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 \(Z=c_1X+c_2Y\) / 正規分布の定数倍同士の和

確率変数の変数変換 \(Z=c_1X+c_2Y\)

$$ \begin{eqnarray} p_{c_1X+c_2Y}(z) &=&\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \delta(z-(c_1x+c_2y))f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y \\&&\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/09/delta-function.html}{\int_{-\infty}^{\infty}\delta(x)f(x)\mathrm{d}x=f(0)} \\&&\;\cdots\;X=x,Y=yの同時確率f(x)g(x)のうちz-\left(c_1x+c_2y\right)=0を満たすものだけを足し合わせる. \\&=&\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \delta(z-(c_1x+c_2y))f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y \\&&\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/09/delta-function.html}{\delta(u(x))=\sum_{\alpha\in u^{-1}(0)}\frac{1}{\left|u^{\prime}(\alpha)\right|}\delta\left(x-\alpha\right)} \\&&\;\cdots\;u(x)=z-(c_1x+c_2y) \\&&\;\cdots\;u(x=\alpha)=0,\;\alpha=\frac{z-c_2y}{c_1} \\&&\;\cdots\;u^\prime=\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{\mathrm{d}}{\mathrm{d}x}\left(z-(c_1x+c_2y)\right)=-c_1 \\&&\;\cdots\;\delta\left(z-(c_1x+c_2y)\right) =\frac{1}{|u^\prime(\alpha)|}\delta\left(x-\alpha\right) =\frac{1}{\left|-c_1\right|}\delta\left(x-\frac{z-c_2y}{c_1}\right) =\frac{1}{\left|c_1\right|}\delta\left(x-\frac{z-c_2y}{c_1}\right) \\&=&\int_{-\infty}^{\infty}\frac{1}{\left|c_1\right|}\delta\left(x-\frac{z-c_2y}{c_1}\right)f_X\left(x\right)g_Y(y)\mathrm{d}y \\&=&\int_{-\infty}^{\infty}\frac{1}{\left|c_1\right|}\delta\left(\frac{z-c_2y}{c_1}-\frac{z-c_2y}{c_1}\right)f_X\left(\frac{z-c_2y}{c_1}\right)g_Y(y)\mathrm{d}y \;\cdots\;z=c_1x+c_2y,\;x=\frac{z-c_2y}{c_1} \\&=&\frac{1}{\left|c_1\right|}\int_{-\infty}^{\infty}\delta(0)f_X\left(\frac{z-c_2y}{c_1}\right)g_Y(y)\mathrm{d}y \;\cdots\;\int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x \\&=&\frac{1}{\left|c_1\right|}\int_{-\infty}^{\infty}f_X\left(\frac{z-c_2y}{c_1}\right)g_Y(y)\mathrm{d}y \end{eqnarray} $$

正規分布の定数倍同士の和の例

$$ \begin{eqnarray} f_X(x)&=&\frac{1}{\sqrt{2\pi\sigma_1^2}}\mathrm{e}^{\frac{-(x-\mu_1)^2}{2\sigma_1^2}} \;\cdots\;\mathrm{N}(\mu_1,\sigma_1^2)(\href{https://shikitenkai.blogspot.com/2019/06/binomial-distributionnormal-distribution.html}{正規分布}) \\g_Y(y)&=&\frac{1}{\sqrt{2\pi\sigma_2^2}}\mathrm{e}^{\frac{-(y-\mu_2)^2}{2\sigma_2^2}} \;\cdots\;\mathrm{N}(\mu_2,\sigma_2^2)(\href{https://shikitenkai.blogspot.com/2019/06/binomial-distributionnormal-distribution.html}{正規分布}) \\p_{c_1X+c_2Y}(z) &=&\frac{1}{\left|c_1\right|}\int_{-\infty}^{\infty}f_X\left(\frac{z-c_2y}{c_1}\right)g_Y(y)\mathrm{d}y \\&=&\frac{1}{\left|c_1\right|}\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi\sigma_1^2}}\mathrm{e}^{\frac{-\left\{\left(\frac{z-c_2y}{c_1}\right)-\mu_1\right\}^2}{2\sigma_1^2}}\frac{1}{\sqrt{2\pi\sigma_2^2}}\mathrm{e}^{\frac{-(y-\mu_2)^2}{2\sigma_2^2}} \mathrm{d}y \\&=&\frac{1}{\left|c_1\right|}\frac{1}{\sqrt{2\pi\sigma_1^2}}\frac{1}{\sqrt{2\pi\sigma_2^2}}\int_{-\infty}^{\infty}\mathrm{e}^{\frac{-\left\{\left(\frac{z-c_2y}{c_1}\right)-\mu_1\right\}^2}{2\sigma_1^2}+\frac{-(y-\mu_2)^2}{2\sigma_2^2}} \mathrm{d}y \;\cdots\;\int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x \\&=&\frac{1}{\left|c_1\right|}\frac{1}{2\pi\sqrt{\sigma_1^2\sigma_2^2}}\int_{-\infty}^{\infty}\mathrm{e}^{\frac{-\left\{\left(\frac{z-c_2y}{c_1}\right)-\mu_1\right\}^2}{2\sigma_1^2}+\frac{-(y-\mu_2)^2}{2\sigma_2^2}} \mathrm{d}y \\&=&\frac{1}{\left|c_1\right|}\frac{1}{2\pi\sqrt{\sigma_1^2\sigma_2^2}} \int_{-\infty}^{\infty} \mathrm{e}^{f_1(c_1,c_2,y,z,\mu_1,\mu_2,\sigma_1,\sigma_2)} \mathrm{d}y \;\cdots\;f_1はyが引数にある． \end{eqnarray} $$

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$$ \begin{eqnarray} f_1(c_1,c_2,y,z,\mu_1,\mu_2,\sigma_1,\sigma_2) &=&\frac{-\left\{\left(\frac{z-c_2y}{c_1}\right)-\mu_1\right\}^2}{2\sigma_1^2}+\frac{-(y-\mu_2)^2}{2\sigma_2^2} \\&=&-\frac{1}{2} \left\{ \frac{\left(\frac{1}{c_1}z-\frac{c_2}{c_1}y-\mu_1\right)^2}{\sigma_1^2} +\frac{(y-\mu_2)^2}{\sigma_2^2} \right\} \\&=&-\frac{1}{2} \left\{ \frac{\sigma_2^2\left(\frac{1}{c_1}z-\frac{c_2}{c_1}y-\mu_1\right)^2 +\sigma_1^2(y-\mu_2)^2}{\sigma_1^2\sigma_2^2} \right\} \\&=&-\frac{1}{2\sigma_1^2\sigma_2^2} \left\{ \sigma_2^2\left(\frac{1}{c_1}z-\frac{c_2}{c_1}y-\mu_1\right)^2 +\sigma_1^2(y-\mu_2)^2 \right\} \\&=&-\frac{1}{2\sigma_1^2\sigma_2^2} \left[ \sigma_2^2\left\{\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\right\} +\sigma_1^2(y^2-2y\mu_2+\mu_2^2) \right] \\&=&-\frac{1}{2\sigma_1^2\sigma_2^2} \left\{ \frac{1}{c_1^2}\sigma_2^2z^2 -2\frac{c_2}{c_1^2}\sigma_2^2zy -2\frac{1}{c_1}\sigma_2^2z\mu_1 +\frac{c_2^2}{c_1^2}\sigma_2^2y^2 +2\frac{c_2}{c_1}\sigma_2^2y\mu_1 +\sigma_2^2\mu_1^2 +\sigma_1^2y^2 -2\sigma_1^2y\mu_2 +\sigma_1^2\mu_2^2 \right\} \\&=&-\frac{1}{2\sigma_1^2\sigma_2^2} \frac{1}{c_1^2} \left\{ \sigma_2^2z^2 -2c_2\sigma_2^2zy -2c_1\sigma_2^2z\mu_1 +c_2^2\sigma_2^2y^2 +2c_1c_2\sigma_2^2y\mu_1 +c_1^2\sigma_2^2\mu_1^2 +c_1^2\sigma_1^2y^2 -2c_1^2\sigma_1^2y\mu_2 +c_1^2\sigma_1^2\mu_2^2 \right\} \\&=&-\frac{1}{2c_1^2\sigma_1^2\sigma_2^2} \left\{ \left( c_1^2\sigma_1^2 +c_2^2\sigma_2^2 \right)y^2 +\left( -2c_2\sigma_2^2z +2c_1c_2\sigma_2^2\mu_1 -2c_1^2\sigma_1^2\mu_2 \right)y +\sigma_2^2z^2 -2c_1\sigma_2^2z\mu_1 +c_1^2\sigma_2^2\mu_1^2 +c_1^2\sigma_1^2\mu_2^2 \right\} \\&=&-\frac{1}{2c_1^2\sigma_1^2\sigma_2^2} \left\{ \left( c_1^2\sigma_1^2 +c_2^2\sigma_2^2 \right)y^2 -2\left( c_2\sigma_2^2z -c_1c_2\sigma_2^2\mu_1 +c_1^2\sigma_1^2\mu_2 \right)y +\sigma_2^2z^2 -2c_1\sigma_2^2z\mu_1 +c_1^2\sigma_2^2\mu_1^2 +c_1^2\sigma_1^2\mu_2^2 \right\} \\&=&-\frac{1}{2c_1^2\sigma_1^2\sigma_2^2} \left\{ \left( c_1^2\sigma_1^2 +c_2^2\sigma_2^2 \right) \left( y^2 -2\frac{ c_2\sigma_2^2z -c_1c_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 \right) +\sigma_2^2z^2 -2c_1\sigma_2^2z\mu_1 +c_1^2\sigma_2^2\mu_1^2 +c_1^2\sigma_1^2\mu_2^2 \right\} \\&=& \\&=&-\frac{1}{2c_1^2\sigma_1^2\sigma_2^2} \left\{ \left( c_1^2\sigma_1^2 +c_2^2\sigma_2^2 \right) \left( y^2 -2\frac{ c_2\sigma_2^2z -c_1c_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 \right) +A-A +\sigma_2^2z^2 -2c_1\sigma_2^2z\mu_1 +c_1^2\sigma_2^2\mu_1^2 +c_1^2\sigma_1^2\mu_2^2 \right\} \\&&\;\cdots\;平方完成させるための項Aを考える \\&=&-\frac{1}{2c_1^2\sigma_1^2\sigma_2^2} \left\{ \left( c_1^2\sigma_1^2 +c_2^2\sigma_2^2 \right) \left( y^2 -2\frac{ c_2\sigma_2^2z -c_1c_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 \right) -A +\sigma_2^2z^2 -2c_1\sigma_2^2z\mu_1 +c_1^2\sigma_2^2\mu_1^2 +c_1^2\sigma_1^2\mu_2^2 \right\} \\&=&-\frac{1}{2c_1^2\sigma_1^2\sigma_2^2} \left[ \left( c_1^2\sigma_1^2 +c_2^2\sigma_2^2 \right) \left\{ y^2 -2\frac{ c_2\sigma_2^2z -c_1c_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{ \left( c_2\sigma_2^2z -c_1c_2\sigma_2^2\mu_1 +c_1^2\sigma_1^2\mu_2 \right)^2 }{c_1^2\sigma_1^2 +c_2^2\sigma_2^2} \right\} -\frac{ \left( c_2\sigma_2^2z -c_1c_2\sigma_2^2\mu_1 +c_1^2\sigma_1^2\mu_2 \right)^2 }{c_1^2\sigma_1^2 +c_2^2\sigma_2^2} +\sigma_2^2z^2 -2c_1\sigma_2^2z\mu_1 +c_1^2\sigma_2^2\mu_1^2 +c_1^2\sigma_1^2\mu_2^2 \right] \\&&\;\cdots\;A=\frac{ \left( c_2\sigma_2^2z -c_1c_2\sigma_2^2\mu_1 +c_1^2\sigma_1^2\mu_2 \right)^2 } { c_1^2\sigma_1^2 +c_2^2\sigma_2^2 } \\&=&-\frac{1}{2c_1^2\sigma_1^2\sigma_2^2} \left\{ \left( c_1^2\sigma_1^2 +c_2^2\sigma_2^2 \right) \left( y^2 -\frac{ c_2\sigma_2^2z -c_1c_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} \right)^2 -\frac{ \left( c_2\sigma_2^2z -c_1c_2\sigma_2^2\mu_1 +c_1^2\sigma_1^2\mu_2 \right)^2 }{c_1^2\sigma_1^2 +c_2^2\sigma_2^2} +\sigma_2^2z^2 -2c_1\sigma_2^2z\mu_1 +c_1^2\sigma_2^2\mu_1^2 +c_1^2\sigma_1^2\mu_2^2 \right\} \\&=& -\frac{\left( c_1^2\sigma_1^2 +c_2^2\sigma_2^2 \right)}{2c_1^2\sigma_1^2\sigma_2^2} \left( y^2 -\frac{ c_2\sigma_2^2z -c_1c_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} \right)^2 -\frac{1}{2c_1^2\sigma_1^2\sigma_2^2} \left\{ -\frac{ \left( c_2\sigma_2^2z -c_1c_2\sigma_2^2\mu_1 +c_1^2\sigma_1^2\mu_2 \right)^2 }{c_1^2\sigma_1^2 +c_2^2\sigma_2^2} +\sigma_2^2z^2 -2c_1\sigma_2^2z\mu_1 +c_1^2\sigma_2^2\mu_1^2 +c_1^2\sigma_1^2\mu_2^2 \right\} \\&=& -f_2(c_1,c_2,\sigma_1^2,\sigma_2^2) \left\{ y-f_3(c_1,c_2,z,\mu_1,\mu_2,\sigma_1^2,\sigma_2^2) \right\}^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はyを引数としない. \\&=& -f_2\left(y-f_3\right)^2-f_4 \;\cdots\;yが平方完成の中の一つだけになっている. \\ \\f_2(c_1,c_2,\sigma_1^2,\sigma_2^2)&=& \frac{\left( c_1^2\sigma_1^2 +c_2^2\sigma_2^2 \right)}{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_2^2z -c_1c_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} \left\{ -\frac{ \left( c_2\sigma_2^2z -c_1c_2\sigma_2^2\mu_1 +c_1^2\sigma_1^2\mu_2 \right)^2 }{c_1^2\sigma_1^2 +c_2^2\sigma_2^2} +\sigma_2^2z^2 -2c_1\sigma_2^2z\mu_1 +c_1^2\sigma_2^2\mu_1^2 +c_1^2\sigma_1^2\mu_2^2 \right\} \\&=&\frac{1}{2c_1^2\sigma_1^2\sigma_2^2} \left\{ \frac{ -\left( c_2\sigma_2^2 \left(z -c_1\mu_1 \right) +c_1^2\sigma_1^2\mu_2 \right)^2 +\left(c_1^2\sigma_1^2+c_2^2\sigma_2^2\right) \left(\sigma_2^2\left( z-c_1\mu_1 \right)^2 +c_1^2\sigma_1^2\mu_2^2\right) }{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{ -\left( c_2\sigma_2^2 B +c_1^2\sigma_1^2\mu_2 \right)^2 +\left(c_1^2\sigma_1^2+c_2^2\sigma_2^2\right) \left(\sigma_2^2B^2 +c_1^2\sigma_1^2\mu_2^2\right) }{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)} \left\{ -\left( c_2\sigma_2^2 B +c_1^2\sigma_1^2\mu_2 \right)^2 +\left(c_1^2\sigma_1^2+c_2^2\sigma_2^2\right) \left(\sigma_2^2B^2 +c_1^2\sigma_1^2\mu_2^2\right) \right\} \\&=&\frac{1}{2c_1^2\sigma_1^2\sigma_2^2(c_1^2\sigma_1^2 +c_2^2\sigma_2^2)} \left( \color{red}{-c_2^2\sigma_2^4B^2} \color{black}{-2c_1^2c_2\sigma_1^2\sigma_2^2\mu_2B} \color{green}{-c_1^4\sigma_1^4\mu_2^2} \color{black}{+c_1^2\sigma_1^2\sigma_2^2B^2} \color{red}{+c_2^2\sigma_2^4B^2} \color{green}{+c_1^4\sigma_1^4\mu_2^2} \color{black}{+c_1^2c_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\left\{ -2 c_2 \mu_2 B + B^2 + c_2^2 \mu_2^2 \right\} \\&=&\frac{1}{2(c_1^2\sigma_1^2+c_2^2\sigma_2^2)}\left(B-c_2\mu_2\right)^2 \\&=&\frac{\left(z-c_1\mu_1-c_2\mu_2\right)^2}{2(c_1^2\sigma_1^2+c_2^2\sigma_2^2)} \;\cdots\;B=z-c_1\mu_1 \\&=&\frac{\left(z-\left(c_1\mu_1+c_2\mu_2\right)\right)^2}{2(c_1^2\sigma_1^2+c_2^2\sigma_2^2)} \end{eqnarray} $$

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$$ \begin{eqnarray} p_{c_1X+c_2Y}(z)&=&\frac{1}{\left|c_1\right|}\frac{1}{2\pi\sqrt{\sigma_1^2\sigma_2^2}} \int_{-\infty}^{\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_{-\infty}^{\infty} \mathrm{e}^{-f_2\left(y-f_3\right)^2-f_4} \mathrm{d}y \\&=&\frac{1}{\left|c_1\right|}\frac{1}{2\pi\sqrt{\sigma_1^2\sigma_2^2}} \int_{-\infty}^{\infty} \mathrm{e}^{-f_2\left(y-f_3\right)^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_{-\infty}^{\infty} \mathrm{e}^{-f_2\left(y-f_3\right)^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_{-\infty}^{\infty} \mathrm{e}^{-u^2} \frac{1}{\sqrt{f_2}}\mathrm{d}u \\&&\;\cdots\;u=\sqrt{f_2}\left(y-f_3\right),\;\frac{\mathrm{d}u}{\mathrm{d}y}=\sqrt{f_2},\;\mathrm{d}y=\frac{1}{\sqrt{f_2}}\mathrm{d}u \\&&\;\cdots\;y:-\infty \rightarrow \infty,\;u:-\infty \rightarrow \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_{-\infty}^{\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\;\href{https://shikitenkai.blogspot.com/2019/06/gaussian-integral.html}{\int_{-\infty}^{\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{\left(c_1^2\sigma_1^2+c_2^2\sigma_2^2\right)}{2c_1^2\sigma_1^2\sigma_2^2}}} \mathrm{e}^{-\frac{\left(z-\left(c_1\mu_1+c_2\mu_2\right)\right)^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}{\left(c_1^2\sigma_1^2+c_2^2\sigma_2^2\right)}} \mathrm{e}^{-\frac{\left(z-\left(c_1\mu_1+c_2\mu_2\right)\right)^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\left(c_1^2\sigma_1^2+c_2^2\sigma_2^2\right)}} \mathrm{e}^{-\frac{\left(z-\left(c_1\mu_1+c_2\mu_2\right)\right)^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\left(c_1^2\sigma_1^2+c_2^2\sigma_2^2\right)}} \mathrm{e}^{-\frac{\left(z-\left(c_1\mu_1+c_2\mu_2\right)\right)^2}{2(c_1^2\sigma_1^2+c_2^2\sigma_2^2)}} \;\cdots\;\sqrt{A^2}= \left\{ \begin{array} \\A&(A\geq0) \\-A&(A\lt0) \end{array} \right. =\left|A\right| \\&=&\frac{1}{\sqrt{2\pi\left(c_1^2\sigma_1^2+c_2^2\sigma_2^2\right)}} \mathrm{e}^{-\frac{\left(z-\left(c_1\mu_1+c_2\mu_2\right)\right)^2}{2(c_1^2\sigma_1^2+c_2^2\sigma_2^2)}} \\&\sim&\mathrm{N}\left(c_1\mu_1+c_2\mu_2,\;c_1^2\sigma_1^2+c_2^2\sigma_2^2\right) \end{eqnarray} $$