# 多变量正态分布

參數 機率多变量函數 Many samples from a multivariate (bivariate) Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.878, 0.478) direction (longer vector) and of 1 in the second direction (shorter vector, orthogonal to the longer vector). 累積分布函數 μ ∈ Rk — location Σ ∈ Rk×k — covariance (nonnegative-definite matrix) x ∈ μ+span(Σ) ⊆ Rk $(2\pi)^{-\frac{k}{2}}|\boldsymbol\Sigma|^{-\frac{1}{2}}\, e^{ -\frac{1}{2}(\mathbf{x}-\boldsymbol\mu)'\boldsymbol\Sigma^{-1}(\mathbf{x}-\boldsymbol\mu) },$ exists only when Σ is positive-definite (no analytic expression) $\mathcal{N}(\boldsymbol\mu,\,\boldsymbol\Sigma)$ μ μ Σ $\frac{1}{2}\ln((2\pi e)^k |\boldsymbol\Sigma|)$ $\exp\!\Big( \boldsymbol\mu'\mathbf{t} + \tfrac{1}{2} \mathbf{t}'\boldsymbol\Sigma \mathbf{t}\Big)$ $\exp\!\Big( i\boldsymbol\mu'\mathbf{t} - \tfrac{1}{2} \mathbf{t}'\boldsymbol\Sigma \mathbf{t}\Big)$

## 一般形式

• 任何线性组合$\ Y = a_1 X_1 + \cdots + a_N X_N$ 服从正态分布
• 存在随机向量 $\ Z = [Z_1, \dots, Z_M]^T$（ 它的每个元素服从独立标准正态分布），向量 $\ \mu = [\mu_1, \dots, \mu_N]^T$$N \times M$ 矩阵 $\ A$ 满足 $\ X = A Z + \mu$.
• 存在 $\mu$ 和一个对称正定阵 $\ \Sigma$ 满足$\ X$特征函数
$\phi_X\left(u;\mu,\Sigma\right) = \exp \left( i \mu^T u - \frac{1}{2} u^T \Sigma u \right).$

$f_X(x_1, \dots, x_N) = \frac {1} {(2\pi)^{N/2}|\Sigma|^{1/2}} \exp \left( -\frac{1}{2} ( x - \mu)^T \Sigma^{-1} (x - \mu) \right)$