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--- science:formal:statistics:pdf-convolution [2025/09/28 01:56] – falsycat
+++ science:formal:statistics:pdf-convolution [2025/09/30 04:49] (current) – falsycat
@@ Line 27: / Line 27: @@
 ^$f_x\left(x\right)$^$f_y\left(y\right)$^$f_z\left(z\right)$^ ^
-|正規分布: $\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{\left(x-\mu\right)^2}{2\sigma^2}\right)$|指数分布: $\lambda \exp\left(-\lambda x\right)$|$\lambda \exp\!\left( \lambda\mu + \frac{1}{2}\lambda^2\sigma^2 - \lambda z \right)\cdot \tfrac{1}{2}\, \mathrm{erfc}\!\left( \frac{\mu + \lambda\sigma^2 - z}{\sqrt{2}\,\sigma} \right)$|[[#正規分布-指数分布|導出手順]]|
+|正規分布: $\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{\left(x-\mu\right)^2}{2\sigma^2}\right)$|指数分布: $\lambda \exp\left(-\lambda x\right)$|ExGaussian: $\lambda \exp\!\left( \lambda\mu + \frac{1}{2}\lambda^2\sigma^2 - \lambda z \right)\cdot \tfrac{1}{2}\, \mathrm{erfc}\!\left( \frac{\mu + \lambda\sigma^2 - z}{\sqrt{2}\,\sigma} \right)$|[[#正規分布-指数分布|導出手順]]|
 ==== 1. 正規分布+指数分布 ====
@@ Line 65: / Line 65: @@
 $$
-以上より、$f_Z\left(z\right)$は誤差関数を用いて以下のように表せる。
+以上より、$f_Z\left(z\right)$は誤差関数$\mathrm{erfc}\left(x\right)$あるいは正規分布の累積分布関数$\Phi\left(x\right)$を用いて以下のように表せる。
+この確率密度関数で表される分布は、ExGaussian分布と呼ばれる。
 $$
 \begin{split}
 f_Z\left(z\right) &= \frac{\lambda}{2}\exp\left(-\lambda z + \frac{\alpha^2-\mu^2}{2\sigma^2}\right) \mathrm{erfc}\left(\frac{\alpha-z}{\sqrt{2}\sigma}\right) \\
-&= \lambda\exp\left(\lambda \mu+\frac{\lambda^2\sigma^2}{2}-\lambda z\right) \cdot \frac{1}{2}\mathrm{erfc}\left(\frac{\mu+\sigma^2\lambda-z}{\sqrt{2}\sigma}\right)
+&= \lambda\exp\left(\lambda \mu+\frac{\lambda^2\sigma^2}{2}-\lambda z\right) \cdot \frac{1}{2}\mathrm{erfc}\left(\frac{\mu+\sigma^2\lambda-z}{\sqrt{2}\sigma}\right) \\
+&= \lambda\exp\left(\lambda \mu+\frac{\lambda^2\sigma^2}{2}-\lambda z\right) \cdot \Phi\left(\frac{z-\mu}{\sigma}-\lambda \sigma\right)
 \end{split}
 $$