 |
途中計算は、こちら。 \[ \begin{align*} g\left(z\right) &= \int_{- \infty}^{\infty} \frac{1}{\sqrt{2\pi{\sigma_1}^2}} \exp\left[- \frac{\left(z-w-\mu_1\right)^2}{2{\sigma_1}^2} \right] \frac{1}{\sqrt{2\pi{\sigma_2}^2}} \exp \left[- \frac{\left(w-\mu_2\right)^2}{2{\sigma_2}^2} \right] dw \\ &= \frac{1}{2\pi\sigma_1 \sigma_2} \int_{- \infty}^{\infty} \exp\left[- \frac{\left(z-w-\mu_1\right)^2}{2{\sigma_1}^2} - \frac{\left(w-\mu_2\right)^2}{2{\sigma_2}^2} \right] dw \end{align*} \] ここで、指数部分は、 \[ \begin{align*} &-\frac{1}{2}\left[\frac{\left(z-w-\mu_1\right)^2}{{\sigma_1}^2} + \frac{\left(w-\mu_2\right)^2}{{\sigma_2}^2}\right] \\ &=-\frac{1}{2}\left[\frac{w^2-2\left(z-\mu_1\right)w+\left(z-\mu_1\right)^2}{{\sigma_1}^2} + \frac{w^2-2\mu_2w+{\mu_2}^2}{{\sigma_2}^2}\right] \\ &=-\frac{1}{2{\sigma_1}^2{\sigma_2}^2}\left[w^2\left({\sigma_1}^2+{\sigma_2}^2\right)-2\left(z{\sigma_2}^2-\mu_1{\sigma_2}^2+\mu_2{\sigma_1}^2\right)w+\left(z-\mu_1\right)^2{\sigma_2}^2 + {\mu_2}^2{\sigma_1}^2\right] \\ &=-\frac{{\sigma_1}^2+{\sigma_2}^2}{2{\sigma_1}^2{\sigma_2}^2}\left[w^2-2\frac{z{\sigma_2}^2-\mu_1{\sigma_2}^2+\mu_2{\sigma_1}^2}{{\sigma_1}^2+{\sigma_2}^2}w+\frac{\left(z-\mu_1\right)^2{\sigma_2}^2 + {\mu_2}^2{\sigma_1}^2}{{\sigma_1}^2+{\sigma_2}^2}\right] \\ &=-\frac{{\sigma_1}^2+{\sigma_2}^2}{2{\sigma_1}^2{\sigma_2}^2}\left[\left(w-\frac{z{\sigma_2}^2-\mu_1{\sigma_2}^2+\mu_2{\sigma_1}^2}{{\sigma_1}^2+{\sigma_2}^2}\right)^2-\left(\frac{z{\sigma_2}^2-\mu_1{\sigma_2}^2+\mu_2{\sigma_1}^2}{{\sigma_1}^2+{\sigma_2}^2}\right)^2+\frac{\left(z-\mu_1\right)^2{\sigma_2}^2 + {\mu_2}^2{\sigma_1}^2}{{\sigma_1}^2+{\sigma_2}^2}\right] \end{align*} \] 更に、右辺の第2項、第3項について、 \[ \begin{align*} &-\left(\frac{z{\sigma_2}^2-\mu_1{\sigma_2}^2+\mu_2{\sigma_1}^2}{{\sigma_1}^2+{\sigma_2}^2}\right)^2+\frac{\left(z-\mu_1\right)^2{\sigma_2}^2 + {\mu_2}^2{\sigma_1}^2}{{\sigma_1}^2+{\sigma_2}^2} \\ &=\frac{-\biggl[\left(z-\mu_1\right){\sigma_2}^2+\mu_2{\sigma_1}^2\biggr]^2+\biggl[\left(z-\mu_1\right)^2{\sigma_2}^2 + {\mu_2}^2{\sigma_1}^2\biggr]\left({\sigma_1}^2+{\sigma_2}^2\right)}{\left({\sigma_1}^2+{\sigma_2}^2\right)^2} \\ &=\frac{-\left(z-\mu_1\right)^2{\sigma_2}^4-2\left(z-\mu_1\right)\mu_2{\sigma_1}^2{\sigma_2}^2-{\mu_2}^2{\sigma_1}^4+\left(z-\mu_1\right)^2{\sigma_1}^2{\sigma_2}^2+{\mu_2}^2{\sigma_1}^4+\left(z-\mu_1\right)^2{\sigma_2}^4+{\mu_2}^2{\sigma_1}^2{\sigma_2}^2}{\left({\sigma_1}^2+{\sigma_2}^2\right)^2} \\ &=\frac{-2\left(z-\mu_1\right)\mu_2{\sigma_1}^2{\sigma_2}^2+\left(z-\mu_1\right)^2{\sigma_1}^2{\sigma_2}^2+{\mu_2}^2{\sigma_1}^2{\sigma_2}^2}{\left({\sigma_1}^2+{\sigma_2}^2\right)^2} \\ &=\frac{{\sigma_1}^2{\sigma_2}^2}{\left({\sigma_1}^2+{\sigma_2}^2\right)^2}\left[\left(z-\mu_1\right)^2-2\left(z-\mu_1\right)\mu_2+{\mu_2}^2\right] = \frac{{\sigma_1}^2{\sigma_2}^2}{\left({\sigma_1}^2+{\sigma_2}^2\right)^2}\left(z-\mu_1-\mu_2\right)^2 \end{align*} \] よって、指数部分は、 \[ \begin{align*} &-\frac{{\sigma_1}^2+{\sigma_2}^2}{2{\sigma_1}^2{\sigma_2}^2}\left[\left(w-\frac{z{\sigma_2}^2-\mu_1{\sigma_2}^2+\mu_2{\sigma_1}^2}{{\sigma_1}^2+{\sigma_2}^2}\right)^2+\frac{{\sigma_1}^2{\sigma_2}^2}{\left({\sigma_1}^2+{\sigma_2}^2\right)^2}\left(z-\mu_1-\mu_2\right)^2\right] \\ &=-\frac{{\sigma_1}^2+{\sigma_2}^2}{2{\sigma_1}^2{\sigma_2}^2}\left(w-\frac{z{\sigma_2}^2-\mu_1{\sigma_2}^2+\mu_2{\sigma_1}^2}{{\sigma_1}^2+{\sigma_2}^2}\right)^2-\frac{\left(z-\mu_1-\mu_2\right)^2}{2\left({\sigma_1}^2+{\sigma_2}^2\right)} \end{align*} \] となるので、 \[ \begin{align*} g\left(z\right) &=\frac{1}{2\pi\sigma_1 \sigma_2} \int_{- \infty}^{\infty} \exp\left[-\frac{{\sigma_1}^2+{\sigma_2}^2}{2{\sigma_1}^2{\sigma_2}^2}\left(w-\frac{z{\sigma_2}^2-\mu_1{\sigma_2}^2+\mu_2{\sigma_1}^2}{{\sigma_1}^2+{\sigma_2}^2}\right)^2-\frac{\left(z-\mu_1-\mu_2\right)^2}{2\left({\sigma_1}^2+{\sigma_2}^2\right)} \right] dw \\ &=\frac{1}{2\pi\sigma_1 \sigma_2} \exp\left[-\frac{\left(z-\mu_1-\mu_2\right)^2}{2\left({\sigma_1}^2+{\sigma_2}^2\right)} \right]\int_{- \infty}^{\infty} \exp\left[-\frac{{\sigma_1}^2+{\sigma_2}^2}{2{\sigma_1}^2{\sigma_2}^2}\left(w-\frac{z{\sigma_2}^2-\mu_1{\sigma_2}^2+\mu_2{\sigma_1}^2}{{\sigma_1}^2+{\sigma_2}^2}\right)^2 \right] dw \\ \end{align*} \] 積分の部分は、Gauss積分の公式が適用でき、 \[ \int_{- \infty}^{\infty} \exp\left[-\frac{{\sigma_1}^2+{\sigma_2}^2}{2{\sigma_1}^2{\sigma_2}^2}\left(w-\frac{z{\sigma_2}^2-\mu_1{\sigma_2}^2+\mu_2{\sigma_1}^2}{{\sigma_1}^2+{\sigma_2}^2}\right)^2 \right] dw =\sqrt{\frac{2\pi{\sigma_1}^2{\sigma_2}^2}{{\sigma_1}^2+{\sigma_2}^2}}=\sigma_1\sigma_2\sqrt{\frac{2\pi}{{\sigma_1}^2+{\sigma_2}^2}} \] よって、 \[ g\left(z\right) =\frac{1}{2\pi\sigma_1 \sigma_2} \exp\left[-\frac{\left(z-\mu_1-\mu_2\right)^2}{2\left({\sigma_1}^2+{\sigma_2}^2\right)} \right]\times\sigma_1\sigma_2\sqrt{\frac{2\pi}{{\sigma_1}^2+{\sigma_2}^2}}=\frac{1}{\sqrt{2\pi}\sqrt{{\sigma_1}^2+{\sigma_2}^2}} \exp\left[-\frac{\left(z-\mu_1-\mu_2\right)^2}{2\left({\sigma_1}^2+{\sigma_2}^2\right)} \right] \] |
| 戻る |