theorem}
���𐔂���^�C�v�̌��ۂ͋ߎ��I�Ƀ|�A�\�����z�ɏ]���Ƃ����Ă���B
���Ƃ��΁A�P�ʎ��Ԃɂ������Ă���d�b�̉񐔁A������������{��
1�y�[�W���̌�A�ȂǁB�L���ȗ�Ƃ��āA���V�A�ł̔n�ɏR���Ď���
���m�̐�(W.Feller vol.I VI��)������B


\begin{comment}
�ΐ����z:\\
LogSeriesDistribution[$\theta$]
$\theta=0$�t�߂�$ \log (1-\theta)$�̋����W�J�ɂ����鍀�́C
�ΐ����z LogSeriesDistribution [$\theta$]
�ɏ]�������U�m���ϐ��̊m���ɔ�Ⴗ��D
�w�肳�ꂽ���ԓ��ɍw������鐻�i���̕��z���C
���̕��z�ɂ���ă��f��������邱�Ƃ�����D

�W�b�v(Zipf)���z:\\
ZipfDistribution[$rho$]�̓[�[�^���z�Ƃ���΂��D
����͍ŏ�����w�Ŏg���C���������ۂ̃��f�����ɂ܂�
�g�����ꂽ�B
\end{comment}
\end{enumerate}

%\newpage
\section{�p�����[�^�ω��̊T�`��`���i2���A�􉽁A����2���A���􉽁j}
�m�����z�ɂ͂������̃p�����[�^���܂܂�Ă���B
���̃p�����[�^���ω������Ƃ��ɂ́A���܂��܂Ȍ`��
�\���ꍇ������A�����ɂ��Ē��ׂĂ������Ƃ�
�d�v�ł���A�O���t�`��̋@�\���悭�i����������ł�
���[�����@�������邩���m��Ȃ��B


\begin{comment}
\textbf{2�����zBinom($50$,$0.3$)��\���������F}
\begin{verbatim}
  ListPlot[
  Table[{k, PDF[BinomialDistribution[50,0.3], k]}, 
  {k, 0, 50}], Filling -> Axis]
\end{verbatim}
\end{comment}


\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  (1) \, �􉽕��z Geom($p$)�� $p=0.1, 0.3, 0.5, 0.8$
  \begin{center}
   \begin{FRAME}
    \noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
    \pmb{\text{PDF}[\text{GeometricDistribution}[p],k]\},}\\
    \pmb{\{p,\{0.1,0.3,0.5,0.8\}\},\{k,0,15\}],}\\
    \pmb{\text{PlotMarkers}\to \text{Automatic},}\\
    \pmb{\text{Joined} \to  \text{True},} \\
    \pmb{\text{PlotRange} \to  \{\{0,15\},\{0,0.8\}\} ]}\)
   \end{FRAME}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr1.eps}
   \caption{�􉽕��z Geom($p$)�̃p�����[�^�ω�)}
   \label{fig:one}
  \end{center}
 \end{minipage}
\end{figure}


\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr2.eps}
   \caption{2�����z Binom($10$,$p$) �ł�$p$�ω�}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  (2) \, 2�����z Binom($10$,$p$) ��$p=0.1,0.3,0.5,0.8$
  \begin{center}
   \begin{FRAME}
    \noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
    \pmb{\text{PDF}[\text{BinomialDistribution}[10,p],k]\},}\\
    \pmb{\{p,\{0.1,0.3,0.5,0.8\}\},\{k,0,10\}],}\\
    \pmb{\text{PlotMarkers}\to \text{Automatic},}\\
    \pmb{\text{Joined} \to  \text{True}, }\\
    \pmb{\text{PlotRange} \to  \text{Automatic} ]}\)
   \end{FRAME}
  \end{center}
 \end{minipage}
\end{figure}



\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  (3) \, 2�����z Binom($n$,$1/2$) �� $n =5,10,20$
  \begin{center}
   \begin{FRAME}
    \noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
    \pmb{\text{PDF}[\text{BinomialDistribution}[n,1/2],k]\},}\\
    \pmb{\{n,\{5,10,20\}\},\{k,0,20\}],}\\
    \pmb{\text{PlotMarkers}\to \text{Automatic},}\\
    \pmb{\text{Joined} \to  \text{True}, }\\
    \pmb{\text{PlotRange} \to  \text{Automatic} ]}\)
   \end{FRAME}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr3.eps}
   \caption{2�����z Binom($n$,$1/2$) �ł�$n$�ω�}
  \end{center}
 \end{minipage}
\end{figure}


\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr4.eps}
   \caption{2�����z Binom($n$,$1/4$) �ł�$n$�ω�}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  (4)\, 2�����z Binom($n$,$1/4$) �� $n =5,10,20$
  \begin{center}
   \begin{FRAME}
    \noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
    \pmb{\text{PDF}[\text{BinomialDistribution}[n,1/4],k]\},}\\
    \pmb{\{n,\{5,10,20\}\},\{k,0,20\}],}\\
    \pmb{\text{PlotMarkers}\to \text{Automatic},}\\
    \pmb{\text{Joined} \to  \text{True}, }\\
    \pmb{\text{PlotRange} \to  \text{Automatic} ]}\)
   \end{FRAME}
  \end{center}
 \end{minipage}
\end{figure}


\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  (5)\, 2�����zBinom($n$,$3/4$) �� $n =5,10,20$
  \begin{center}
   \begin{FRAME}
    \noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
    \pmb{\text{PDF}[\text{BinomialDistribution}[n,3/4],k]\},}\\
    \pmb{\{n,\{5,10,20\}\},\{k,0,20\}],}\\
    \pmb{\text{PlotMarkers}\to \text{Automatic},}\\
    \pmb{\text{Joined} \to  \text{True}, }\\
    \pmb{\text{PlotRange} \to  \text{Automatic} ]}\)
   \end{FRAME}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr5.eps}
   \caption{2�����z Binom($n$,$3/4$) �ł�$n$�ω�}
  \end{center}
 \end{minipage}
\end{figure}

\vspace{2ex}


\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr6.eps}
   \caption{����2�����z NegBinom($8$,$p$), $p$�̕ω�}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  (6)\, ����2�����z NegBinom($8$,$p$), $p=0.1, 0.3, 0.5, 0.8$
  \begin{center}
   \begin{FRAME}
    \noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
    \pmb{\text{PDF}[\text{NegativeBinomialDistribution}[5,p],k]\},}\\
    \pmb{\{p,\{0.1,0.3,0.5,0.8\}\},\{k,0,30\}],}\\
    \pmb{\text{PlotMarkers}\to \text{Automatic},}\\
    \pmb{\text{Joined} \to  \text{True}, }\\
    \pmb{\text{PlotRange} \to  \{\{0,30\},\{0,0.35\} ]}\)
   \end{FRAME}
  \end{center}
 \end{minipage}
\end{figure}


\vspace{2ex}

\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  (7)\, ����2�����z NegBinom($n$,$0.3$), $n=1,5,8,10$
  \begin{center}
   \begin{FRAME}
    \noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
    \pmb{\text{PDF}[\text{NegativeBinomialDistribution}[n,0.3],k]\},}\\
    \pmb{\{n,\{1,5,8,10\}\},\{k,0,30\}],}\\
    \pmb{\text{PlotMarkers}\to \text{Automatic},}\\
    \pmb{\text{Joined} \to  \text{True},}\\
    \pmb{\text{PlotRange} \to  \{\{0,30\},\{0,0.35\}\} ]}\)
   \end{FRAME}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr7.eps}
   \caption{����2�����z NegBinom($n$,$0.3$), $n$�̕ω�}
  \end{center}
 \end{minipage}
\end{figure}


\begin{comment}
(���􉽕��z HyperGeom($n$,$n_{succ}$,$n_{tot}$))
\noindent
$\text{�m��}:n=5, \dfrac{n_{succ}}{n_{tot}} = \dfrac{2}{10}, 
  \dfrac{5}{10}, \dfrac{7}{10} (n < n_{tot})$

\noindent
\(
 \pmb{\text{Table}[
 \text{PDF}[
 \text{HypergeometricDistribution}
 [5,\text{ns},10],k],\{\text{ns},\{2,5,7\}\}]}
\)

\noindent
\(
  \left\{\dfrac{1}{252} 
   \binom{2}{k} \binom{8}{5-k},
   \dfrac{1}{252} \binom{5}{5-k} \binom{5}{k},
   \dfrac{1}{252} \binom{3}{5-k} \binom{7}{k}
  \right\}
\)
\end{comment}


\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr8.eps}
   \caption{���􉽕��z HyperGeom($n$,$n_{succ}$,$n_{tot}$), $n$�̕ω�}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  (8) \, ���􉽕��z HyperGeom($5$,$n_{succ}$,$n_{tot}$)\\
  $\dfrac{n_{succ}}{n_{tot}} = \dfrac{2}{10}, 
  \dfrac{5}{10}, \dfrac{7}{10} \, (n < n_{tot})$
  \begin{center}
   \begin{FRAME}
    \noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
    \pmb{\text{PDF}[\text{HypergeometricDistribution}[5,\text{ns},10],k]}\\
    \pmb{\},\{\text{ns},\{2,5,7\}\},\{k,0,5\}],}\\
    \pmb{\text{PlotMarkers}\to \text{Automatic},}\\
    \pmb{\text{Joined} \to  \text{True}, }\\
    \pmb{\text{PlotRange} \to  \text{Automatic} ]}\)
   \end{FRAME}
  \end{center}
 \end{minipage}
\end{figure}


\begin{comment}}
���􉽕��z(HypergeometricDistribution[$n$,$n_{succ}$,$n_{tot}$])

\noindent
\(
\pmb{\text{�m��}:n=15, \dfrac{n_s}{n_t} = \dfrac{4}{20}, 
\dfrac{10}{20}, \dfrac{14}{20}\,  (n < n_t)}
\)

\noindent
\(
\pmb{\text{Table}[\text{PDF}[\text{HypergeometricDistribution}[15,\text{ns},20],k],\{\text{ns},\{4,10,14\}\}]}
\)

\noindent
\(
 \left\{
  \dfrac{\binom{4}{k} \binom{16}{15-k}}{15504},
  \dfrac{\binom{10}{15-k} \binom{10}{k}}{15504},
  \dfrac{\binom{6}{15-k} \binom{14}{k}}{15504}
 \right\}
\)
\end{comment}}


\vspace{2ex}


\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
   (9)\, ���􉽕��z\\
   (HypergeometricDistribution[$15$,$n_{succ}$,$n_{tot}$])\\
   $\frac{n_s}{n_t} = \frac{4}{20}, 
   \frac{10}{20}, \frac{14}{20}$
  \begin{center}
   \begin{FRAME}
    \noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
    \pmb{\text{PDF}[\text{HypergeometricDistribution}[15,\text{ns},20],k]}\\
    \pmb{\},\{\text{ns},\{4,10,14\}\},\{k,0,15\}],}\\
    \pmb{\text{PlotMarkers}\to \text{Automatic},}\\
    \pmb{\text{Joined} \to  \text{True}, }\\
    \pmb{\text{PlotRange} \to  \text{Automatic} ]}\)
   \end{FRAME}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr9.eps}
   \caption{���􉽕��z Hypergeom[$n$,$n_{succ}$,$n_{tot}$], $n$�̕ω�}
  \end{center}
 \end{minipage}
\end{figure}


\vspace{2ex}

\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr10.eps}
   \caption{���􉽕��z Hypergeom[$n$,$n_{succ}$,$n_{tot}$], $n$�̕ω�}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
   (10)\, ���􉽕��z\\
   (HypergeometricDistribution[$15$,$n_{succ}$,$n_{tot}$])\\
   $\frac{n_s}{n_t} = \frac{40}{200}, 
   \frac{100}{200}, \frac{140}{200}$
  \begin{center}
   \begin{FRAME}
    \noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
    \pmb{\text{PDF}[\text{HypergeometricDistribution}[15,\text{ns},20],k]}\\
    \pmb{\},\{\text{ns},\{40,100,140\}\},\{k,0,15\}],}\\
    \pmb{\text{PlotMarkers}\to \text{Automatic},}\\
    \pmb{\text{Joined} \to  \text{True}, }\\
    \pmb{\text{PlotRange} \to  \text{Automatic} ]}\)
   \end{FRAME}
  \end{center}
 \end{minipage}
\end{figure}




\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
   (11) \, ���􉽕��z���Q�����z�ɋ߂Â�\\
   ($\dfrac{n_{succ}}{n_{tot}} \to p$)
   ���r����B\\
   $n=15$ �ł�
   2�����z(BinomialDistribution[$15$,$p$,]
   $p= \frac{n_s}{n_t} = \frac{40}{200}, 
   \frac{100}{200}, \frac{140}{200}$ ��`���B
  \begin{center}
   \begin{FRAME}
    \noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
    \pmb{\text{PDF}[\text{BinomialDistribution}[15,p],k]}\\
    \pmb{\},\{\text{ns},\{40,100,140\}\},\{k,0,15\}],}\\
    \pmb{\text{PlotMarkers}\to \text{Automatic},}\\
    \pmb{\text{Joined} \to  \text{True}, }\\
    \pmb{\text{PlotRange} \to  \text{Automatic} ]}\)
   \end{FRAME}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr11.eps}
   \caption{2�����z Binom[$n$,$p$}
  \end{center}
 \end{minipage}
\end{figure}

\newpage
\section{�x�[�^���z�ɂ���}
�Ɨ��Ɉ�l���z $\text{U}(0,1)$ �ɏ]�� $p+q-1$ �̊m���ϐ���傫���̏���
���בւ����Ƃ��C������������ $p$ �Ԃ߁i�傫��������� $q$ �Ԗځj��
�m���ϐ� $X$ �̕��z���x�[�^���z Beta[$p$,$q$] �ƂȂ�B�������v�ʂ�
�v�Z���邱�ƂŎ������B�����ł͕ʂ̊ϓ_�ŏd�v�������B�x�C�Y���v�w
�ł���B
�x�[�^���z�͓񍀕��z�̋������O���z�ł���B�����ڂ����q�ׂ�ƁA
���v�w�I�ȁi�����̂��̂������j��Ƃ��āA�񍀕��z�̕ꐔ�̎��㕪�z��
�v�Z���邱�Ƃ��l���悤�B
�ώ@���ʂ��A���� $m$ ��A���s $n$ ��ƂȂ����Ƃ���B��̓I�ɂ�
�����E���s�̃R�C�������Ƃ��A��Ăɑ΂���^���E���΂ł��悢�B
�����m���̕ꐔ$p$�ɂ��Ď��O�m���ŕ\�����Ƃ���B
�^����ꂽ $p$ �̒l�ɑ΂��āA�S $m+n$ ��̎��s�̂��������� $m$ ��
�ƂȂ�m���́A2�����z
$ \binom{n+m}{m} p^m (1-p)^n$,
$m$ �� $n$ �͌Œ肳��A $a$ �͖��m�ł���B
�x�C�Y�̒藝�i�A�����z�̌`�j����A����m���̌v�Z�������Ȃ��ƁA
���m��$p$ �ɑ΂��鎖�O���z�� �x�[�^���zBeta[$m_0$,$n_0$]�Ȃ�΁A
�x�C�Y�̒藝����v�Z�ł��鎖�㕪�z���x�[�^���z�ŁA
Beta[ $m+m_0$, $n+n_0$] �ƂȂ�B
���̃x�[�^���z�̂悤�ɁA���㕪�z�������^�C�v�̕��z�ɂȂ�悤��
���O���z���������O���z�Ƃ����B
�x�C�Y����i�x�C�Y�����Ă��j�Ƃ́A����؋��Ɋ�Â��āA���̌�����
�Ȃ������ۂ𐄒肷�邽�߂̊m���_�I���@�ł���B�܂����v�w�ɉ��p�����
�x�C�Y���v�w�̑�\�I�ȕ��@�ƂȂ��Ă���B

%\textbf{�x�[�^���z:}
%BetaDistribution[$\alpha$,$\beta$]\\
�p�����[�^�̒l$\alpha$, $\beta$ �͐��̎����ł悢�B�ɂ߂čL�͈͂�
�`��ƂȂ镪�z�`�B$\alpha = \beta=1$�ł͘A���^��l���z�B
$0 < \alpha < 1, \alpha =1, 1 <\alpha < \infty $�ƕω���
����ɏ]���A���܂��܂Ȍ`��ƂȂ�B$\beta$�ɂ��Ă�
���l�ł���B

\[
  \text{Beta}[\alpha,\beta] \,\, \sim \,\, \dfrac{1}{B(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1}, \, \,\, 0< x< 1
\]

\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \begin{FRAME}
   (12)\, 5�{�̃x�[�^���z��`�����߂Ƃ��̌��ʁF
   \noindent\(\pmb{\text{Show}[\text{Plot}[\{}\\
   \pmb{\text{PDF}[\text{BetaDistribution}[1/2,1/2],x],}\\
   \pmb{\text{PDF}[\text{BetaDistribution}[1,2],x],}\\
   \pmb{\text{PDF}[\text{BetaDistribution}[3,3],x],}\\
   \pmb{\text{PDF}[\text{BetaDistribution}[5,2],x],}\\
   \pmb{\text{PDF}[\text{BetaDistribution}[1,1],x]\},}\\
   \pmb{\{x,0,1\}],}\\
   \pmb{\text{Graphics}[\{}\\
   \pmb{\text{Text}[\text{$\texttt{"}$�x�[�^���z�̖��x�֐�$\texttt{"}$},
     \{0.5,3.5\}],}\\
   \pmb{\text{Text}[\text{$\texttt{"}\alpha $=$\beta $=1/2$\texttt{"}$},
     \{0.1,3.2\}],}\\
   \pmb{\text{Text}[\text{$\texttt{"}\alpha $=1, $\beta $=2$\texttt{"}$},
     \{0.25,1.9\}],}\\
   \pmb{\text{Text}[\text{$\texttt{"}\alpha $=$\beta $=3$\texttt{"}$},
     \{0.5,2.2\}],}\\
   \pmb{\text{Text}[\text{$\texttt{"}\alpha $=5,$\beta $=3$\texttt{"}$},
     \{0.8,2.8\}],}\\
   \pmb{\text{Text}[\text{$\texttt{"}\alpha $=$\beta $=1$\texttt{"}$},
     \{0.1,0.8\}]\}]]}\\
   \)
   \end{FRAME}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01beta_gr14.eps}
   \caption{5�{�̃x�[�^���z Beta[$\alpha$,$\beta$]}
  \end{center}
 \end{minipage}
\end{figure}


%  PDF[BetaDistribution[$\alpha$, $\beta$]], x]
%  Plot[PDF[BetaDistribution[5, 2], x], {x, 0, 1}]




\section{�􉽕��z�Ǝw�����z}

�􉽕��z�Ǝw�����z�́A���U�^�^�A���^�̈Ⴂ�͂���܂����A
�ƂĂ����̂悢�Z�핪�Ƃ��Ă̐����������܂��B
(1)�������[���X�A(2)���ς╪�U�A
(3)�����t�����Ғl�̊֌W���Ȃǂ��������܂��B
�����̘a�Ƃ��āA����2�����z�A�K���}���z���Ή����Ă��܂��B

\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr12.eps}
   \caption{�􉽕��z Geom[$p$]}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr13.eps}
   \caption{�w�����z Expo[$\lambda$]}
  \end{center}
 \end{minipage}
\end{figure}


% \textbf{�􉽕��z�Ǝw�����z:}\\
% \pmb{\text{GeometricDistribution}[p], 0<p <1}\\
% \pmb{\text{ExponentialDistribution}[\text{lm}], \text{lm} >0}\)
�􉽕��z $\text{Geom}(p)$ �̃p�����[�^�ω��̏󋵂𒲂ׂ邽�߂ɁA
�􉽕��z�̊m�����x��\�����閽��
%
%\noindent
\textbf{PDF[GeometricDistribution[p],k]}
����͂���ƁA
���ʂ�\((1-p)^k p\)�ƂȂ�B$p$��$\{0.1,0.3,0.5,0.8\}$�ƕω�������
�`�����߂͈ȉ��̂悤�ɂȂ�܂��B
\begin{FRAME}
\noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}
\pmb{\text{PDF}[\text{GeometricDistribution}[p],k]\},}\\
\pmb{\{p,\{0.1,0.3,0.5,0.8\}\},\{k,0,15\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \{\{0,15\},\{0,0.8\}\} ]}\)
\end{FRAME}
�܂��w�����z�̃p�����[�^�ω��𒲂ׂ�ɂ�
�w�����z�̊m�����x�F
\textbf{PDF[ExponentialDistribution[lm],x]}�Ɠ��͂���
\(e^{-\text{lm} x} \text{lm}\)�𓾂�B���l��$lm =\{0.3, 0.5, 0.8\}$
�̏ꍇ��`���Ă݂܂��B2�̌��ʂ��ׂĂ݂Ă��������B�قړ����悤��
�Ȑ��ɂȂ��Ă��܂��B
\begin{FRAME}
\noindent\(\pmb{\text{Plot}[}
\pmb{\{\text{PDF}[\text{ExponentialDistribution}[0.3],x],}\\
\pmb{\text{PDF}[\text{ExponentialDistribution}[0.5],x],}\\
\pmb{\text{PDF}[\text{ExponentialDistribution}[0.8],x]\},}
\pmb{\{x,0,15\},\text{PlotRange} \to  \{\{0,15\},\{0,0.8\}\}]}\)
\end{FRAME}

\section{2�����z�Ɛ��K���z}
����2�̕��z���r���邱�Ƃ́A���K���z�𔭌����邱�Ƃ�
���������ɂȂ����A�h���A�u�������v���X�̒藝�A����ɂ�
�K�E�X�ɂ�钆�S�Ɍ��藝�Ɣ��W���Ă����������̊m���_
�ł̍ł��d�v�Ȍ��ʂ̂ЂƂł��B

2�����z$Binom[n,p]$�ɂ����āA$n$�����񂾂�傫�����āA
�O���t��`���ƁA�͂��߂�$p$�̕΂�ɉe�����݂��Ă�
$p$���ω����Ȃ��Ă��A���X�ɏ��Ȃ��Ȃ��āA
����ȂЂƂR�ɋ߂Â��B�O�߂Ńp�����[�^�ω��ł݂�
�Ƃ���ł���B
����͈�̒��S�Ɍ��藝�̕\��ł����āA�h���A�u�������v���X�̒藝
�Ƃ�΂��B����$p$�̒l��$0$�ɋ߂Â��A�ς����l�Ȃ��
$(1-p)^n \rightarrow 1^{\infty} = e$����|�A�\�����z
�ƂȂ�B�܂��X�^�[�����O�̌����F
\[
   n! \,\, \sim \,\, \sqrt{2 \pi n} \, n^n e^{-n}
\]
���瓱�����2���W����
\[
   \binom{2n}{n} \,\, \sim \, \, \dfrac{1}{\sqrt{\pi n}}4^n
\]
���K���z�̖��x�̊֘A�����l�ł���B

\begin{comment}
\noindent\(\pmb{\text{�Q�����z�Ɛ��K���z}}\\
\pmb{2\text{�����z�ƃ|�A�\�����z}}\\
\pmb{\text{��l���z�̘a�Ɛ��K���z}}\\
\pmb{\text{��l���z�ƃR�[�V�[���z}}\)

\noindent\(\pmb{}\\
\pmb{\text{Tally}[\text{list}]}\\
\pmb{\text{Tally}[\{a,a,b,a,c,b,a\}]\text{   }\text{--}\to >>}\\
\pmb{\text{     }\{\{a,4\},\{b,2\},\{c,1\}\}}\)

\noindent\(\pmb{}\\
\pmb{\text{�������v��}}\\
\pmb{\text{Median}[\text{list}]}\\
\pmb{\text{Median}[\{1,2,3,4,5,6,7\}]\text{     }\text{--}\to >>\text{  }4}\\
\pmb{\text{Median}[\{1,2,3,4,5,6,7,8\}]\text{  }\text{--} \to >> \text{9/2}}\\
\pmb{\text{���ʎl���ʓ_����я�ʎl���ʓ_�F}}\\
\pmb{\text{Quantile}[\{1,2,3,4,5,6,7\},\{1/4,3/4\}]\text{  }\text{--} \to >>}\\
\pmb{\{2,6\}}\)

\noindent\(\pmb{\text{Histogram}[\text{RandomReal}[\text{NormalDistribution}[0,1],200]]}\)

\end{comment}







\section{���K���z�̂Ƃ�}

�W�����K���z$N(0,1)$�̖��x�֐��́A�����ȒP�Ȃ��̂ł͂Ȃ����A
�T�`�͓��v�w�̃e�L�X�g�ł͂��Ȃ炸�T�`���`�����B
�x���^�C�v�Ƃ�΂�A


����$\mu$ �ƕ��U$\sigma^2$���������ꍇ��$N(\mu,\sigma^2)$
���x�֐��Ƃ��̃O���t�B���ہA���x�ϐ�$x$�̕ψ��
$x_{min}= - \infty,\, x_{max} = \infty$
�ł����āA�R���s���[�^�̕`��̂��߂ɐݒ肵�Ă���B
\[
\begin{array}{l}
  \text{PDF} [\text{NormalDistribution}[\mu,\sigma^2],
  \{x,x_{min},x_{max}\}]\\[2ex]
  \qquad\qquad \,\, 
  \sim \, \, \dfrac{1}{\sqrt{2 \pi \sigma^2}} 
    \exp \left\{- \dfrac{(x-\mu)^2}{2 \sigma^2}\right\},\,\,
    x_{min} < x < x_{max}
\end{array}
\]

�W�����K���z$N(0,1)$�̖��x�֐���\������:
\noindent\(\pmb{\text{PDF}[\text{NormalDistribution}[0,1],x]}\)
����͂���
\noindent\(\dfrac{e^{-\frac{x^2}{2}}}{\sqrt{2 \pi }}\)
�Əo�͂����B���ςƕ��U��ω�������
5��ނ̃O���t��`�����߂͈ȉ��̒ʂ�B

\vspace{2ex}
\begin{minipage}{.35\textwidth}
%\begin{FRAME}
\(\pmb{\text{Plot}[\{}\\
\pmb{\hspace{1.5em} \text{PDF}[\text{NormalDistribution}[0,1],x],}\\
\pmb{\hspace{1.5em} \text{PDF}[\text{NormalDistribution}[-1.5,1],x],}\\
\pmb{\hspace{1.5em} \text{PDF}[\text{NormalDistribution}[-1/4,1/2],x],}\\
\pmb{\hspace{1.5em} \text{PDF}[\text{NormalDistribution}[0,1.5],x],}\\
\pmb{\hspace{1.5em} \text{PDF}[\text{NormalDistribution}[1,0.8],x]\}}\\
\pmb{,\{x,-3,3\}]}\)
%\end{FRAME}
\end{minipage}
\begin{minipage}{.1\textwidth}
    \hfill
\end{minipage}
\begin{minipage}{.55\textwidth}
\includegraphics[width=80mm]{���K�̃O���t_gr1.eps}
\end{minipage}

\section{�J�C2�敪�z�ɂ���}
���v�w�Ő��K���z�͏d�v�ł��邪�A���������ɔ��W�����
�J�C2�敪�z��m���Ă����Ȃ���΂Ȃ�Ȃ��B
�J�C��敪�z�i�����ɂ��傤�Ԃ�ՁA�J�C�����傤�Ԃ�Ձj�A�܂���$\chi^2$���z��
�������v�w�ōL�����p�����B�w�������g
(Helmert, F. R. (1875),Zeit Math Phy, 20,300-303)�ɂ�蔭������AK.�s�A�\��
�ɂ�薽�����ꂽ�Ƃ�����B
�J�C��挟��A�K���x����A�Ɨ����̌���ȂǑ����̌���@�ɗ��p�����B
\[
\begin{array}{l}
  \text{PDF} [\text{ChiSquareDistribution}[\nu ],x],
  \{x,x_{min},x_{max}\}]\\[2ex]
  \qquad\qquad \,\, 
  \sim \, \, 
  \dfrac{2^{-\nu /2} e^{-x/2} x^{-1 + \frac{\nu }{2}}}
  {\text{Gamma}\left[\frac{\nu }{2}\right]},\,\,
    x_{min} < x < x_{max}
\end{array}
\]

\noindent
�J�C2�敪�z�̖��x�֐���\������B�K���}�֐�
$\text{Gamma}\left[\nu \right]= 
\Gamma(\nu)= \displaystyle\int_0^{\infty} x^{\nu-1}e^{-x} dx\, (\nu > 0)$ ����������B
�K��̊g���ł���A���R���ł�
$\Gamma(n) = (n-1)!$�ƂȂ�A�Ƃ���$\Gamma(1/2) = \sqrt{\pi}$
���o���Ă����Ƃ悢�B

\noindent\(\pmb{\text{PDF}[\text{ChiSquareDistribution}[\nu ],x]}\)

\noindent\(\frac{2^{-\nu /2} e^{-x/2} x^{-1+\frac{\nu }{2}}}{\text{Gamma}\left[\frac{\nu }{2}\right]}
\)
\[
\text{Gam}(\nu) \, \, \sim \, \, 
  \dfrac{2^{-\nu /2} e^{-x/2} x^{-1 
   + \dfrac{\nu }{2}}}{\Gamma\left[\frac{\nu }{2}\right]}, 
  \, x > 0 
\]
\noindent
�p�����[�^$\nu = 1�C2�C3�C5$ �̏ꍇ���O���t�ŕ\������.

\begin{minipage}{.35\textwidth}
\noindent
\(\pmb{\text{Plot}[}\\
\pmb{\{\text{PDF}[\text{ChiSquareDistribution}[1],x],}\\
\pmb{\text{PDF}[\text{ChiSquareDistribution}[2],x],}\\
\pmb{\text{PDF}[\text{ChiSquareDistribution}[3],x],}\\
\pmb{\text{PDF}[\text{ChiSquareDistribution}[5],x],\{x,0,20\}]}\)
\end{minipage}
\begin{minipage}{.1\textwidth}
    \hfill
\end{minipage}
\begin{minipage}{.55\textwidth}
\includegraphics[width=80mm]{�J�C2��̃O���t_gr1.eps}

\end{minipage}

\vspace{4ex}

\noindent
\begin{minipage}{.4\textwidth}
�p�����[�^�̒l��傫�����ăv���b�g�����

\noindent\(\pmb{\text{Plot}[}\\
\pmb{\text{  }\{\text{PDF}[\text{ChiSquareDistribution}[2],x],}\\
\pmb{\text{PDF}[\text{ChiSquareDistribution}[5],x],}\\
\pmb{\text{PDF}[\text{ChiSquareDistribution}[10],x],}\\
\pmb{\text{PDF}[\text{ChiSquareDistribution}[20],x],\},}\\
\pmb{\{x,0,50\}]}\)
\end{minipage}
\begin{minipage}{.1\textwidth}
    \hfill
\end{minipage}
\begin{minipage}{.4\textwidth}
\includegraphics{�J�C2��̃O���t_gr2.eps}
\end{minipage}

\section{���U�^���z�̂܂Ƃ�}
�ł��L���g���Ă��闣�U���z�ł���C���̖��x�C���ρC���U�C���̑��̊֘A�������v�Z���邱�Ƃ��ł���D���z���̂� name[$param_1$,$param_2$,$\cdot$]�Ƃ����L���`���ŕ\�����D���v���z�̓�����^����Mean�̂悤�Ȋ֐��́C�����Ƃ��ĕ��z�̋L���\�������D

\vspace{2ex}
\begin{tabular}{lp{24zw}}\hline
BernoulliDistribution[$p$]	
&
�����̊m�� $p$ �̎��s��1��s�����ʁA�x���k�[�C(Bernoulli)���z \\
BetaBinomialDistribution[$\alpha$,$beta$,$n$]	
&
�����m����BetaDistribution[$\alpha$,$\beta$]
�m���ϐ��ł���񍀕��z\\
BetaNegativeBinomialDistribution[$\alpha$,$\beta$,$n$]	
&
�����m����BetaDistribution[$\alpha$,$\beta$]�m���ϐ���
���镉�̓񍀕��z \\
BinomialDistribution[$n$,$p$]	
&
�e���s�ɂ����鐬������ $p$ �Ƃ����Ƃ��ɁC$n$ ��̎��s�̂����ŋN��
�����̉�\\
DiscreteUniformDistribution[\{$i_{min}$,$i_{max}$\}]
&
$i_{min}$ ���� $i_{max}$ �܂ł̐����l��ł̗��U��l���z\\
GeometricDistribution[p]&	�e���s�ɂ����鐬������ $p$ �Ƃ����Ƃ��́C�ŏ��̐����܂ł̎��s�񐔂̊􉽕��z\\
HypergeometricDistribution[$n$,$n_{succ}$,$n_{tot}$]
&
������ $n_{succ}$ ���܂ޑ傫�� $n_{tot}$ �̕�W�c����
�W�{�� $n$ �����o�����Ƃ��̐������̒��􉽕��z\\
LogSeriesDistribution[$\theta$]
&
�p�����[�^ $\theta$ �̑ΐ��������z\\
NegativeBinomialDistribution[$n$,$p$]&
	�p�����[�^ $n$ ����� $p$ �̕��̓񍀕��z\\
PoissonDistribution[$\mu$]&
	���� $\mu$ �̃|�A�\�� (Poisson) ���z\\
%ZipfDistribution[$\rho$]&
%	�p�����[�^ $\rho$ �̃W�b�v(Zipf) ���z\\
\hline
\end{tabular}

\vspace{2ex}
��ʓI�ȗ��U���z�̂قƂ�ǂ́C�u�����Ǝ��s�v�̂悤��2�ʂ�̌��ʂ��l�����鎎�s�̌J��Ԃ��ƍl���邱�Ƃŗ����ł���D

$\bullet$
�x���k�[�C���zBernoulliDistribution[$p$]�Ƃ́C�����i�l1�ɑΉ��j�� $p$ �̊m���ŋN��C���s�i�l0�ɑΉ��j��$1-p$ �̊m���ŋN��$1$��(�P��)���s�̊m�����z�ł���D

$\bullet$
�񍀕��zBinomialDistribution[$n,p$]�Ƃ́C�e���s�ɂ����鐬���m���� $p$ �Ƃ����Ƃ��ɁC$n$ ��̓Ɨ��������s�ŋN�鐬�����W�v�����񐔒l�̕��z�ł���D

$\bullet$
���̐��� $n$ �ɑ΂��镉�̓񍀕��zNegativeBinomialDistribution[$n,p$]�Ƃ́C�e���s�ɂ����鐬���m���� $p$ �Ƃ����Ƃ��ɁC$n$ �񐬌�����܂łɋN�������s�񐔂̕��z�ł���D���̕��z�͔C�ӂ̐��� $n$ �ɂ��Ē�`�����D�������C$n$ �������łȂ��Ƃ��́C$n$ ���������C$p$ �������m���Ƃ������߂͓��Ă͂܂�Ȃ��D

$\bullet$
�x�[�^�񍀕��zBetaBinomialDistribution[$\alpha$,$\beta$,$n$]�͓񍀕��z�ƃx�[�^���z��g�ݍ��������̂ł���DBetaBinomialDistribution[$\alpha$,$\beta$,$n$]�̊m���ϐ��́C�����m�� $p$ ���̂��m���ϐ���
�x�[�^���zBetaDistribution[$\alpha$,$\beta$]�ɏ]�����̂̂Ƃ��A
BinomialDistribution[$n,p$]�ɏ]���D�x�[�^���̓񍀕��zBetaNegativeBinomialDistribution[$\alpha$,$\beta$,$n$]�̓x�[�^���z�ƕ��̓񍀕��z��g�ݍ��������̂ł���D
$p$�̒l�����m�̂��̂Ƃ���̂ł͂Ȃ��A�ϓ�������̂ƍl����B

$\bullet$
�􉽕��zGeometricDistribution[$p$]�Ƃ́C�e���s�̐������� $p$ �Ƃ����Ƃ��ɁC�ŏ��ɐ������N��܂łɗv�������s���̕��z�̂��Ƃł���D

$\bullet$
���􉽕��zHypergeometricDistribution[$n,n_{succ}],n_{tot}]$]�� $n$ ��̎��s���C�����\�� $n_{succ}$ ��ŃT�C�Y $n_{tot}$ �̕�W�c����
���ɖ߂����ɃT���v�����O����i�񕜌����o�j���Ƃ�
������������ŁC�񍀕��z�ɑ��g�p�������̂ł���D

$\bullet$
���U��l���zDiscreteUniformDistribution[$i_{min},i_{max}$]�Ƃ́C
�����l�̊m���i�������m���j�ŋN�錋�ʂ� $i_min$ ���� 
$i_{max}$ �܂ł̐����ŕ\�������s��\�����̂ł���D

$\bullet$
�|�A�\�����zPoissonDistribution[$\mu$]�Ƃ́C�w�肳�ꂽ���ԓ��ɋN�鎖�ې����L�q����B������ $\mu$ �͈�莞�Ԃɑ΂��鎖�ۂ̕��ω񐔂ł���B2�����z��
�Ɍ��Ƃ��ē�����A�H�Ȍ��ۂ��L�q����Ƃ��ɗp�����A�d�v�ȕ��z�ł���B

$\bullet$
$\theta=0$�t�߂�$\log (1-\theta)$�̋����W�J�ɂ����鍀�́C�ΐ����zLogSeriesDistribution[$\theta$]�ɏ]�������U�m���ϐ��̊m���ɔ�Ⴗ��B�w�肳�ꂽ���ԓ��ɍw������鐻�i���̕��z���C���̕��z�ɂ���ă��f��������邱�Ƃ�����B

%$\bullet$
%�W�b�v(Zipf)���zZipfDistribution[$\rho$]�̓[�[�^���z�Ƃ�΂��B
%����͍ŏ�����w�Ŏg���C���������ۂ̃��f�����ɂ܂Ŋg�����ꂽ�Ƃ����B

\vspace{2ex}
��ʓI�Ȗ��x�A���z�A���ρA���U�Ȃǂ����߂閽�ߌ`�F

\begin{tabular}{lp{24zw}}\hline
PDF[dist,$x$]&	���z��dist�̕ϐ�$x$�ɂ�����m�����x�֐�
                (probabilisty density function)\\
CDF[dist,$x$]&	$x$ �ɂ�����ݐϕ��z�֐�(cumulative distribution dunction)�A\\
InverseCDF[dist,$q$]&	CDF[dist,$x$]���ő�� $q$ �ƂȂ�悤�ȍő吮�� $x$\\
Quantile[dist,q]&	�� $q$ ���ʐ�\\
Mean[dist]	&����(mean)\\
Variance[dist]	&���U(variance)\\
StandardDeviation[dist]	& �W���΍�(standard deviation)\\
Skewness[dist]	&�c�x�W��(skewness)\\
Kurtosis[dist]	&��x�W��(kurtosis)\\
CharacteristicFunction[dist,$t$] &�����֐� $\phi(t)= E[e^{iXt}]$,
                 $X\sim dist$ \\
ExpectedValue[$f$,dist]	& dist ��(��)�֐� $f$ �̊��Ғl\\
ExpectedValue[$f[x]$,dist,$x$]	&
  dist �̕ϐ� $x$ �ɑ΂��� �֐�$f[x]$�̊��Ғl\\
Median[dist]	&�����l\\
Quartiles[dist]	& 
dist �ɑ΂���1/4�C1/2�C3/4�Ԗڂ̎l���ʐ�(quartile)\\
InterquartileRange[dist]&
    ��1�l���ʐ�(first)�Ƒ�3�l���ʐ�(third)�Ƃ̍���\\
QuartileDeviation[dist]	&�l���ʔ͈͂̔����l\\
QuartileSkewness[dist]	&�l���ʐ��x�[�X�̘c�x\\
RandomInteger[dist]	&�w�肳�ꂽ���z�̋[������\\
RandomInteger[dist,dims]&
������ dims �ŁC�w�肳�ꂽ���z�̗v�f�����[�������z��\\
\hline
\end{tabular}


\section{�A���^���z�̂܂Ƃ�}
�����ɋL�ڂ���Ă���֐��́C�A�����z�ōł���ʓI�Ɏg�p�������̂ł���C�A�����z�̖��x�C���ρC���U�C���̑��̊֘A�������v�Z���邱�Ƃ��ł���D���z�� 
name[$param_1$],$param_2$,$\ldots$]�Ƃ����L���`���ŕ\�������D
���v���z�̓�����^����Mean�̂悤�Ȋ֐��́C�����Ƃ��ĕ��z�̋L���\�������D�u���U���z�v�͑����̗��U���z�ɂ��ċL�q����D

\vspace{2ex}

\begin{tabular}{lp{24zw}} \hline
UniformDistribution[$\{\min,\max\}$]	&���$(\min,\max)$�ɂ������l���z\\
NormalDistribution[$\mu$,$\sigma$]&	���� $\mu$�C�W���΍� $\sigma$
(���U$\sigma^2$) �̐��K�i�K�E�X�j���z\\
HalfNormalDistribution[$\theta$]&	�ړx���p�����[�^ $\theta$ ��
����Ⴗ�锼���K���z\\
LogNormalDistribution[$\mu$,$\sigma$]&	���� $\mu$�C�W���΍� $\sigma$
(���U$\sigma^2$) �̐��K���z�Ɋ�Â����ΐ����K���z\\
InverseGaussianDistribution[$\mu$,$\lambda$]
&	
���� $\mu$�C�ړx $\lambda$ �̋t�K�E�X���z\\
\hline
\end{tabular}

\vspace{2ex}

$\bullet$
��l���zUniformDistribution[{min,max}]��, ��`���z�ƌĂ΂�邱�Ƃ�����C�l���ǂ��ɂł�����������悤�Ȋm���ϐ�������t����D��l���z�̊m���ϐ��̗�ɂ́Cmin ���� max �܂ł̒�����Ń����_���ɑI�񂾓_�̏ꏊ������D

$\bullet$
�ΐ����K���zLogNormalDistribution[$\mu$,$\sigma$]��
���K���z�Ɋ�Â��m���ϐ��̎w���֐����]�����z�ł���D

���̕��z�͑����̓Ɨ��m���ϐ�����@�I�ɓ������ꂽ�ꍇ�ɋN��D

�����K���zHalfNormalDistribution[$\theta$]�͗̈�[$0,\infty$])��
����ꂽ���zNormalDistribution[$0,1/(\theta \sqrt{2/\pi})$]�ɔ�Ⴗ��

$\bullet$
�t�K�E�X���zInverseGaussianDistribution[$\mu,\lambda$]��
�����h(Wald)���z�Ƃ��Ă΂�C���̃h���t�g�t���u���E���^����
�����ʉߎ��Ԃ̕��z�ł���D

\vspace{2ex}

\begin{tabular}{lp{24zw}} \hline
ChiSquareDistribution[$\nu$] & 
���R�x $\nu$ �� ${\chi}^2$���z\\
InverseChiSquareDistribution[$\nu$] &
���R�x $\nu$ �̋t ${\chi}^2$���z\\
FRatioDistribution[n,m]	&
���q�̎��R�x�� n�C����̎��R�x�� m �ł��� F ���z\\
StudentTDistribution[$\nu$] & 
���R�x�� $\nu$ �̃X�`���[�f���g�� t ���z\\
NoncentralChiSquareDistribution[$\nu$,$\lambda$] &
���R�x�� $\nu$�C��S�p�����[�^�� $\lambda$ �̔�S ${\chi}^2$���z\\
NoncentralStudentTDistribution[$\nu$,$\delta$] &
���R�x�� $\nu$�C��S�p�����[�^�� $\delta$ �̔�S�X�`���[�f���g�� t ���z\\
NoncentralFRatioDistribution[n,m,$\lambda$] &
���q�̎��R�x�� n�C����̎��R�x�� m�C��S�p�����[�^�� $\lambda$ 
�ł����S F ���z \\
\hline
\end{tabular}

\vspace{2ex}
���K���z�ɏ]���W�{�Ɋ֘A���镪�z�F

$\bullet$
$X_1,\cdots,X_{\nu}$���P�ʕ��U�Ɨ땽�ς����Ɨ�����
���K�m���ϐ��Ȃ�΁C\(\sum_{i = 1}^{\nu} X_i^2\)
�͎��R�x $\nu$ �� $\chi^2$���z�ɂ��������D
���K�ϐ����炻�̕��ς������W���΍��Ŋ��邱�Ƃɂ��W���������ƁC���̐��ʂ̕����a�͂��̕��z�ɏ]���D$\chi^2$���z�͐��K���z����W�{�̕��U���L�q����ꍇ�ɍł�
�悭�p������D

$\bullet$
���� $Y$ �����R�x $\nu$ �� $\chi^2$���z�ɏ]���Ȃ�΁C$1/Y$ �͋t $\chi^2$���z
InverseChiSquareDistribution[$\nu$]�ɏ]���D���R�x�� $\nu$ �ŃX�P�[���� $\xi$ �̃X�P�[�����ꂽ�t $\chi^2$ ���z��
InverseChiSquareDistribution[$\nu$,$\xi$]�Ƃ��ė^���邱�Ƃ��ł���D�t $\chi^2$���z�́C���K���z�ɏ]���W�{�̃x�C�Y���͂ɂ����镪�U�̎��O���z�Ƃ��čL���g���Ă���D

$\bullet$
�X�`���[�f���g�� t ���z�ɂȂ�ϐ��́C���K�m���ϐ��̊֐��Ƃ��ċL�q���邱�Ƃ��ł���DX �� Z ���Ɨ������m���ϐ��ł���Ƃ���D������ X �͕W�����K���z�CZ �͎��R�x 
$\nu$ �� $\chi^2$�ϐ��ł���D���̏ꍇ�C$X/\sqrt{Z/\nu}$ �͎��R�x
 $\nu$ �� t ���z�����D�X�`���[�f���g��t ���z�͏c���ɂ��đΏ̂ł���C���K�ϐ��̂��̕W���΍��ɑ΂��銄��������t����D$\nu=1$�Ȃ�΁Ct ���z�̓R�[�V�[���z��
��v����D

$\bullet$
F ���z��2�̓Ɨ����� $\chi^2$�ϐ������ꂼ��̎��R�x�Ŋ������Ƃ��̔�̕��z�ł���D����͉��茟���2�̕�W�c�̕��U���r����Ƃ��ɍL���g����D

���ς����̐��K���z����h�o����镪�z�́u��S���z�v�ƌĂ΂��D

$\bullet$
��S�X�`���[�f���g�� t ���z: NoncentralStudentTDistribution[$\nu$,$\delta$]�́C
�� $X/\sqrt{\chi_\nu^2/\nu}$���L�q����D������ $\chi_\nu^2$ �͎��R�x $\nu$ ��
 $\chi^2$�m���ϐ��CX �͕��� $\delta$, ���U $\sigma^2=1$ �̐��K���z��
�Ɨ������m���ϐ��ł���D

$\bullet$
��S F ���z: NoncentralFRatioDistribution[n,m,$\lambda$]�� 
$1/m \chi_m^2$ �ɑ΂��� $1/n \chi_n^2$ �̔�̕��z�ł���D
������ $ \chi_n^2(\lambda)$�͔�S�p�����[�^ $\lambda$�C
���R�x $n$�̔�S $\chi^2$�m���ϐ��ł���C $ \chi_m^2(\lambda)$�͎��R�x $m$ �� 
$\chi^2$ �m���ϐ��ł���D

\vspace{2ex}

\begin{tabular}{ll}\hline
BetaDistribution[$\alpha,\beta$]&	
�`��p�����[�^�� $\alpha$ �� $\beta$ �̘A���x�[�^���z \\
CauchyDistribution[a,b]&
�ʒu�p�����[�^�� a�C�ړx�p�����[�^�� b �̃R�[�V�[���z \\
ChiDistribution[$\nu$]&	
���R�x $\nu$ �� $\chi$ ���z \\
ExponentialDistribution[$\lambda$]&	
�ړx���p�����[�^ $\lambda$ �ɔ���Ⴗ��w�����z\\
ExtremeValueDistribution[$\alpha,\beta$]&	
�ʒu�p�����[�^ $\alpha$ �C�ړx�p�����[�^ $\beta$ �̋ɒl(Fisher-Tippett)���z\\
GammaDistribution[$\alpha,\beta$]&	
�`��p�����[�^ $\alpha$�C�ړx�p�����[�^ $\lambda$ �̃K���}���z\\
GumbelDistribution[$\alpha,\beta$]&	
�ʒu�p�����[�^ $\alpha$�C�ړx�p�����[�^ $\beta$ �̃K���x���̍ŏ��ɒl���z\\
InverseGammaDistribution[$\alpha,\beta$]&	
�`��p�����[�^ $\alpha$�C�ړx�p�����[�^ $\lambda$ �̋t�K���}���z\\
LaplaceDistribution[$\mu$,$\beta$]&	
���� $\mu$�C�ړx�p�����[�^ $\beta$ �̃��v���X�i��d�w���j���z\\
LevyDistribution[$\mu,\sigma$]&	
�ʒu�p�����[�^ $\mu$�C�g�U�p�����[�^ $\sigma$ �̃����B���z\\
LogisticDistribution[$\mu$,$\beta$]&	
���� $\mu$�C�ړx�p�����[�^ $\beta$ �̃��W�X�e�B�b�N���z\\
MaxwellDistribution[$\sigma$]&	
�ړx�p�����[�^ $\sigma$ �̃}�b�N�X�E�F���i�}�b�N�X�E�F���E�{���c�}���j���z\\
ParetoDistribution[k,$\beta$]&	
�ŏ��l�p�����[�^ k �C�`��p�����[�^ $\alpha$ �̃p���[�g���z\\
RayleighDistribution[$\sigma$]&	
�ړx�p�����[�^ $\sigma$ �̃��C���[(Rayleigh)���z\\
WeibullDistribution[$\alpha$,$\beta$]&	
�`��p�����[�^ $\alpha$ �C�ړx�p�����[�^ $\beta$ �̃��C�u��(Weibull)���z\\
\hline
\end{tabular}

\vspace{2ex}
$\bullet$
$X$ ��$[-\pi, \pi]$��ň�l�ɕ��z���Ă���Ȃ�C�m���ϐ�$\tan(X)$��$a=0$�C$b=1$
�R�[�V�[ (Cauchy) ���z: CauchyDistribution[a,b]�ɏ]���D

$\bullet$
$\alpha=n/2$�� $\lambda=2$�̂Ƃ��C
�K���}���z: GammaDistribution[$\alpha$,$\lambda$]�� n ���j�b�g��
���K�m���ϐ��̕����a�ɕ��z���L�q����D���̌`���̃K���}���z�͎��R�x $\nu$ �� 
$\chi^2$���z�ƌĂ΂��D$\alpha=1$�̂Ƃ��C�K���}���z�́C�C�x���g�Ԃ̑҂����Ԃ�
�L�q����̂ɂ悭�g����w�����zExponentialDistribution[$\lambda$]�̌`�����D

$\bullet$
�����m���ϐ� $X$ ���K���}���z: GammaDistribution[$\alpha,\beta$]�ɏ]���Ȃ�΁C
$1/X$ �͋t�K���}���z: InverseGammaDistribution[$\alpha$,$1/\beta$]�ɏ]���D
�m���ϐ� X ��InverseGammaDistribution[$1/2$,$\sigma/2$]�ɏ]���Ȃ�΁C$X+\mu$ �̓����B���zLevyDistribution[$\mu$,$\sigma$]�ɏ]���D

$\bullet$
$X_1$�� $X_2$���C�ړx�p�����[�^���������ʁX�̃K���}���z�ł���Ƃ��C
�m���ϐ�$X_1/(X_1 + X_2)$�̓x�[�^���z:
BetaDistribution[$\alpha$,$\beta$]�ɏ]���D

$\bullet$
$\chi$ ���z: ChiDistribution[$\nu$]�́C$\chi^2$ �m���ϐ��̕��������]�����z�ł���D$n=1$�̂Ƃ��́C$\chi$ ���z��$\theta=\sqrt{\pi/2}$��HalfNormalDistribution[$\theta$]�ɓ������D$n=2$�̂Ƃ��́C$\chi$ ���z�� $\sigma=1$�̃��C���[ (Rayleigh) 
���z: RayleighDistribution[$\sigma$]�ɓ������D$n=3$�̂Ƃ��� $\chi$ ���z��
$\sigma=1$�̃}�b�N�X�E�F���E�{���c�}��(Maxwell-Boltzmann)���z: 
MaxwellDistribution[$\sigma$]�ɓ������D

$\bullet$
���v���X (Laplace) ���z: LaplaceDistribution[$\mu$,$\beta$] �́C����̎w�����z������2�̓Ɨ������m���ϐ��̍��̕��z�ł���D���W�X�e�B�b�N���z: LogisticDistribution[$\mu$,$\beta$]�͐��̒������z��]�ޏꍇ�ɁC���K���z�̑��ɕp�ɂɎg����D

$\bullet$
�p���[�g���z (Pareto) ���z: ParetoDistribution[$k$,$\alpha$]�͎�����\�����߂Ɏg����D$k$ �͍ŏ��̎����������D

$\bullet$
���C�u�� (Weibull) ���z: WeibullDistribution[$\alpha$,$\beta$]�́C���̂̎������L�q���邽�߂ɍH�w�ōL���g���Ă���D�ɒl���z: ExtremeValueDistribution[$\alpha$,$\beta$]�́C���K���z���܂ޑ��l�ȕ��z���璊�o���ꂽ�傫���W�{�ɂ�����ő�l�ɑ΂���Ɍ����z�ł���D���̂悤�ȕW�{�̍ŏ��l�ɑ΂���Ɍ����z�́C
�K���x��(Gumbel)���z: GumbelDistribution[$\alpha$,$\beta$]�ł���D�ő�ɒl�ƍŏ��ɒl�́C�ϐ��̐��`�ω��ɂ��֘A���Ă���̂ŁC�ɒl���z�ƃK���x�����z�͓����Ӗ��Ŏg���邱�Ƃ�����D�܂��ɒl���z�����m���ϐ��ƁC�K�؂ɃV�t�g�E�X�P�[�����ꂽ���C�u�����z�����m���ϐ��Ƃ̊Ԃɂ͑ΐ��֌W������̂ŁC�ɒl���z�͑ΐ����C�u�����z�ƌĂ΂�邱�Ƃ�����D

\end{document}


\section{�O���t�̃v���b�g�p

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\end{minipage}
\begin{minipage}{.4\textwidth}
   \includegraphics{yyy_gr2.eps}
\end{minipage}

\end{document}
