確率変数の変数変換 Z=X+Y その2 / 正規分布の再生性

確率変数の変数変換 Z=X+Y その2 / 正規分布の再生性

Z=X+Y 正規分布同士の例

$$\begin{array}{rcl}{f}_X(x)& =& \frac{1}{\sqrt{2\pi {\sigma }_1^2}}{\mathrm{e}}^{\frac{-(x-{\mu }_1{)}^2}{2{\sigma }_1^2}}\cdots \mathrm{N}({\mu }_1,{\sigma }_1^2)(\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ g_Y(x)& =& \frac{1}{\sqrt{2\pi {\sigma }_2^2}}{\mathrm{e}}^{\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\cdots \mathrm{N}({\mu }_2,{\sigma }_2^2)(\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ p_{X+Y}(z)& =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (z-(x+y))f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (z-((z-y)+y))f_X(z-y)g_Y(y)\mathrm{d}y\cdots z=x+y,x=z-y\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (0)f_X(z-y)g_Y(y)\mathrm{d}y\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_X(z-y)g_Y(y)\mathrm{d}y\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{\sqrt{2\pi {\sigma }_1^2}}{\mathrm{e}}^{\frac{-(z-y-{\mu }_1{)}^2}{2{\sigma }_1^2}}\frac{1}{\sqrt{2\pi {\sigma }_2^2}}{\mathrm{e}}^{\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\mathrm{d}y\\ & =& \frac{1}{\sqrt{2\pi {\sigma }_1^2}}\frac{1}{\sqrt{2\pi {\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\frac{-(z-y-{\mu }_1{)}^2}{2{\sigma }_1^2}+\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\mathrm{d}y\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\frac{-(z-y-{\mu }_1{)}^2}{2{\sigma }_1^2}+\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\mathrm{d}y\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{f_1(y,z,{\mu }_1,{\mu }_2,{\sigma }_1,{\sigma }_2)}\mathrm{d}y\cdots f_1\mathrm{は}y\mathrm{が}\mathrm{引}\mathrm{数}\mathrm{に}\mathrm{あ}\mathrm{る}．\end{array}$$

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$$\begin{array}{rcl}{p}_{X+Y}(z)& =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{f_1(y,z,{\mu }_1,{\mu }_2,{\sigma }_1,{\sigma }_2)}\mathrm{d}y\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-f_2{(y-f_3)}^2-f_4}\mathrm{d}y\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-f_2{(y-f_3)}^2}{\mathrm{e}}^{-f_4}\mathrm{d}y\cdots A^{B+C}=A^B{A}^C\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-f_2{(y-f_3)}^2}\mathrm{d}y\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-f_2{(y-f_3)}^2}\mathrm{d}y\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-u^2}\left(\frac{1}{\sqrt{f_2}}\right)\mathrm{d}u\\ & & \cdots u=\sqrt{f_2}(y-f_3),\frac{\mathrm{d}u}{\mathrm{d}y}=\sqrt{f_2},\mathrm{d}y=\frac{1}{\sqrt{f_2}}\mathrm{d}u,\\ & & \cdots y:-\mathrm{\infty }\to \mathrm{\infty },u:-\mathrm{\infty }\to \mathrm{\infty }\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}\frac{1}{\sqrt{f_2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-u^2}\mathrm{d}u\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}\frac{1}{\sqrt{f_2}}{\mathrm{e}}^{-f_4}\sqrt{\pi }\cdots {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-u^2}\mathrm{d}u=\sqrt{\pi }\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}\frac{1}{\sqrt{\frac{{\sigma }_1^2+{\sigma }_2^2}{2{\sigma }_1^2{\sigma }_2^2}}}{\mathrm{e}}^{-\frac{{\{z-({\mu }_1+{\mu }_2)\}}^2}{2({\sigma }_1^2+{\sigma }_2^2)}}\sqrt{\pi }\\ & =& \frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}\sqrt{\frac{2\pi {\sigma }_1^2{\sigma }_2^2}{{\sigma }_1^2+{\sigma }_2^2}}{\mathrm{e}}^{-\frac{{\{z-({\mu }_1+{\mu }_2)\}}^2}{2({\sigma }_1^2+{\sigma }_2^2)}}\\ & =& \frac{1}{\sqrt{2\pi ({\sigma }_1^2+{\sigma }_2^2)}}{\mathrm{e}}^{-\frac{{\{z-({\mu }_1+{\mu }_2)\}}^2}{2({\sigma }_1^2+{\sigma }_2^2)}}\cdots \mathrm{N}({\mu }_1+{\mu }_2,{\sigma }_1^2+{\sigma }_2^2)(\mathrm{再}\mathrm{生}\mathrm{性})\end{array}$$ 再生性は，同じ確率分布族に含まれる確率分布$F_1$,$F_2$に対して $X_1\sim F_1$,$X_2\sim F_2$とする互いに独立な確率変数に対して， $X_1+X_2$がやはり同一の確率分布族に含まれる性質．