### 统计代写|工程统计代写engineering statistics代考|Continuous Random Variables

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|工程统计代写engineering statistics代考|CONTINUOUS MODELS

Continuous distributions are more important in industrial experiments and research studies. Measurement of quantities (like height, weight, length, temperature, conductivity, resistance, etc.) on the ratio scale is continuous or quantitative data.

Definition 1.1 The stochastic variable that underlies quantitative data is called a continuous random variable, as they can take a continuum of possible values in a finite or infinite interval with an associated probability.

This can be thought of as the limiting form of a point probability function, as the possible values of the underlying continuous random variable become more and more of fine granularity. Thus, the mark in an exam (say between 0 and 100 ) is assumed to be a continuous random variable, even if fractional marks are not permitted. In other words, marks can be modeled by a continuous law even though it is not measured at the finest possible granularity level of fractions. If all students scored between 50 and 100 in an exam, the observed range for that exam is of course $50 \leq x \leq 100$. This range may vary from exam to exam, so that the lower limit could differ from 50 , and the upper limit of 100 is never achieved (nobody got a perfect 100 ). This range is in fact immaterial in several statistical procedures.

All continuous variables need not follow a statistical law. But there are many chance phenomena and physical laws that can be approximated by one of the continuous distributions like the normal law, if not exact. For instance, errors in various measurements are assumed to be normally distributed with zero mean. Similarly, symmetric measurement variations in physical properties like diameter, size of manufactured products, exceedences of dams and reservoirs, and so on, are assumed to follow a continuous uniform law centered around an ideal value $\theta$. This is because they can vary in both directions from an ideal value called its central value.

## 统计代写|工程统计代写engineering statistics代考|STANDARD DISTRIBUTIONS

Most of the statistical distributions have one or more parameters. These parameters describe the location (central tendency), spread (dispersion), and other shape characteristics of the distribution. There exist several distributions for which the location information is captured by one, and scale information by another parameter. These are called location-and-scale (LaS) distributions (page 7). There are some distributions called standard probability distributions (SPD) for which the parameters are universally fixed. This applies not only to LaS distributions, but to others as well.

Definition 1.2 A standard probability distribution is a specific member of a parametric family in which all the parameters are fixed so that every member of the family can be obtained by arithmetic transformations of variates.

These are also called “parameter-free” distributions (although location parameter is most often 0 , and scale parameter is 1). Examples in univariate case are the standard normal $\mathrm{N}(0,1)$ with PDF $f(z)=(1 / \sqrt{2 \pi}) \exp \left(-z^{2} / 2\right)$, for which location parameter is 0 , and scale parameter is 1 ; unit rectangular $\mathrm{U}(0,1)$, standard exponential distribution (SED) with PDF $f(x)=$ $\exp (-x)$, standard Laplace distribution with PDF $f(x)=\frac{1}{2} \exp (-|x|)$, standard Cauchy distribution with PDF $f(x)=1 /\left(\pi\left(1+x^{2}\right)\right)$, standard lognormal distribution with PDF $f(x)=$ $\exp \left(-(\log (x))^{2} / 2\right) /(\sqrt{2 \pi} x)$, and so on. This concept can easily be extended to the bivariate and multivariate probability distributions too. Simple change of origin and scale transformation can be used on the SPD to obtain all other members of its family as $X=\mu+\sigma Z$. Not all statistical distributions have meaningful SPD forms, however. Examples are $\chi^{2}, F$, and $T$ distributions that depend on one or more degrees of freedom (DoF) parameters, and gamma distribution with two parameters that has PDF $f(x ; a, m)=a^{m} x^{m-1} \exp (-a x) / \Gamma(m)$. This is because setting special values to the respective parameters results in other distributions. ${ }^{2}$ As examples, the $\mathrm{T}$ distribution becomes Cauchy distribution for DoF $n=1$, and $\chi^{2}$ distribution with $n=2$ becomes exponential distribution with parameter $1 / 2$.

The notion of SPD is important from many perspectives: (i) tables of the distributions are easily developed for standard forms; (ii) all parametric families of a distribution can be obtained from the SPD form using appropriate variate transformations; (iii) asymptotic convergence of various distributions are better understood using the SPD (for instance, the Student’s $t$ distribution tends to the standard normal when the DoF parameter becomes large); and (iv) test statistics and confidence intervals used in statistical inference are easier derived using the respective SPD.

## 统计代写|工程统计代写engineering statistics代考|TAIL AREAS

The area from the lower limit to a particular value of $x$ is called the CDF (left-tail area). It is called “probability content” in physics and some engineering fields, although statisticians seem to use “probability content” to mean the volume under bivariate or multivariate distributions. The PDF is usually denoted by lowercase English letters, and the CDF by uppercase letters. Thus, $f(x ; \mu)$ denotes the PDF (called Lebesque density in some fields), and $F(x ; \mu)=$ $\int_{l l}^{x} f(y) d y=\int_{l l}^{x} d F(y)$, where $l l$ is the lower limit, the CDF ( $\mu$ denotes unknown parameters). It follows that $(\partial / \partial x) F(x)=f(x)$, and $\operatorname{Pr}[a<X \leq b]=F(b)-F(a)=\int_{a}^{b} f(x) d x$. The differential operator $d x, d y$, etc. are written in the beginning in some non-mathematics fields (especially physics, astronomy, etc.) as $F(x ; \mu)=\int_{l l}^{x} d y f(y)$. Although a notational issue, we will use it at the end of an integral, especially in multiple integrals involving $d x d y$, etc. The quantity $f(x) d x$ is called probability differential in physical sciences. Note that $f(x)$ (density function evaluated at a particular value of $x$ within its domain) need not represent a probability, and in fact could sometimes exceed one in magnitude. For instance, Beta-I $(p, q)$ for $p=8, q=3$ evaluated at $x=0.855$ returns $2.528141$. However, $f(x) d x$ always represents the probability $\operatorname{Pr}(x-d x / 2 \leq X \leq x+d x / 2)$, which is in $[0,1]$.

Alternate notations for the PDF are $f(x \mid \mu), f_{x}(\mu)$, and $f(x ; \mu) d x$, and corresponding $\mathrm{CDF}$ are $F(x \mid \mu)$ and $F_{x}(\mu)$. These are written simply as $f(x)$ and $F(x)$ when general statements (without regard to the parameters) are made that hold for all continuous distributions. If $X$ is any continuous random variable with CDF $F(x)$, then $U=F(x) \sim U[0,1]$ (Chapter 2). This fact is used to generate random numbers from continuous distributions when the CDF or SF has closed form. The right-tail area (i.e., SF) is denoted by $S(x)$. As the total area is unity, we get $F(x)+S(x)=1$. Many other functions are defined in terms of $F(x)$ or $S(x)$. The hazard function used in reliability is defined as
$$h(x)=f(x) /(1-F(x))=f(x) / S(x)$$

## 统计代写|工程统计代写engineering statistics代考|TAIL AREAS

PDF 的替代符号是F(X∣μ),FX(μ)， 和F(X;μ)dX, 和对应的CDF是F(X∣μ)和FX(μ). 这些简单地写成F(X)和F(X)当做出适用于所有连续分布的一般陈述（不考虑参数）时。如果X是任何具有 CDF 的连续随机变量F(X)， 然后在=F(X)∼在[0,1]（第2章）。当 CDF 或 SF 具有闭合形式时，此事实用于从连续分布中生成随机数。右尾区域（即SF）表示为小号(X). 由于总面积是一单位，我们得到F(X)+小号(X)=1. 许多其他功能是根据以下定义的F(X)或者小号(X). 可靠性中使用的风险函数定义为

H(X)=F(X)/(1−F(X))=F(X)/小号(X)

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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。