Aritalab:Lecture/Basic/Distribution/Normal

正規分布の確率密度関数は

$\mbox{Pr}(X=k) = \frac{1}{\sqrt{2\pi}\sigma} \exp\Big( -\frac{(x-\mu)^2}{2\sigma^2} \Big)$

で与えられる。確率母関数は

$\begin{align}\textstyle G(z) &= \int \frac{1}{\sqrt{2\pi}\sigma} \exp\Big( -\frac{(x-\mu)^2}{2\sigma^2} \Big) z^x dx \\ &= \int \frac{1}{\sqrt{2\pi}\sigma} \exp\Big( -\frac{(x-\mu)^2}{2\sigma^2} \Big) \exp(x\log z) dx \end{align}$

ここで$u = (x-\mu)/\sigma$とおくと$dx = \sigma du$

$\begin{align}\textstyle G(z) &= \int \frac{1}{\sqrt{2\pi}} \exp\Big( (\log z)(\mu + \sigma u) \Big) \exp\Big( -\frac{u^2}{2}\Big) \sigma du\\ &= e^{\mu\log z} \int \frac{1}{\sqrt{2\pi}} \exp\Big( (\log z)\sigma u - \frac{u^2}{2} \Big) du\\ &= e^{\mu\log z + \frac{\sigma^2 (\log z)^2}{2} } \int \frac{1}{\sqrt{2\pi}} \exp\Big( - \frac{(u - \sigma \log z)^2}{2} \Big) du\\ &= z^{\mu} \exp{(\sigma^2 (\log z)^2 / 2 )} \end{align}$

これより

$\begin{align} G'(1) &= \mu \\ G''(1) &= \end{align}$