\section{附录 A: 二元正态分布}%Bivariate Normal Distribution
%If the joint density function of two random variables has the form
如果随机变量 和 具有二元标准正态分布,则 的联合密度函数为:
% and are said to have a bivariate standard normal distribution. The marginal distributions are standard normal: and Normal . The correlation between and is given by The basic rule for conditional densities is that Since
它们的边际分布是标准正态分布: 和 。二者的相关性由参数 决定。
我们通常更为关心条件密度函数:。由于
\beginequation*}\label{eq:limdep-Winkel-7-77}
f\left(u_{2}\right)=\frac{1}{\sqrt{2 \pi}} \exp \left{-\frac{1}{2} u_{2}^{2}\right}
\end{equation*}
再做一些简化处理,可知 可以写成
%and using some simplifications, we find that the ratio can be written as
%In other words, has a normal distribution as well, and . The conditional expectation function of the standard bivariate normal distribution is given by
可见,$u_{1}u_{1} | u_{2} \sim N\left(\rho u_{2}, 1-\rho^{2}\right)z_{1} \sim N\left(m_{1}, \sigma_{1}^{2}\right)z_{2} \sim N \left(m_{2}, \sigma_{2}^{2}\right)\left(z_{1}, z_{2}\right) \sim Bivariate Normal(\left.m_{1}, m_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}, \rho\right)z_{1} \sim N\left(m_{1}, \sigma_{1}^{2}\right)z_{2} \sim N \left(m_{2}, \sigma_{2}^{2}\right)\left(z_{1}, z_{2}\right) \sim N(\left.m_{1}, m_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}, \rho\right)\mathrm{E}\left(z_{1} \mid z_{2}\right)=\alpha+\beta z_{2}\beta=\sigma_{12} / \sigma_{2}^{2}\alpha=m_{1}-\beta m_{2}$. Thus, the conditional expectation function of the bivariate normal distribution is a linear function, or a "linear regression".