# 正規分布

\begin{align*}
f(x;\mu, \sigma^2)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} \exp \left[-\frac{(x-\mu)^{2}}{2 \sigma^{2}}\right] \quad(-\infty<x<\infty)
\end{align*}

そして、次の性質がある。

\begin{align*}
&E[X] = \mu \\
&V[X] = \sigma^2 \\
&M_{X}(t) = \exp \left[\mu t+\frac{\sigma^{2}}{2} t^{2}\right]
\end{align*}

\begin{align*}
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\end{align*}

## 積率母関数

\begin{align*}
&M_{X}(t) = \exp \left[\mu t+\frac{\sigma^{2}}{2} t^{2}\right]
\end{align*}

確率変数 $Z \sim \mathcal{N}(0, 1)$ を考えて、

\begin{align*}
M_{Z}(t) &= E[e^{tZ}] \n
& = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{t z} \cdot \exp \left[-\frac{z^{2}}{2}\right] d z \n
& = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \exp \left[-\frac{1}{2}\left(z^{2}-2 t z\right)\right] d z \n
& = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \exp $-\f{1}{2} \{(z – t)^2 – t^2 \}$ dz \n
&= \exp$\f{t^2}{2}$ \cdot \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \exp $-\f{(z – t)^2}{2}$ dz \n
&= \exp $\f{t^2}{2}$.
\end{align*}

$X = \sigma Z + \mu$ であるから、

\begin{align*}
M_X(t) &= E[e^{tX}] \n
&= E[e^{t(\sigma Z + \mu)}] \n
&= E[e^{t \mu}] \cdot E[e^{t\sigma Z}] \n
&= \exp$\mu t + \f{\sigma^2}{2}t^2$.
\end{align*}

## 期待値

\begin{align*}
E[X] = \mu
\end{align*}

確率変数 $Z \sim \mathcal{N}(0, 1)$ を考えて、$M_Z(t) = e^{\f{t^2}{2}}$ より、

\begin{align*}
E[Z] &= \left. \d{}{t} M_Z(t) \right|_{t=0} \n
&= \left. t e^{\f{t^2}{2}} \right|_{t=0} \n
&= 0.
\end{align*}

$X = \sigma Z + \mu$ であるから、

\begin{align*}
E[X] &= \sigma E[Z] + \mu \n
&= \mu.
\end{align*}

## 分散

\begin{align*}
V[X] = \sigma^2
\end{align*}

確率変数 $Z \sim \mathcal{N}(0, 1)$ を考えて、$M_Z(t) = e^{\f{t^2}{2}}$ より、

\begin{align*}
E[Z^2] &= \left. \f{{\rm d}^2}{{\rm d}t^2} M_Z(t) \right|_{t=0} \n
&= \left. \d{}{t} $t e^{\f{t^2}{2}}$ \right|_{t=0} \n
&= \left. $e^{\f{t^2}{2}} + t^2 e^{\f{t^2}{2}}$ \right|_{t=0} \n
&= 1.
\end{align*}

よって、

\begin{align*}
V[Z] &= E[Z^2] – E[Z]^2 \n
&= 1.
\end{align*}

$X = \sigma Z + \mu$ であるから、

\begin{align*}
V[X] &= V[\sigma Z + \mu] \n
&= \sigma^2 V[Z] \n
&= \sigma^2.
\end{align*}

Yukkuri Machine Learning
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