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\begin{document}
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  ���v���x�̃v���b�g
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\section{��l���z�̘a}

�Q�l�����F��l���z�̘a Feller Vol .2, p .26

\(\text{xp}[\text{x$\_$}]=(x+\text{Abs}[x])/2\)

\(\frac{1}{2} (x+\text{Abs}[x])\)

xp[x]


2 �̘a�F2�{�̈ꎟ�� �i�O�p�`���z�j $(0 < x < 1, 1 < x < 2)$

\(\text{usma2}[\text{x$\_$}]=\text{Sum}[(-1)^{}\{k\}*\text{Binomial}[2,k]*(\text{xp}[x-k]){}^{}\{2-1\},\{k,0,2\}]/(2-1)!\)

\(\left\{1-x+\frac{1}{2} (-2+x+\text{Abs}[-2+x])-\text{Abs}[-1+x]+\frac{1}{2} (x+\text{Abs}[x])\right\}\)

Plot[usma2[x],\{x,0,2\}]

\includegraphics{sum_of_Unif02.eps}



\text{3 �̘a�F3�{�̂Q���Ȑ� (0 $<$ x $<$ 1, 1 $<$ x $<$ 2, 2 $<$ x $<$ 3)}

\(\text{usma3}[\text{x$\_$}]=\text{Sum}[(-1){}^{\wedge}k*\text{Binomial}[3,k]*(\text{xp}[x-k]){}^{\wedge}\{3-1\},\{k,0,3\}]/(3-1)!\)

\(\left\{\frac{1}{2} \left(-\frac{1}{4} (-3+x+\text{Abs}[-3+x])^2+\frac{3}{4} (-2+x+\text{Abs}[-2+x])^2-\frac{3}{4} (-1+x+\text{Abs}[-1+x])^2+\frac{1}{4} (x+\text{Abs}[x])^2\right)\right\}\)

Plot[usma3[x],\{x,0,4\}]

\includegraphics{sum_of_Unif3.eps}



�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}



�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}


\text{���K���z N (0, 1) �̋ߎ��� sum$\_$ $\{$i = 1$\}{}^{\wedge}\{$12$\}$ X$\_$i - 6 , �P�Q����������̂U�����������́A���ςO�A���U�P�ƂȂ�}

\text{Plot}[\text{usma12}[x+6],\{x,-4,4\}]

\includegraphics{sum_of_Unif12.eps}

\end{document}
