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