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\begin{document}

\title{���U�^�m�����z�̐����\�\�\�p�����[�^�ω��Ƃ̊֘A}
\author{}
\date{}
\maketitle


\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}

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



\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
\[ 
\text{Binom}[1,p] \,\, \sim \,\,
\binom{1}{x}p^x (1-p)^{1-x}
=\left\{ \begin{array}{ll} p & x=1 \\ 1-p & x=0 \end{array} \right.
\]

\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�x���k�[�C���z�̘a�ƂȂ��Ă���B\\
\[ 
\text{Binom}[n,p] \,\, \sim \,\,
\binom{n}{x}p^x (1-p)^{n-x}, x=0,1,2,\cdots,n
\]

\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�񐔂̕��z�ł���D
�􉽕��z�̘a�ƍl����΂悢�B
�͂��߂Đ������N����܂ł̑҂����Ԃ��􉽕��z�ŁA�����$n$�񐬌�����
�܂ł̉񐔒l�̕��z�ł���B$q=1-p$
\[ 
\text{NegBinom}[n,p] \,\, \sim \,\, 
\binom{-n}{x}p^n (-q))^{x}
=\binom{n +x -1}{x}p^n q^{x} , x=0,1,2,\cdots,
\]




\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
\[ 
\text{BetaBinom}[\alpha,\beta,n] \,\, \sim \,\, 
\binom{n}{x} \dfrac{B(x+\alpha, n + \beta-x)}{B(\alpha, \beta)}, x=0,1,2,\cdots,n
\]



\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
\[
\text{BetaNegBinom}(\alpha,\beta,n) \, \, \sim \, \,
 \binom{-1 + n + x}{ -1 + n}\dfrac{B[\alpha + n, \beta + x]}{B[\alpha, \beta]},
 \, x=0,1,2,\cdots,n
\]


\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
\[
   \text{Geom}(p) \, \, \sim \, \, p q^x, \, x=0,1,2,\cdots
\]

\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$n_{tot}= n_{succ} + n_{fail}$,
$\dfrac{n_{succ}}{n_{tot}} \to p, \dfrac{n_{fail}}{n_{tot}} \to q $
�Ȃ�΁A$\text{HyperGeom}[n,n_{succ},n_{tot}] \doteqdot \text{Binom}[n,p]$
�ƂȂ�Ή�������B
\[ 
\text{HyperGeom}[n,n_{succ},n_{tot}] \, \, \sim \,\,
  \dfrac{\binom{n_{succ}}{x} \binom{n_{fail}}{n-x}}{\binom{n_{tot}}{n}}
= \binom{n}{x} 
   \dfrac{(n_{succ})_x \, (n_{fail})_{n-x}}{(n_{tot})_n}, \, x=0,1,2,\cdots,n
\]



\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�̋Ɍ��Ƃ��ē�����A�H�Ȍ��ۂ��L�q����Ƃ��ɗp�����A
�d�v�ȕ��z�ł���B�w�����o�Ă��闝�R�� $\lim_n(1+p_n)^n = \lambda$, 
�������A$n p_n \to \lambda (n \to \infty)$����ł���B
\[\text{Po}[\lambda] \,\, \sim \,\,
  e^{- \lambda} \dfrac{\lambda^x}{x!}, \, x= 0,1,2,\cdots
\]


\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}

\vspace{4ex}
\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}

\vspace{4ex}

\begin{figure}[htbp]
 \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}
 \begin{minipage}{0.5\hsize}
  �􉽕��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}
\end{figure}

\vspace{2ex}

\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�����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}


\vspace{2ex}

\begin{figure}[htbp]
 \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}
 \begin{minipage}{0.5\hsize}
  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}
\end{figure}

\vspace{2ex}


\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}
  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}

\vspace{2ex}


\begin{figure}[htbp]
 \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}
 \begin{minipage}{0.5\hsize}
  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}
\end{figure}

\vspace{2ex}


\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr6.eps}
   \caption{���̊􉽕��z NegBinom($8$,$p$), $p$�̕ω�}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  ���̊􉽕��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}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr7.eps}
   \caption{���̊􉽕��z NegBinom($n$,$0.3$), $n$�̕ω�}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
  ���̊􉽕��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}
\end{figure}



(���􉽕��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
\(
  \left\{\dfrac{1}{252} 
   \binom{2}{k} \binom{8}{5-k},
   \dfrac{1}{252} \binom{5}{5-k} \binom{5}{k},
   \dfrac{1}{252} \binom{3}{5-k} \binom{7}{k}
  \right\}
\)

\vspace{2ex}


\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}
  ���􉽕��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}


���􉽕��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
\(
 \left\{
  \dfrac{\binom{4}{k} \binom{16}{15-k}}{15504},
  \dfrac{\binom{10}{15-k} \binom{10}{k}}{15504},
  \dfrac{\binom{6}{15-k} \binom{14}{k}}{15504}
 \right\}
\)

\vspace{2ex}


\begin{figure}[htbp]
 \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}
 \begin{minipage}{0.5\hsize}
   ���􉽕��z(HypergeometricDistribution[$15$,$n_{succ}$,$n_{tot}$])
   $\dfrac{n_s}{n_t} = \dfrac{4}{20}, 
   \dfrac{10}{20}, \dfrac{14}{20}\,  (n < n_t)$
  \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}
\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}
   ���􉽕��z(HypergeometricDistribution[$15$,$n_{succ}$,$n_{tot}$])
   $\dfrac{n_s}{n_t} = \dfrac{40}{200}, 
   \dfrac{100}{200}, \dfrac{140}{200}\,  (n < n_t)$
  \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}
  \begin{center}
   \includegraphics[width=70mm]{kakubun01_gr11.eps}
   \caption{2�����z Binom[$n$,$p$}
  \end{center}
 \end{minipage}
 \begin{minipage}{0.5\hsize}
   ���܂܂ł̒��􉽕��z�͂Q�����z�ɋ߂Â��󋵂��������߂ɁA
   $n=15$, �ł�
   2�����z(BinomialDistribution[$15$,$p$,]
   $p= \dfrac{n_s}{n_t} = \dfrac{40}{200}, 
   \dfrac{100}{200}, \dfrac{140}{200}$ ��`���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}
\end{figure}


\textbf{�x�[�^���z:}
BetaDistribution[$\alpha$,$\beta$]\\
$\alpha$, $\beta$ �͐��̎����ł悢�B�ɂ߂čL�͈͂�
�`��ƂȂ镪�z�`�B$\alpha = \beta=1$�ł͘A���^��l���z�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}
   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���̂悢�Z�핪�Ƃ��Ă̐�����
�����܂��B�i�P�j�������[���X�A�i�Q�j���ς╪�U
�i�R�j�����t�����Ғl�̊֌W���Ȃǂ��������܂��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}

\noindent\(\pmb{\text{�􉽕��z�Ǝw�����z}}\\
\pmb{\text{GeometricDistribution}[p], 0<p <1}\\
\pmb{\text{ExponentialDistribution}[\text{lm}], \text{lm} >0}\)

\noindent\(\pmb{\text{�􉽕��z�̃p�����[�^�ω�}}\)
\noindent\(\pmb{\text{�􉽕��z�̊m�����x�F}}\)
\noindent\(\pmb{\text{PDF}[\text{GeometricDistribution}[p],k]}\)
\noindent\((1-p)^k p\)
\noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
\pmb{\text{PDF}[\text{GeometricDistribution}[p],k]\},}\\
\pmb{\{p,\{0.1,0.3,0.5,0.8\}\},\{k,0,15\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \{\{0,15\},\{0,0.8\}\} ]}\)


\noindent\(\pmb{\text{�w�����z�̃p�����[�^�ω�}}
\pmb{\text{�w�����z�̊m�����x�F}}\)
\noindent\(\pmb{\text{PDF}[\text{ExponentialDistribution}[\text{lm}],x]}\)
\noindent\(e^{-\text{lm} x} \text{lm}\)
\noindent\(\pmb{\text{Plot}[}\\
\pmb{\{\text{PDF}[\text{ExponentialDistribution}[0.3],x],}\\
\pmb{\text{PDF}[\text{ExponentialDistribution}[0.5],x],}\\
\pmb{\text{PDF}[\text{ExponentialDistribution}[0.8],x]\},}\\
\pmb{\{x,0,15\},\text{PlotRange} \to  \{\{0,15\},\{0,0.8\}\}]}\)


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

\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{document}



