# 多変量正規分布
###### tags: `probability-theory`

互いに独立ではない１組の計測値の確率モデル。

## 2次元

### 確率密度関数

$$
\begin{align}
& f\left(x_1, x_2; \mu_1, \mu_2, \sigma_1^2, \sigma_2^2, \rho_{12}\right) \notag \\
= & \frac{1}{2\pi\sqrt{\sigma_1^2\sigma_2^2-\rho_{12}^2\sigma_1^2\sigma_2^2}}\exp\left\{-\frac{\left(\left(x_1-\mu_1\right)^2\sigma_2^2+\left(x_2-\mu_2\right)^2\sigma_1^2-2\left(x_1-\mu_1\right)\left(x_2-\mu_2\right)\rho_{12}\sigma_1\sigma_2\right)}{2\left(\sigma_1^2\sigma_2^2-\rho_{12}^2\sigma_1^2\sigma_2^2\right)}\right\} \notag \\
= & \frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho_{12}^2}}\exp\left\{-\frac{\left(\left(x_1-\mu_1\right)^2\sigma_2^2+\left(x_2-\mu_2\right)^2\sigma_1^2-2\left(x_1-\mu_1\right)\left(x_2-\mu_2\right)\rho_{12}\sigma_1\sigma_2\right)}{2\sigma_1^2\sigma_2^2\left(1-\rho_{12}^2\right)}\right\} \notag
\end{align}
$$

## 一般
### パラメータ

$$
\boldsymbol{\mu} = \left(
\begin{array}{c}
\mu_1 \\
\mu_2 \\
\vdots \\
\mu_p
\end{array}
\right)
$$

$$
\Sigma = \left(
\begin{array}{cccc}
\sigma_1^2 & \rho_{12}\sigma_1\sigma_2 & \cdots & \rho_{1p}\sigma_1\sigma_p \\
\rho_{21}\sigma_2\sigma_1 & \sigma_2^2 & \cdots & \rho_{2p}\sigma_2\sigma_p \\
\vdots & \vdots & \ddots & \vdots \\
\rho_{p1}\sigma_p\sigma_1 & \rho_{p2}\sigma_p\sigma_2 & \cdots & \sigma_p^2
\end{array}
\right)
$$

### 確率ベクトル

$$
\boldsymbol{X} = \left(\begin{array}{c}
X_1 \\
X_2 \\
\vdots \\
X_p
\end{array}\right) \sim N\left(\boldsymbol{\mu}, \Sigma\right)
$$

### 確率密度関数

$$
f\left(\boldsymbol{x}; \boldsymbol{\mu}, \Sigma\right)
= \frac{1}{\sqrt{\left|2\pi\Sigma\right|}}\exp\left\{-\frac{1}{2}\left(\boldsymbol{x}-\boldsymbol{\mu}\right)^\top\Sigma\left(\boldsymbol{x}-\boldsymbol{\mu}\right)\right\}
$$