Gaussian Distribution

又称为 Normal Distribution，其中均值为0，标准差为1($\mathcal{N}(x|0,1)$)的正太分布又称为 Standard Normal Distribution

Univariate Formular

$\hspace{4mm}\mathcal{N}(x|\mu, \sigma^2) = \frac{1}{(2\pi\sigma^2)^{1/2}}exp\{-\frac{1}{2\sigma^2}(x-\mu)^2\}$

Multivariate Formular

当其为n维时，其形式如下:
$\hspace{4mm}\mathcal{N}(\boldsymbol{x|\mu, |\Sigma|^{-1}}) = \frac{1}{(2\pi)^{n/2}|\Sigma|^{1/2}}exp\{-\frac{1}{2}\boldsymbol{(x-\mu)^T\Sigma^{-1}(x-\mu)}\}$

Properties

$1. \hspace{4mm} \int_{-\infty}^{\infty} \mathcal{N}(x|\mu, \sigma^2)dx=1 \\ 2. \hspace{4mm} E[x] = \int_{-\infty}^{\infty} \mathcal{N}(x|\mu, \sigma^2)xdx = \mu \\ 3. \hspace{4mm} E[x^2] = \int_{-\infty}^{\infty} \mathcal{N}(x|\mu, \sigma^2)x^2dx = \mu^2 + \sigma^2 \\ 4. \hspace{4mm} var[x] = E[x^2] - E[x]^2 = \sigma^2$