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\usepackage{amsmath, amssymb, graphics}

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\begin{document}

\title{}
\author{���U�^���z}
\date{}
\maketitle

\noindent\(\pmb{}\\
\pmb{\text{�x���k�[�C���z}}\\
\pmb{\text{BernoulliDistribution}[\text{$\$$p$\$$}]}\\
\pmb{\text{�Ƃ́C����}�i\text{�l1�ɑΉ�}�j�� \text{$\$$p$\$$} \text{�̊m���ŋN��C���s}�i\text{�l0�ɑΉ�}�j\text{��$\$$1}-}\\
\pmb{\text{p$\$$} \text{�̊m���ŋN��$\$$1$\$$��}(\text{�P��})\text{���s�̊m�����z�ł���D}}\\
\pmb{}\\
\pmb{\text{�񍀕��z}}\\
\pmb{\text{BinomialDistribution}[\text{$\$$n},\text{p$\$$}]}\\
\pmb{\text{�Ƃ́C�e���s�ɂ����鐬���m����} \text{$\$$p$\$$} \text{�Ƃ����Ƃ��ɁC$\$$n$\$$} }\\
\pmb{\text{��̓Ɨ��������s�ŋN�鐬�����W�v�����񐔒l�̕��z�ł���D�x���k�[�C���z�̘a�ƂȂ��Ă���B}}\\
\pmb{}\\
\pmb{\text{���̓񍀕��z}}\\
\pmb{\text{NegativeBinomialDistribution}[\text{$\$$n},\text{p$\$$}]}\\
\pmb{\text{���̐���} \text{$\$$n$\$$} \text{�ɑ΂��镉�̓񍀕��zNegativeBinomialDistribution}[\text{$\$$n},\text{p$\$$}]}\\
\pmb{\text{�Ƃ́C�e���s�ɂ����鐬���m����} \text{$\$$p$\$$} \text{�Ƃ����Ƃ��ɁC$\$$n$\$$} }\\
\pmb{\text{��̐�������܂łɋN�������s�񐔂̕��z�ł���D�􉽕��z�̘a�ƍl����΂悢�B}}\\
\pmb{\text{�͂��߂Đ������N����܂ł̑҂����Ԃ��􉽕��z�ŁA�����$\$$n$\$$�񐬌�����܂ł�}}\\
\pmb{\text{�񐔒l�̕��z�ł���B}}\\
\pmb{}\\
\pmb{\text{�x�[�^�񍀕��z}}\\
\pmb{\text{BetaBinomialDistribution}[\$\backslash \text{alpha$\$$},\$\backslash \text{beta$\$$},\text{$\$$n$\$$}]}\\
\pmb{\text{�͓񍀕��z�ƃx�[�^���z��g�ݍ��������̂ł���D���̊m���ϐ��́C�����m��} \text{$\$$p$\$$} }\\
\pmb{\text{���̂��m���ϐ��Ńx�[�^���zBetaDistribution}[\$\backslash \text{alpha$\$$},\$\backslash \text{beta$\$$}]}\\
\pmb{\text{�ɏ]�����̂̂Ƃ��ABinomialDistribution}[\text{$\$$n},\text{p$\$$}]\text{�ɏ]���D}}\\
\pmb{}\\
\pmb{}\\
\pmb{\text{�x�[�^���̓񍀕��z}}\\
\pmb{\text{BetaNegativeBinomialDistribution}[\backslash \text{alpha},\backslash \text{beta},n]}\\
\pmb{\text{�̓x�[�^���z�ƕ��̓񍀕��z��g�ݍ��������̂ł���D$\$$p$\$$�̒l�����m�̂��̂Ƃ���̂ł͂Ȃ��A}}\\
\pmb{\text{�ϓ�������̂ƍl����B}}\\
\pmb{}\\
\pmb{\text{�􉽕��z}}\\
\pmb{\text{GeometricDistribution}[p]}\\
\pmb{\text{�Ƃ́C�e���s�̐�������} \text{$\$$p$\$$�Ƃ����Ƃ��ɁC�ŏ��ɐ������N��܂łɕK�v�Ƃ������s����}}\\
\pmb{\text{���z�ł���D}}\\
\pmb{}\\
\pmb{\text{���􉽕��z}}\\
\pmb{\text{HypergeometricDistribution}[n,\text{n$\_$}\{\text{succ}\},\text{n$\_$}\{\text{tot}\}]}\\
\pmb{\text{    }�� \text{$\$$n$\$$} \text{��̎��s�ɂ����āC}}\\
\pmb{\text{�����̌�}�i\text{�T�C�Y}�j \text{$\$$n$\_$}\{\text{tot}\} \$ \text{�̕�W�c�Ƃ��āA�ŏ���$\$$n$\_$}\{\text{tot}\}}\\
\pmb{\text{$\$$�̖����̂���������A���̓��̓����肭����}(\text{�����Ƃ����}) \text{$\$$n$\_$}\{\text{succ}\}\$ }\\
\pmb{\text{��ƊO�ꂭ����}�i\text{���s�Ƃ����}�j\text{$\$$n$\_$}\{\text{fail}\}=}\\
\pmb{\text{n$\_$}\{\text{tot}\}-}\\
\pmb{\text{n$\_$}\{\text{succ}\}\text{$\$$�ō\������Ă���B��x���o���������͔j������2�x�Ǝ��o����邱�Ƃ͂Ȃ��B}}\\
\pmb{\text{���Ȃ킿�A���ɖ߂����ɃT���v�����O����}�i\text{�񕜌����o}�j�B}\\
\pmb{\text{�͂��߂ɂ́A����$\$$n$\_$}\{\text{tot}\}\text{$\$$�̂����������萔$\$$n$\_$}\{\text{succ}\}\text{$\$$�ƊO�ꐔ}}\\
\pmb{\text{$\$$n$\_$}\{\text{fail}\}=\text{n$\_$}\{\text{tot}\}-\text{n$\_$}\{\text{succ}\}\text{$\$$�ł���A}}\\
\pmb{\text{�񕜌����o�������s���B���􉽕��z�Ƃ́A�������������݂���$\$$n$\$$��̂�����}}\\
\pmb{\text{�����肭�����������񐔂�\���B���̂���Ԃ����s�͓Ɨ��ł͂Ȃ����Ƃɒ��ӁB}}\\
\pmb{\text{�񍀕��z�̓x���k�[�C���z�̂���Ԃ����s�œƗ��ł���ƂƂɈႢ������B}}\\
\pmb{\text{�����肭���������m��}�i\text{�����肭���ƊO�ꂭ���̔䗦}�j��}\\
\pmb{\text{���}�i\text{�������o}�j\text{�̏ꍇ�ł���B}}\\
\pmb{}\\
\pmb{\text{���U��l���z}}\\
\pmb{\text{DiscreteUniformDistribution}[\text{$\$$i$\_$}\{\min \}\$,\text{$\$$i$\_$}\{\max \}\$]}\\
\pmb{\text{�Ƃ́C�����l�̊m��}�i\text{�������m��}�j\text{�ŋN�錋�ʂ�} \text{$\$$i$\_$min$\$$} \text{����}}\\
\pmb{\text{$\$$i$\_$}\{\max \}\$ \text{�܂ł̐����ŕ\�������s��\�����̂ł���D}}\\
\pmb{}\\
\pmb{\text{�|�A�\�����z}}\\
\pmb{\text{PoissonDistribution}[\$\backslash \text{mu$\$$}]\text{�Ƃ́C�w�肳�ꂽ���ԓ��ɋN�鎖�ې����L�q����D������} }\\
\pmb{\$\backslash \text{mu$\$$} \text{�͈�莞�Ԃɑ΂��鎖�ۂ̕��ω񐔂ł���D2�����z��}}\\
\pmb{\text{�Ɍ��Ƃ��ē�����A�H�Ȍ��ۂ��L�q����Ƃ��ɗp�����A�d�v�ȕ��z�ł���B}}\\
\pmb{}\\
\pmb{\text{�ΐ����z}}\\
\pmb{\text{LogSeriesDistribution}[\$\backslash \text{theta$\$$}]}\\
\pmb{\$\backslash \text{theta}=0\text{$\$$�t�߂�$\$$}\backslash \log  (1-\backslash \text{theta})}\\
\pmb{\text{$\$$�̋����W�J�ɂ����鍀�́C�ΐ����zLogSeriesDistribution}[\$\backslash \text{theta$\$$}]}\\
\pmb{\text{�ɏ]�������U�m���ϐ��̊m���ɔ�Ⴗ��D�w�肳�ꂽ���ԓ��ɍw������鐻�i���̕��z���C��}}\\
\pmb{\text{�̕��z�ɂ���ă��f��������邱�Ƃ�����D}}\\
\pmb{}\\
\pmb{\text{�W�b�v}(\text{Zipf})\text{���z}}\\
\pmb{\text{ZipfDistribution}[\$\text{$\backslash $r}\text{ho$\$$}]}\\
\pmb{\text{�̓[�[�^���z�ƌ����邱�Ƃ�����D����͍ŏ�����w�Ŏg���C���������ۂ̃��f�����ɂ܂�}}\\
\pmb{\text{�g�����ꂽ}}\)

\noindent\(\pmb{}\)

���x�֐��̊֐��`��\�������閽�߁F\\
PDF[���z��[�p�����[�^],�ϐ�],\\
$\{$�ϐ��A�ŏ��l�A�ő�l$\}$\\
\\
�O���t��`�����߁F\\
Plot[PDF[���z��[�p�����[�^],�ϐ�],\\
$\{$�ϐ��A�ŏ��l�A�ő�l$\}$]\\
\\
ListPlot[Table[$\{$k, PDF[BinomialDistribution[50, 0.3], k]$\}$, $\{$k, 0, 50$\}$], Filling -$>$ Axis]\\


\noindent\(\pmb{\text{�􉽕��z} \text{Geom}(p) \text{�̃p�����[�^}(p=0.1,0.3,0.5,0.8)\text{�ω�}}\)

\noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
\pmb{\text{PDF}[\text{GeometricDistribution}[p],k]\},}\\
\pmb{\{p,\{0.1,0.3,0.5,0.8\}\},\{k,0,15\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \{\{0,15\},\{0,0.8\}\} ]}\)

\includegraphics{kakubun01_gr1.eps}

\noindent\(\pmb{2\text{�����z} \text{Binom}(n,p) �́@n=8\text{�ł�}=0.1,0.3,0.5,0.8\text{�̕ω�}}\)

\noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
\pmb{\text{PDF}[\text{BinomialDistribution}[10,p],k]\},}\\
\pmb{\{p,\{0.1,0.3,0.5,0.8\}\},\{k,0,10\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \text{Automatic} ]}\)

\includegraphics{kakubun01_gr2.eps}

\noindent\(\pmb{2\text{�����z} \text{Binom}(n,p) �́@p=1/2 �� n =5,10,20\text{�̕ω�}}\)

\noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
\pmb{\text{PDF}[\text{BinomialDistribution}[n,1/2],k]\},}\\
\pmb{\{n,\{5,10,20\}\},\{k,0,20\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \text{Automatic} ]}\)

\includegraphics{kakubun01_gr3.eps}

\noindent\(\pmb{2\text{�����z} \text{Binom}(n,p) �́@p=1/4 =0.25�� n =5,10,20\text{�̕ω�}}\)

\noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
\pmb{\text{PDF}[\text{BinomialDistribution}[n,1/4],k]\},}\\
\pmb{\{n,\{5,10,20\}\},\{k,0,20\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \text{Automatic} ]}\)

\includegraphics{kakubun01_gr4.eps}

\noindent\(\pmb{2\text{�����z} \text{Binom}(n,p) �́@p=3/4=0.75 �� n=5,10,20 \text{�̕ω�}}\)

\noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
\pmb{\text{PDF}[\text{BinomialDistribution}[n,3/4],k]\},}\\
\pmb{\{n,\{5,10,20\}\},\{k,0,20\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \text{Automatic} ]}\)

\includegraphics{kakubun01_gr5.eps}

\noindent\(\pmb{\text{���̊􉽕��z} \text{NegBinom}(n,p), n=8, p=0.1, 0.3, 0.5, 0.8}\)

\noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
\pmb{\text{PDF}[\text{NegativeBinomialDistribution}[5,p],k]\},}\\
\pmb{\{p,\{0.1,0.3,0.5,0.8\}\},\{k,0,30\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \{\{0,30\},\{0,0.35\}\} ]}\)

\includegraphics{kakubun01_gr6.eps}

\noindent\(\pmb{\text{���̊􉽕��z} \text{NegBinom}(n,p), n=1,5,8,10, p= 0.3}\)

\noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
\pmb{\text{PDF}[\text{NegativeBinomialDistribution}[n,0.3],k]\},}\\
\pmb{\{n,\{1,5,8,10\}\},\{k,0,30\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \{\{0,30\},\{0,0.35\}\} ]}\)

\includegraphics{kakubun01_gr7.eps}

\noindent\(\pmb{\text{���􉽕��z}\text{  }\text{HyperGeom}(n,\text{n$\_$s},\text{n$\_$t})}\)

\noindent\(\pmb{\text{HypergeometricDistribution}[n,\text{n$\_$}\{\text{succ}\},\text{n$\_$}\{\text{tot}\}]}\)

\noindent\(\pmb{\text{�m��}:�@n=5,\text{n$\_$s}/\text{n$\_$t} = 2/10, 5/10,7/10\text{  }(n < \text{n$\_$t})}\)

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

\noindent\(\left\{\frac{1}{252} \text{Binomial}[2,k] \text{Binomial}[8,5-k],\frac{1}{252} \text{Binomial}[5,5-k] \text{Binomial}[5,k],\frac{1}{252}
\text{Binomial}[3,5-k] \text{Binomial}[7,k]\right\}\)

\noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
\pmb{\text{PDF}[\text{HypergeometricDistribution}[5,\text{ns},10],k]}\\
\pmb{\},\{\text{ns},\{2,5,7\}\},\{k,0,5\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \text{Automatic} ]}\)

\includegraphics{kakubun01_gr8.eps}

\noindent\(\pmb{\text{�m��}:�@n=15, \text{n$\_$s}/\text{n$\_$t} = 4/20, 10/20,14/20 (n < \text{n$\_$t})}\)

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

\noindent\(\left\{\frac{\text{Binomial}[4,k] \text{Binomial}[16,15-k]}{15504},\frac{\text{Binomial}[10,15-k] \text{Binomial}[10,k]}{15504},\frac{\text{Binomial}[6,15-k]
\text{Binomial}[14,k]}{15504}\right\}\)

\noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
\pmb{\text{PDF}[\text{HypergeometricDistribution}[15,\text{ns},20],k]}\\
\pmb{\},\{\text{ns},\{4,10,14\}\},\{k,0,15\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \text{Automatic} ]}\)

\includegraphics{kakubun01_gr9.eps}

\noindent\(\pmb{\text{�m��}:n=15, \text{n$\_$s}/\text{n$\_$t} = 40/200, 100/200,140/200(n < \text{n$\_$t})}\)

\noindent\(\pmb{\text{Table}[\text{PDF}[\text{HypergeometricDistribution}[15,\text{ns},200],k],\{\text{ns},\{40,100,140\}\}]}\)

\noindent\(\left\{\frac{\text{Binomial}[40,k] \text{Binomial}[160,15-k]}{14629416353818682834880},\frac{\text{Binomial}[100,15-k] \text{Binomial}[100,k]}{14629416353818682834880},\frac{\text{Binomial}[60,15-k]
\text{Binomial}[140,k]}{14629416353818682834880}\right\}\)

\noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
\pmb{\text{PDF}[\text{HypergeometricDistribution}[15,\text{ns},200],k]}\\
\pmb{\},\{\text{ns},\{40,100,140\}\},\{k,0,15\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \text{Automatic} ]}\)

\includegraphics{kakubun01_gr10.eps}

\noindent\(\pmb{2\text{�����z�ɋ߂Â��An}=15, p= 40/200, 100/200, 140/200}\)

\noindent\(\pmb{\text{Table}[\text{PDF}[\text{BinomialDistribution}[15,p],k],\{p,\{40/200,100/200,140/200\}\}]}\)

\noindent\(\left\{\frac{4^{15-k} \text{Binomial}[15,k]}{30517578125},\frac{\text{Binomial}[15,k]}{32768},\frac{3^{15-k} 7^k \text{Binomial}[15,k]}{1000000000000000}\right\}\)

\noindent\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
\pmb{\text{PDF}[\text{BinomialDistribution}[15,p],k]}\\
\pmb{\},\{p,\{40/200,100/200,140/200\}\},\{k,0,15\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \text{Automatic} ]}\)

\includegraphics{kakubun01_gr11.eps}

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

\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\(\pmb{\text{ListPlot}[\text{Table}[\{k,}\\
\pmb{\text{PDF}[\text{GeometricDistribution}[p],k]\},}\\
\pmb{\{p,\{0.1,0.3,0.5,0.8\}\},\{k,0,15\}],}\\
\pmb{\text{PlotMarkers}\to \text{Automatic},\text{Joined} \to  \text{True}, \text{PlotRange} \to  \{\{0,15\},\{0,0.8\}\} ]}\)

\includegraphics{kakubun01_gr12.eps}

\noindent\(\pmb{}\\
\pmb{\text{�w�����z�̊m�����x�F}}\)

\noindent\(\pmb{\text{PDF}[\text{ExponentialDistribution}[\text{lm}],x]}\)

\noindent\(e^{-\text{lm} x} \text{lm}\)

\noindent\(\pmb{\text{Plot}[}\\
\pmb{\{\text{PDF}[\text{ExponentialDistribution}[0.3],x],}\\
\pmb{\text{PDF}[\text{ExponentialDistribution}[0.5],x],}\\
\pmb{\text{PDF}[\text{ExponentialDistribution}[0.8],x]\},}\\
\pmb{\{x,0,15\},\text{PlotRange} \to  \{\{0,15\},\{0,0.8\}\}]}\)

\includegraphics{kakubun01_gr13.eps}

\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}
