確率変数の変数変換 Z=c1X+c2Y / 正規分布の定数倍同士の和

確率変数の変数変換 $Z=c_{1}X+c_{2}Y$ / 正規分布の定数倍同士の和

確率変数の変数変換 $Z=c_{1}X+c_{2}Y$

$$\begin{array}{rcl}{p}_{c_{1}X+c_{2}Y}(z)& =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (z-(c_{1}x+c_{2}y))f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y\\ & & \cdots {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)f(x)\mathrm{d}x=f(0)\\ & & \cdots X=x,Y=y\mathrm{の}\mathrm{同}\mathrm{時}\mathrm{確}\mathrm{率}f(x)g(x)\mathrm{の}\mathrm{う}\mathrm{ち}z-(c_{1}x+c_{2}y)=0\mathrm{を}\mathrm{満}\mathrm{た}\mathrm{す}\mathrm{も}\mathrm{の}\mathrm{だ}\mathrm{け}\mathrm{を}\mathrm{足}\mathrm{し}\mathrm{合}\mathrm{わ}\mathrm{せ}\mathrm{る}.\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (z-(c_{1}x+c_{2}y))f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y\\ & & \cdots \delta (u(x))=\sum _{\alpha \in u^{-1}(0)}\frac{1}{|u^{\prime }(\alpha )|}\delta (x-\alpha )\\ & & \cdots u(x)=z-(c_{1}x+c_{2}y)\\ & & \cdots u(x=\alpha )=0,\alpha =\frac{z-c_{2}y}{c_1}\\ & & \cdots u^{\prime }=\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{\mathrm{d}}{\mathrm{d}x}(z-(c_{1}x+c_{2}y))=-c_1\\ & & \cdots \delta (z-(c_{1}x+c_{2}y))=\frac{1}{|u^{\prime }(\alpha )|}\delta (x-\alpha )=\frac{1}{|-c_1|}\delta (x-\frac{z-c_{2}y}{c_1})=\frac{1}{\left|c_1\right|}\delta (x-\frac{z-c_{2}y}{c_1})\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{\left|c_1\right|}\delta (x-\frac{z-c_{2}y}{c_1})f_X\left(x\right)g_Y(y)\mathrm{d}y\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{\left|c_1\right|}\delta (\frac{z-c_{2}y}{c_1}-\frac{z-c_{2}y}{c_1})f_X\left(\frac{z-c_{2}y}{c_1}\right)g_Y(y)\mathrm{d}y\cdots z=c_{1}x+c_{2}y,x=\frac{z-c_{2}y}{c_1}\\ & =& \frac{1}{\left|c_1\right|}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (0)f_X\left(\frac{z-c_{2}y}{c_1}\right)g_Y(y)\mathrm{d}y\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{\left|c_1\right|}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_X\left(\frac{z-c_{2}y}{c_1}\right)g_Y(y)\mathrm{d}y\end{array}$$

正規分布の定数倍同士の和の例

$$\begin{array}{rcl}{f}_X(x)& =& \frac{1}{\sqrt{2\pi {\sigma }_1^2}}{\mathrm{e}}^{\frac{-(x-{\mu }_1{)}^2}{2{\sigma }_1^2}}\cdots \mathrm{N}({\mu }_1,{\sigma }_1^2)(\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ g_Y(y)& =& \frac{1}{\sqrt{2\pi {\sigma }_2^2}}{\mathrm{e}}^{\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\cdots \mathrm{N}({\mu }_2,{\sigma }_2^2)(\mathrm{正}\mathrm{規}\mathrm{分}\mathrm{布})\\ p_{c_{1}X+c_{2}Y}(z)& =& \frac{1}{\left|c_1\right|}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_X\left(\frac{z-c_{2}y}{c_1}\right)g_Y(y)\mathrm{d}y\\ & =& \frac{1}{\left|c_1\right|}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{\sqrt{2\pi {\sigma }_1^2}}{\mathrm{e}}^{\frac{-{\{\left(\frac{z-c_{2}y}{c_1}\right)-{\mu }_1\}}^2}{2{\sigma }_1^2}}\frac{1}{\sqrt{2\pi {\sigma }_2^2}}{\mathrm{e}}^{\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\mathrm{d}y\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{\sqrt{2\pi {\sigma }_1^2}}\frac{1}{\sqrt{2\pi {\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\frac{-{\{\left(\frac{z-c_{2}y}{c_1}\right)-{\mu }_1\}}^2}{2{\sigma }_1^2}+\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\mathrm{d}y\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\frac{-{\{\left(\frac{z-c_{2}y}{c_1}\right)-{\mu }_1\}}^2}{2{\sigma }_1^2}+\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}}\mathrm{d}y\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{f_1(c_1,c_2,y,z,{\mu }_1,{\mu }_2,{\sigma }_1,{\sigma }_2)}\mathrm{d}y\cdots f_1\mathrm{は}y\mathrm{が}\mathrm{引}\mathrm{数}\mathrm{に}\mathrm{あ}\mathrm{る}．\end{array}$$

---

$$\begin{array}{rcl}{f}_1(c_1,c_2,y,z,{\mu }_1,{\mu }_2,{\sigma }_1,{\sigma }_2)& =& \frac{-{\{\left(\frac{z-c_{2}y}{c_1}\right)-{\mu }_1\}}^2}{2{\sigma }_1^2}+\frac{-(y-{\mu }_2{)}^2}{2{\sigma }_2^2}\\ & =& -\frac{1}{2}\{\frac{{(\frac{1}{c_1}z-\frac{c_2}{c_1}y-{\mu }_1)}^2}{{\sigma }_1^2}+\frac{(y-{\mu }_2{)}^2}{{\sigma }_2^2}\}\\ & =& -\frac{1}{2}\left\{\frac{{\sigma }_2^2{(\frac{1}{c_1}z-\frac{c_2}{c_1}y-{\mu }_1)}^2+{\sigma }_1^2(y-{\mu }_2{)}^2}{{\sigma }_1^2{\sigma }_2^2}\right\}\\ & =& -\frac{1}{2{\sigma }_1^2{\sigma }_2^2}\{{\sigma }_2^2{(\frac{1}{c_1}z-\frac{c_2}{c_1}y-{\mu }_1)}^2+{\sigma }_1^2(y-{\mu }_2{)}^2\}\\ & =& -\frac{1}{2{\sigma }_1^2{\sigma }_2^2}[{\sigma }_2^2\{{\left(\frac{1}{c_1}z\right)}^2-2\left(\frac{1}{c_1}z\right)\left(\frac{c_2}{c_1}y\right)-2\left(\frac{1}{c_1}z\right){\mu }_1+{\left(\frac{c_2}{c_1}y\right)}^2+2\left(\frac{c_2}{c_1}y\right){\mu }_1+{\mu }_1^2\}+{\sigma }_1^2(y^2-2y{\mu }_2+{\mu }_2^2)]\\ & =& -\frac{1}{2{\sigma }_1^2{\sigma }_2^2}\{\frac{1}{c_1^2}{\sigma }_2^2{z}^2-2\frac{c_2}{c_1^2}{\sigma }_2^{2}zy-2\frac{1}{c_1}{\sigma }_2^{2}z{\mu }_1+\frac{c_2^2}{c_1^2}{\sigma }_2^2{y}^2+2\frac{c_2}{c_1}{\sigma }_2^{2}y{\mu }_1+{\sigma }_2^2{\mu }_1^2+{\sigma }_1^2{y}^2-2{\sigma }_1^{2}y{\mu }_2+{\sigma }_1^2{\mu }_2^2\}\\ & =& -\frac{1}{2{\sigma }_1^2{\sigma }_2^2}\frac{1}{c_1^2}\{{\sigma }_2^2{z}^2-2c_2{\sigma }_2^{2}zy-2c_1{\sigma }_2^{2}z{\mu }_1+c_2^2{\sigma }_2^2{y}^2+2c_1{c}_2{\sigma }_2^{2}y{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{y}^2-2c_1^2{\sigma }_1^{2}y{\mu }_2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)y^2+(-2c_2{\sigma }_2^{2}z+2c_1{c}_2{\sigma }_2^2{\mu }_1-2c_1^2{\sigma }_1^2{\mu }_2)y+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)y^2-2(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)y+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)(y^2-2\frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}y)+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& \\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)(y^2-2\frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}y)+A-A+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & & \cdots \mathrm{平}\mathrm{方}\mathrm{完}\mathrm{成}\mathrm{さ}\mathrm{せ}\mathrm{る}\mathrm{た}\mathrm{め}\mathrm{の}\mathrm{項}A\mathrm{を}\mathrm{考}\mathrm{え}\mathrm{る}\\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)(y^2-2\frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}y+\frac{1}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}A)-A+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}[(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)\{y^2-2\frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}y+\frac{1}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}\frac{{(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)}^2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}\}-\frac{{(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)}^2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2]\\ & & \cdots A=\frac{{(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)}^2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}\\ & =& -\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2){(y^2-\frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2})}^2-\frac{{(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)}^2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& -\frac{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}{2c_1^2{\sigma }_1^2{\sigma }_2^2}{(y^2-\frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2})}^2-\frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{-\frac{{(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)}^2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& -f_2(c_1,c_2,{\sigma }_1^2,{\sigma }_2^2){\{y-f_3(c_1,c_2,z,{\mu }_1,{\mu }_2,{\sigma }_1^2,{\sigma }_2^2)\}}^2-f_4(c_1,c_2,z,{\mu }_1,{\mu }_2,{\sigma }_1^2,{\sigma }_2^2)\cdots f_2,f_3,f_4\mathrm{は}y\mathrm{を}\mathrm{引}\mathrm{数}\mathrm{と}\mathrm{し}\mathrm{な}\mathrm{い}.\\ & =& -f_2{(y-f_3)}^2-f_4\cdots y\mathrm{が}\mathrm{平}\mathrm{方}\mathrm{完}\mathrm{成}\mathrm{の}\mathrm{中}\mathrm{の}\mathrm{一}\mathrm{つ}\mathrm{だ}\mathrm{け}\mathrm{に}\mathrm{な}\mathrm{っ}\mathrm{て}\mathrm{い}\mathrm{る}.\\ \\ f_2(c_1,c_2,{\sigma }_1^2,{\sigma }_2^2)& =& \frac{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\\ \\ f_3(c_1,c_2,z,{\mu }_1,{\mu }_2,{\sigma }_1^2,{\sigma }_2^2)& =& \frac{c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}\\ \\ f_4(c_1,c_2,z,{\mu }_1,{\mu }_2,{\sigma }_1^2,{\sigma }_2^2)& =& \frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\{-\frac{{(c_2{\sigma }_2^{2}z-c_1{c}_2{\sigma }_2^2{\mu }_1+c_1^2{\sigma }_1^2{\mu }_2)}^2}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}+{\sigma }_2^2{z}^2-2c_1{\sigma }_2^{2}z{\mu }_1+c_1^2{\sigma }_2^2{\mu }_1^2+c_1^2{\sigma }_1^2{\mu }_2^2\}\\ & =& \frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\left\{\frac{-{(c_2{\sigma }_2^2(z-c_1{\mu }_1)+c_1^2{\sigma }_1^2{\mu }_2)}^2+(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)({\sigma }_2^2{(z-c_1{\mu }_1)}^2+c_1^2{\sigma }_1^2{\mu }_2^2)}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}\right\}\\ & =& \frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2}\frac{-{(c_2{\sigma }_2^{2}B+c_1^2{\sigma }_1^2{\mu }_2)}^2+(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)({\sigma }_2^2{B}^2+c_1^2{\sigma }_1^2{\mu }_2^2)}{c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2}\cdots B=z-c_1{\mu }_1\\ & =& \frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}\{-{(c_2{\sigma }_2^{2}B+c_1^2{\sigma }_1^2{\mu }_2)}^2+(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)({\sigma }_2^2{B}^2+c_1^2{\sigma }_1^2{\mu }_2^2)\}\\ & =& \frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}\left(-c_2^2{\sigma }_2^4{B}^2-2c_1^2{c}_2{\sigma }_1^2{\sigma }_2^2{\mu }_{2}B-c_1^4{\sigma }_1^4{\mu }_2^2+c_1^2{\sigma }_1^2{\sigma }_2^2{B}^2+c_2^2{\sigma }_2^4{B}^2+c_1^4{\sigma }_1^4{\mu }_2^2+c_1^2{c}_2^2{\sigma }_1^2{\sigma }_2^2{\mu }_2^2\right)\\ & =& \frac{1}{2c_1^2{\sigma }_1^2{\sigma }_2^2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}{c}_1^2{\sigma }_1^2{\sigma }_2^2\{-2c_2{\mu }_{2}B+B^2+c_2^2{\mu }_2^2\}\\ & =& \frac{1}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}{(B-c_2{\mu }_2)}^2\\ & =& \frac{{(z-c_1{\mu }_1-c_2{\mu }_2)}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}\cdots B=z-c_1{\mu }_1\\ & =& \frac{{(z-(c_1{\mu }_1+c_2{\mu }_2))}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}\end{array}$$

---

$$\begin{array}{rcl}{p}_{c_{1}X+c_{2}Y}(z)& =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{f_1(c_1,c_2,y,z,{\mu }_1,{\mu }_2,{\sigma }_1,{\sigma }_2)}\mathrm{d}y\\ & =& \frac{1}{\left|c_1\right|}\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}{\left|c_1\right|}\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\\ & =& \frac{1}{\left|c_1\right|}\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}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-u^2}\frac{1}{\sqrt{f_2}}\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}{\left|c_1\right|}\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\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}\frac{1}{\sqrt{f_2}}\sqrt{\pi }\cdots {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-u^2}\mathrm{d}u=\sqrt{\pi }\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}{\mathrm{e}}^{-f_4}\frac{\sqrt{\pi }}{\sqrt{f_2}}\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}\sqrt{\frac{\pi }{f_2}}{\mathrm{e}}^{-f_4}\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}\sqrt{\frac{\pi }{\frac{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}{2c_1^2{\sigma }_1^2{\sigma }_2^2}}}{\mathrm{e}}^{-\frac{{(z-(c_1{\mu }_1+c_2{\mu }_2))}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}\\ & =& \frac{1}{\left|c_1\right|}\frac{1}{2\pi \sqrt{{\sigma }_1^2{\sigma }_2^2}}\sqrt{\frac{2\pi c_1^2{\sigma }_1^2{\sigma }_2^2}{(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}{\mathrm{e}}^{-\frac{{(z-(c_1{\mu }_1+c_2{\mu }_2))}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}\\ & =& \frac{1}{\left|c_1\right|}\frac{\sqrt{c_1^2}}{\sqrt{2\pi (c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}{\mathrm{e}}^{-\frac{{(z-(c_1{\mu }_1+c_2{\mu }_2))}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}\\ & =& \frac{1}{\left|c_1\right|}\frac{\left|c_1\right|}{\sqrt{2\pi (c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}{\mathrm{e}}^{-\frac{{(z-(c_1{\mu }_1+c_2{\mu }_2))}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}\cdots \sqrt{A^2}=\{\begin{array}{}A& (A\ge 0)\\ -A& (A<0)\end{array}=\left|A\right|\\ & =& \frac{1}{\sqrt{2\pi (c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}{\mathrm{e}}^{-\frac{{(z-(c_1{\mu }_1+c_2{\mu }_2))}^2}{2(c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)}}\\ & \sim & \mathrm{N}(c_1{\mu }_1+c_2{\mu }_2,c_1^2{\sigma }_1^2+c_2^2{\sigma }_2^2)\end{array}$$