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# 统计代写|统计推断代考Statistical Inference代写|ContinuouS Distributions

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## 统计代写|统计推断代考Statistical Inference代写|ContinuouS Distributions

In this section we will discuss some of the more common families of continuous distributions, those with well-known names. The distributions mentioned here by no means constitute all of the distributions used in statistics. Indeed, as was seen in Section 1.6, any nonnegative, integrable function can be transformed into a pdf.
Uniform Distribution
The continuous uniform distribution is defined by spreading mass uniformly over an interval $[a, b]$. Its pdf is given by
$$f(x \mid a, b)= \begin{cases}\frac{1}{b-a} & \text { if } x \in[a, b] \ 0 & \text { otherwise. }\end{cases}$$
It is easy to check that $\int_a^b f(x) d x=1$. We also have
\begin{aligned} \mathrm{E} X & =\int_a^b \frac{x}{b-a} d x=\frac{b+a}{2} ; \ \operatorname{Var} X & =\int_a^b \frac{\left(x-\frac{b+a}{2}\right)^2}{b-a} d x=\frac{(b-a)^2}{12} \end{aligned}
Gamma Distribution
The gamma family of distributions is a flexible family of distributions on $[0, \infty)$ and can be derived by the construction discussed in Section 1.6. If $\alpha$ is a positive constant, the integral
$$\int_0^{\infty} t^{\alpha-1} e^{-t} d t$$
is finite. If $\alpha$ is a positive integer, the integral can be expressed in closed form; otherwise, it cannot. In either case its value defines the gamma function,
$$\Gamma(\alpha)=\int_0^{\infty} t^{\alpha-1} e^{-t} d t$$
The gamma function satisfies many useful relationships, in particular,
$$\Gamma(\alpha+1)=\alpha \Gamma(\alpha), \quad \alpha>0$$
which can be verified through integration by parts. Combining (3.3.3) with the easily verified fact that $\Gamma(1)=1$, we have for any integer $n>0$,
$$\Gamma(n)=(n-1) !$$

## 统计代写|统计推断代考Statistical Inference代写|Normal Distribution

The normal distribution (sometimes called the Gaussian distribution) plays a central role in a large body of statistics. There are three main reasons for this. First, the normal distribution and distributions associated with it are very tractable analytically (although this may not seem so at first glance). Second, the normal distribution has the familiar bell shape, whose symmetry makes it an appealing choice for many population models. Although there are many other distributions that are also bellshaped, most do not possess the analytic tractability of the normal. Third, there is the Central Limit Theorem (see Chapter 5 for details), which shows that, under mild conditions, the normal distribution can be used to approximate a large variety of distributions in large samples.

The normal distribution has two parameters, usually denoted by $\mu$ and $\sigma^2$, which are its mean and variance. The pdf of the normal distribution with mean $\mu$ and variance $\sigma^2$ (usually denoted by $\mathrm{n}\left(\mu, \sigma^2\right)$ ) is given by
$$f\left(x \mid \mu, \sigma^2\right)=\frac{1}{\sqrt{2 \pi} \sigma} e^{-(x-\mu)^2 /\left(2 \sigma^2\right)}, \quad-\infty<x<\infty .$$
If $X \sim \mathrm{n}\left(\mu, \sigma^2\right)$, then the random variable $Z=(X-\mu) / \sigma$ has a $\mathrm{n}(0,1)$ distribution, also known as the standard normal. This is easily established by writing
\begin{aligned} P(Z \leq z) & =P\left(\frac{X-\mu}{\sigma} \leq z\right) \ & =P(X \leq z \sigma+\mu) \ & =\frac{1}{\sqrt{2 \pi} \sigma} \int_{-\infty}^{z \sigma+\mu} e^{-(x-\mu)^2 /\left(2 \sigma^2\right)} d x \ & =\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^z e^{-t^2 / 2} d t, \quad\left(\text { substitute } t=\frac{x-\mu}{\sigma}\right) \end{aligned}
showing that $P(Z \leq z)$ is the standard normal cdf.

# 统计推断代写

## 统计代写|统计推断代考Statistical Inference代写|ContinuouS Distributions

$$f(x \mid a, b)= \begin{cases}\frac{1}{b-a} & \text { if } x \in[a, b] \ 0 & \text { otherwise. }\end{cases}$$

\begin{aligned} \mathrm{E} X & =\int_a^b \frac{x}{b-a} d x=\frac{b+a}{2} ; \ \operatorname{Var} X & =\int_a^b \frac{\left(x-\frac{b+a}{2}\right)^2}{b-a} d x=\frac{(b-a)^2}{12} \end{aligned}

gamma族分布是$[0, \infty)$上的一个灵活的分布族，可以通过1.6节中讨论的构造推导出来。如果$\alpha$是一个正常数，积分
$$\int_0^{\infty} t^{\alpha-1} e^{-t} d t$$

$$\Gamma(\alpha)=\int_0^{\infty} t^{\alpha-1} e^{-t} d t$$

$$\Gamma(\alpha+1)=\alpha \Gamma(\alpha), \quad \alpha>0$$

$$\Gamma(n)=(n-1) !$$

## 统计代写|统计推断代考Statistical Inference代写|Normal Distribution

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

\begin{aligned} P(Z \leq z) & =P\left(\frac{X-\mu}{\sigma} \leq z\right) \ & =P(X \leq z \sigma+\mu) \ & =\frac{1}{\sqrt{2 \pi} \sigma} \int_{-\infty}^{z \sigma+\mu} e^{-(x-\mu)^2 /\left(2 \sigma^2\right)} d x \ & =\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^z e^{-t^2 / 2} d t, \quad\left(\text { substitute } t=\frac{x-\mu}{\sigma}\right) \end{aligned}

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## 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。