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# 经济代写|计量经济学代写Introduction to Econometrics代考|The Normal Distribution

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## 经济代写|计量经济学代写Introduction to Econometrics代考|The Normal Distribution

We say that a random variable $Z$ has the standard normal distribution, or Gaussian, written $Z \sim$ $\mathrm{N}(0,1)$, if it has the density
$$\phi(x)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{x^2}{2}\right), \quad-\infty<x<\infty$$
The standard normal density is typically written with the $\operatorname{symbol} \phi(x)$ and the corresponding distribution function by $\Phi(x)$. Plots of the standard normal density function $\phi(z)$ and distribution function $\Phi(x)$ are displayed in Figure 5.1.

If $Z \sim \mathrm{N}(0,1)$ and $X=\mu+\sigma Z$ for $\mu \in \mathbb{R}$ and $\sigma \geq 0$ then $X$ has the univariate normal distribution, written $X \sim \mathrm{N}\left(\mu, \sigma^2\right)$. By change-of-variables $X$ has the density
$$f(x)=\frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left(-\frac{(x-\mu)^2}{2 \sigma^2}\right), \quad-\infty<x<\infty .$$
The mean and variance of $X$ are $\mu$ and $\sigma^2$, respectively.
The normal distribution and its relatives (the chi-square, student t, F, non-central chi-square and F) are frequently used for inference to calculate critical values and $\mathrm{p}$-values. This involves evaluating the normal cdf $\Phi(x)$ and its inverse. Since the $\operatorname{cdf} \Phi(x)$ is not available in closed form statistical textbooks have traditionally provided tables for this purpose. Such tables are not used currently as these calculations are embedded in modern statistical software. For convenience, we list the appropriate commands in MATLAB, R, and Stata to compute the cumulative distribution function of commonly used statistical distributions.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Multivariate Normal Distribution

We say that the $k$-vector $Z$ has a multivariate standard normal distribution, written $Z \sim \mathrm{N}\left(0, \boldsymbol{I}_k\right)$, if it has the joint density
$$f(x)=\frac{1}{(2 \pi)^{k / 2}} \exp \left(-\frac{x^{\prime} x}{2}\right), \quad x \in \mathbb{R}^k$$
The mean and covariance matrix of $Z$ are 0 and $\boldsymbol{I}_k$, respectively. The multivariate joint density factors into the product of univariate normal densities, so the elements of $Z$ are mutually independent standard normals.

If $Z \sim \mathrm{N}\left(0, \boldsymbol{I}_k\right)$ and $X=\mu+\boldsymbol{B} Z$ then the $k$-vector $X$ has a multivariate normal distribution, written $X \sim \mathrm{N}(\mu, \Sigma)$ where $\Sigma=\boldsymbol{B} \boldsymbol{B}^{\prime} \geq 0$. If $\Sigma>0$ then by change-of-variables $X$ has the joint density function
$$f(x)=\frac{1}{(2 \pi)^{k / 2} \operatorname{det}(\Sigma)^{1 / 2}} \exp \left(-\frac{(x-\mu)^{\prime} \Sigma^{-1}(x-\mu)}{2}\right), \quad x \in \mathbb{R}^k$$

The mean and covariance matrix of $X$ are $\mu$ and $\Sigma$, respectively. By setting $k=1$ you can check that the multivariate normal simplifies to the univariate normal.

An important property of normal random vectors is that affine functions are multivariate normal.
Theorem 5.2 If $X \sim \mathrm{N}(\mu, \Sigma)$ and $Y=\boldsymbol{a}+\boldsymbol{B} X$, then $Y \sim \mathrm{N}\left(\boldsymbol{a}+\boldsymbol{B} \mu, \boldsymbol{B} \Sigma \boldsymbol{B}^{\prime}\right)$.
One simple implication of Theorem 5.2 is that if $X$ is multivariate normal then each component of $X$ is univariate normal.

Another useful property of the multivariate normal distribution is that uncorrelatedness is the same as independence. That is, if a vector is multivariate normal, subsets of variables are independent if and only if they are uncorrelated.

## 经济代写|计量经济学代写Introduction to Econometrics代考|The Normal Distribution

$$\phi(x)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{x^2}{2}\right), \quad-\infty<x<\infty$$

$$f(x)=\frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left(-\frac{(x-\mu)^2}{2 \sigma^2}\right), \quad-\infty<x<\infty .$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Multivariate Normal Distribution

$$f(x)=\frac{1}{(2 \pi)^{k / 2}} \exp \left(-\frac{x^{\prime} x}{2}\right), \quad x \in \mathbb{R}^k$$

$$f(x)=\frac{1}{(2 \pi)^{k / 2} \operatorname{det}(\Sigma)^{1 / 2}} \exp \left(-\frac{(x-\mu)^{\prime} \Sigma^{-1}(x-\mu)}{2}\right), \quad x \in \mathbb{R}^k$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。