正态分布_高斯分布_Normal_Distribution-Gaussian_Distribution

Last updated Sep 16, 2021 Edit Source

• All
• Math statistics

Tags: #Math/Statistics

正态分布, 概率密度函数: $$f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{\Large -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}$$ or $$f(x)=\frac{1}{\sigma \sqrt{2 \pi}} \exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right)$$

• Mean $(\mu)$ and standard deviation $(\sigma)$ $$ \begin{aligned} &\mu=E(X)=\int_{-\infty}^{\infty} x p(x) d x \\ &\sigma^{2}=E\left{(X-\mu)^{2}\right}=\int_{-\infty}^{\infty}(x-\mu)^{2} p(x) d x \end{aligned} $$

• Probability within any particular number of standard deviations of $\mu$ $$ \begin{aligned} p{\mu-k \sigma \leq x \leq \mu+k \sigma} &=\int_{\mu-k \sigma}^{\mu+k \sigma} \frac{1}{\sqrt{2 \pi} \sigma} \exp \left[-\frac{(x-\mu)^{2}}{2 \sigma^{2}}\right] d x \\ &=\frac{1}{\sqrt{2 \pi}} \int_{-k}^{k} \exp \left[-\frac{y^{2}}{2}\right] d y \end{aligned} $$

• 线性变换后: 如果 $X \sim N\left(\mu,\sigma^{2}\right)$ 且 $a, b$是实数, 那么 $$a X+b \sim N\left(a \mu+b,(a \sigma)^{2}\right)$$

• 正态分布的和还是正态分布¹ $$\begin{aligned} &X \sim N\left(\mu_{X}, \sigma_{X}^{2}\right) \\ &Y \sim N\left(\mu_{Y}, \sigma_{Y}^{2}\right) \\ &Z=X+Y \end{aligned}$$ 则 $$Z \sim N\left(\mu_{X}+\mu_{Y}, \sigma_{X}^{2}+\sigma_{Y}^{2}\right)$$

• 注意$\sigma$在根号外面
• 指数是负的($x=\mu$的时候等于0, 同时取得最大值)

拉普拉斯分布与高斯分布的联系_Relation_of_Laplace_distribution _and_Gaussian_distribution

$$\begin{aligned} &p{x}=\frac{1}{(\sqrt{2 \pi})^{n / 2}|C|^{1 / 2}} \exp \left[-\frac{1}{2}(x-\mu)^{T} C^{-1}(x-\mu)\right] \\ &x=\left[\begin{array}{c} x_{1} \\ \cdots \\ x_{n} \end{array}\right] \quad \mu=\left[\begin{array}{c} \mu_{1} \\ \cdots \\ \mu_{n} \end{array}\right] \quad C=\left[\begin{array}{ccc} \sigma_{11}^{2} & \ldots & \sigma_{1 n}^{2} \\ \cdots & & \ldots \\ \sigma_{m 1}^{2} & \ldots & \sigma_{m n}^{2} \end{array}\right] \end{aligned}$$