二元正态分布

一元正态分布X\sim N(\mu,\sigma^2)

密度函数f(x)=-\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}

分布函数\Phi(x)=\int_{-\infty}^x\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(t-\mu)^2}{2\sigma^2}}dt

二元正态分布(X_1,X_2)\sim N(\mu_1,\mu_2,\sigma_1^2,\sigma_2^2,\rho)

令\Sigma=\left[ \begin{array}{cccc} ~~\sigma_1^2~~~~~\rho \sigma_1\sigma_2~~\\ \rho\sigma_1\sigma_2~~~~~\sigma_2^2 \end{array} \right] ，\mu=(\mu_1,\mu_2)^T

则\Sigma^{-1}=\frac{1}{(1-\rho^2)}\left[ \begin{array}{cccc} ~~~\frac{1}{\sigma_1^2}~~~\frac{\rho}{\sigma_1\sigma_2}~\\ ~\frac{\rho}{\sigma_1\sigma_2}~~~\frac{1}{\sigma_1^2}~ \end{array} \right]

其中-\frac{1}{2(1-\rho^2)}\left[ \left(\frac{x_1-\mu_1}{\sigma_1}\right)^2 -2\frac{(x_1-\mu_1)(x_2-\mu_2)}{\sigma_1\sigma_2} +\left(\frac{x_2-\mu_2}{\sigma_2}\right)^2 \right {-\frac{1}{2(1-\rho^2)}\left[ \left(\frac{x_1-\mu_1}{\sigma_1}\right)^2 -2\rho\frac{(x_1-\mu_1)(x_2-\mu_2)}{\sigma_1\sigma_2} +\left(\frac{x_2-\mu_2}{\sigma_2}\right)^2 \right]}

= -\frac{1}{2}\left(\frac{x_1-\mu_1}{\sigma_1}\right)^2 -\frac{1}{2(1-\rho^2)}\left[\frac{x_2-\mu_2}{\sigma_2}-\rho\frac{x_1-\mu_1}{\sigma_1}\right]^2

=-\frac{1}{2}\left(\frac{x_2-\mu_2}{\sigma_2}\right)^2 -\frac{1}{2(1-\rho^2)}\left[\frac{x_1-\mu_1}{\sigma_1}-\rho\frac{x_2-\mu_2}{\sigma_2}\right]^2

密度函数f(x,y)=\frac{1}{(2\pi)^\frac{2}{2}\sigma_1\sigma_2}e^{-\frac{1}{2(1-\rho^2)}\left[ \left(\frac{x_1-\mu_1}{\sigma_1}\right)^2 -2\rho\frac{(x_1-\mu_1)(x_2-\mu_2)}{\sigma_1\sigma_2} +\left(\frac{x_2-\mu_2}{\sigma_2}\right)^2 \right]}

分布函数为\Phi(X,Y)=\int_{-\infty}^X\int_{-\infty}^Yf(x,y)dxdy