# Stata：缺失值与多重补漏-misstable-D204

\section{附录 A: 二元正态分布}%Bivariate Normal Distribution %If the joint density function of two random variables $\left({u}_{1},{u}_{2}\right)$ has the form 如果随机变量 ${u}_{1}$ 和 ${u}_{2}$ 具有二元标准正态分布，则 $\left({u}_{1},{u}_{2}\right)$ 的联合密度函数为：

%${u}_{1}$ and ${u}_{2}$ are said to have a bivariate standard normal distribution. The marginal distributions are standard normal: ${u}_{1}\sim \mathrm{N}\left(0,1\right)$ and ${u}_{2}\sim$ Normal $\left(0,1\right)$. The correlation between ${u}_{1}$ and ${u}_{2}$ is given by $\rho .$ The basic rule for conditional densities is that $f\left({u}_{1}\mid {u}_{2}\right)=f\left({u}_{1},{u}_{2}\right)/f\left({u}_{2}\right).$ Since 它们的边际分布是标准正态分布：${u}_{1}\sim \mathrm{N}\left(0,1\right)$ 和 ${u}_{2}\sim N\left(0,1\right)$。二者的相关性由参数 $\rho$ 决定。

%In other words, ${u}_{1}\mid {u}_{2}$ has a normal distribution as well, and ${u}_{1}\mid {u}_{2}\sim \mathrm{N}\left(\rho {u}_{2},1-{\rho }^{2}\right)$. The conditional expectation function of the standard bivariate normal distribution is given by 可见，u_{1}u_{2$也服从正态分布，即$u_{1} | u_{2} \sim N\left(\rho u_{2}, 1-\rho^{2}\right)\text{。由此可得标准二元正态分布的的条件期望函数为： $$\label{eq:limdep-Winkel-7-79} \mathrm{E}\left(u_{1} \mid u_{2}\right)=\rho u_{2}$$ %The results can be generalized to allow for marginal normal distributions with non-zero means and non-unit variances. Let 进一步考虑“非零均值”和“非单位方差”的一般化情形： \label{eq:limdep-Winkel-7-80} \begin{aligned} &z_{1}=m_{1}+\sigma_{1} u_{1} \ &z_{2}=m_{2}+\sigma_{2} u_{2} \end{aligned} %so that}z_{1} \sim N\left(m_{1}, \sigma_{1}^{2}\right)$and$z_{2} \sim N \left(m_{2}, \sigma_{2}^{2}\right)$,and$\left(z_{1}, z_{2}\right) \sim Bivariate Normal(\left.m_{1}, m_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}, \rho\right)$denotesthegeneralbivariatenormaldistribution.Fortheconditionalexpectation,weobtain简记为$z_{1} \sim N\left(m_{1}, \sigma_{1}^{2}\right)$，$z_{2} \sim N \left(m_{2}, \sigma_{2}^{2}\right)$，以及$\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}$,where$\beta=\sigma_{12} / \sigma_{2}^{2}$and$\alpha=m_{1}-\beta m_{2}. Thus, the conditional expectation function of the bivariate normal distribution is a linear function, or a "linear regression".