Abstract: 本文介绍第一个多变量连续分布——双变量正态分布(本篇内有未证明定理，需要后续要补充 )
Keywords: The Bivariate Normal Distributions

# 二维正态分布

## 二维正态分布的定义和来源 Definition and Derivation of Bivariate Normal Distributions

Theorem Suppose that $Z_1$ and $Z_2$ are independent random variables,each of which has the standard normal distribution.Let $\mu_1,\mu_2,\sigma_1,\sigma_2$ ,and $\rho$ be constants such that $-\infty<\mu_i<\infty(i=1,2)$ , $\sigma_i>0(i=1,2)$ ,and $-1<\rho<1$ . Define two new random variables $X_1$ and $X_2$ as follows:
$$X_1=\sigma_1Z_1+\mu_1\\ X_2=\sigma_2[\rho Z_1+(1-\rho^2)^{\frac{1}{2}}Z_2]+\mu_2 \tag{5.10.1}$$
The joint p.d.f. of $X_1$ and $X_2$ is
$$f(x_1,x_2)=\frac{1}{2\pi(1-\rho^2)^{\frac{1}{2}}\sigma_1\sigma_2}e^{-\frac{1}{2(1-\rho^2)}[(\frac{x_1-\mu_1}{\sigma_1})^2-2\rho(\frac{x_1-\mu_1}{\sigma_1})(\frac{x_2-\mu_2}{\sigma_2})+(\frac{x_2-\mu_2}{\sigma_2})^2]} \tag{5.10.2}$$

Theorem Suppose that $X_1$ and $X_2$ have the joint distribution whose p.d.f. is given by Eq.(5.10.2) Then there exist independent standard normal random variables $Z_1$ and $Z_2$ such that Eqs (5.10.1) hold .Also,the mean of $X_i$ is $\mu_i$ and the variance of $X_i$ is $\sigma_i^2$ for $i=1,2$ .Furthermore the correlation between $X_1$ and $X_2$ is $\rho$ .Finally,the marginal distribution of $X_i$ is the normal distribution with mean $\mu_i$ and variance $\sigma_i^2$ for $i=1,2$

Definition Bivariate Normal Distributions.When the joint p.d.f. of two random variables $X_1$ and $X_2$ is of the form in Eq(5.10.2),it is said that $X_1$ and $X_2$ have the bivariate normal distribution with mean $\mu_1$ and $\mu_2$ variance $\sigma_1^2$ and $\sigma_2^2$ ,and correlation $\rho$

## 二维正态分布的性质 Properties of Bivariate Normal Distributions

Theorem Independence and Correlation.Two random variables $X_1$ and $X_2$ that have a bivariate normal distribution are independent if and only if they are uncorrelated.

Theorem Conditional Distribution.Let $X_1$ and $X_2$ have the bivariate normal distribution whose p.d.f. is Eq.(5.10.2) .The conditional distribution of $X_2$ given that $X_1=x_1$ is the normal distribution with mean and variance given by
\begin{aligned} E(X_2|x_1)&=\mu_2+\rho\sigma_2(\frac{x_1-\mu_1}{\sigma_1})\\ Var(X_2|x_1)&=(1-\rho^2)\sigma_2^2 \end{aligned}

1. 给定条件 $X_1=x_1$ 等价于给定 $Z_1=\frac{x_1-\mu_1}{\sigma_1}$
2. 那么我们只需要证明给定条件 $Z_1=\frac{x_1-\mu_1}{\sigma_1}$ 下的 $X_2$ 的分布。
3. 那么把 $Z_1=\frac{x_1-\mu_1}{\sigma_1}$ 带入到式子 5.10.1中的第二个公式。
4. 那么给定条件 $X_1=x_1$ 下的 $X_2$ 的分布，等价于给定条件 $Z_1=\frac{(x_1-\mu_1)}{\sigma_1}$ 下，以下关系式的分布：
$$(1-\rho^2)^{1/2}\sigma_2Z_2+\mu_2+\rho\sigma_2(\frac{x_1-\mu_1}{\sigma_1})\tag{5.10.7}$$
5. 上式可见 $Z_2$ 是唯一的随机变量，并且 $Z_1$ 和 $Z_2$ 是独立的，所以 $X_2$ 在给定 $X_1=x_1$ 的条件下是 5.10.7 的边缘分布
6. 所以条件期望和条件方差如 5.10.6 所写。
7. 证毕

$$E(X_1|x_2)=\mu_1+\rho\sigma_1(\frac{x_2-\mu_2}{\sigma_2})\\ Var(X_1|x_2)=(1-\rho^2)\sigma_1^2$$

## Linear Combination

Theorem Linear Combination of Bivariate Normals.Suppose that two random variables $X_1$ and $X_2$ have a bivariate normal distribution ,for which the p.d.f is specified by Eq.(5.10.2).Let $Y=a_1X_1+a_2X_2+b$ ,where $a_1,a_2$ and $b$ are arbitrary given constants .Then $Y$ has the normal distribution with mean $a_1\mu_1+a_2\mu_2+b$ and variance
$$a_1^2\sigma_1^2+a_2^2\sigma_2^2+2a_1a_2\rho\sigma_1\sigma_2$$

1. 依据双变量正态分布的定义，我们可以用 $Z_1$ 和 $Z_2$ 的线性组合来表示 $X_1$ 和 $X_2$ 的线性组合。
2. $Z_1$ 和 $Z_2$ 独立（已知条件）
3. $Y$ 可以表示成$Z_1$ 和 $Z_2$ 的线性组合
4. 根据5.6中推论 Y还是正态分布，并且期望为：
\begin{aligned} E(Y)&=a_1E(X_1)+a_2E(X_2)+b\\ &=a_1\mu_1+a_2\mu_2+b \end{aligned}
5. 根据4.6 中的推论： $Var(Y)=a_1^2 Var(X_1)+a_2^2 Var(X_2)+2a_1a_2 Cov(X_1,X_2)$
6. 证毕

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