Section04_二维均匀分布和二维正态分布

二维均匀分布

1. 定义 若 $f(x,y) = \begin{cases} \frac{1}{S_{\mathbb{G}}}, & (x,y)\in \mathbb{G} \\ 0, & \text{其他} \end{cases}$，则 $(X,Y)$ 为区域 $\mathbb{G}$ 上的均匀分布
2. 求概率 若 $\mathbb{D}\subset \mathbb{G}$，则 $$\begin{split} \mathbb{P}\{(X,Y)\in \mathbb{D}\} & = \iint\limits_{\mathbb{D}}\frac{1}{S_{\mathbb{G}}}\cdot d\sigma \\ & = \frac{S_{\mathbb{D}}}{S_{\mathbb{G}}} \quad \text{(面积之比)} \end{split}$$

二维正态分布 $(X,Y)\sim \mathcal{N}(\mu_{1}, \mu_{2}; \sigma^{2}_{1}, \sigma^{2}_{2}; \rho)$

$$\begin{split} f(x,y) &= \frac{1}{\sqrt{2\pi}\sigma_{1}\sqrt{2\pi}\sigma_{2}\sqrt{1-\rho^{2}}} \times\\ & \exp\bigg\{-\frac{1}{1-\rho^{2}}\bigg[\frac{(x-\mu_{1})^{2}}{2\sigma_{1}^{2}} + \frac{(y-\mu_{2})^{2}}{2\sigma^{2}_{2}} - \rho \frac{x-\mu_{1}}{\sigma_{1}}\frac{y-\mu_{2}}{\sigma_{2}}\bigg]\bigg\} \end{split}$$

• 其中 $\begin{cases} X\sim \mathcal{N}(\mu_{1},\sigma_{1}^{2}) \\ Y\sim \mathcal{N}(\mu_{2},\sigma_{2}^{2}) \\ \rho \text{为}X,Y \text{的相关系数} \end{cases}$
• 注
  • 若 $\rho = 0$，则 $f(x,y) = \frac{1}{\sqrt{2\pi}\sigma_{1}\sqrt{2\pi}\sigma_{2}}\times \exp[- \frac{(x-\mu_{1})^{2}}{2\sigma_{1}^{2}}-\frac{(y-\mu_{1})^{2}}{2\sigma_{2}^{2}}]=f_{X}(x)f_{Y}(y)$，
  • 故若 $X\sim \mathcal{N}(\mu_{1},\sigma_{1}^{2}), Y\sim \mathcal{N}(\mu_{2},\sigma_{2}^{2})$，且 $X,Y$ 独立，则 $(X,Y)\sim \mathcal{N}(\mu_{1},\mu_{2};\sigma_{1}^{2},\sigma_{2}^{2};0)$