## 统计代写|工程统计代写engineering statistics代考|STA 3032

statistics-lab™ 为您的留学生涯保驾护航 在代写工程统计engineering statistics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写工程统计engineering statistics代写方面经验极为丰富，各种代写工程统计engineering statistics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|ALTERNATE REPRESENTATIONS

The standard arcsine distribution SASD-I has range $(0,1)$, and SASD-II has range $(-1,+1)$. They are also written as $\frac{1}{\pi}(x(1-x))^{-1 / 2}\left(\right.$ or $\left.\left(\pi^{2} x(1-x)\right)^{-1 / 2},\left(\pi^{2} x(1-x)\right)^{-0.5}\right)$ and $\frac{1}{\pi}(1-$ $\left.x^{2}\right)^{-1 / 2}\left(\right.$ or $\left.\left(\pi^{2}\left(1-x^{2}\right)\right)^{-1 / 2},\left(\pi^{2}\left(1-x^{2}\right)\right)^{-0.5}\right)$, respectively. As $\Gamma(1 / 2)=\sqrt{\pi}$, it can also be written as $(x(1-x))^{-1 / 2} / \Gamma(1 / 2)^{2}$. Arcsine distribution with PDF
$$f(x ; R)=1 /[\pi \sqrt{x(R-x)}] \text {, for } 00 whose CDF is the inverse hyperbolic function (1 / R) \sinh ^{-1}(x / R). Geometrically, the density on [0,1] (SASD-I) gives the distribution of the projection of a random point on a circle of radius half centered at (0.50,0) to the continuous interval [0,1] on the \mathrm{X}-axis, and as projection of a random point on a centered circle (at origin) with appropriate radius for symmetric versions (e.g., SASD-II). When the domain is [-R,+R] this circle is origin-centered with radius R. Shifts of the circle on the horizontal axis results in other displaced distributions discussed below. By assuming that the circle rolls continuously at constant speed horizontally, it can be used to model the position of a particle moving in simple harmonic motion with amplitude R at a random time t. It is also used in von Neumann algebra theory. The two-parameter ASD-I has PDF$$
f(x ; a, b)=1 /[b \pi \sqrt{((x-a) / b)(1-(x-a) / b)}] \quad \text { for } \quad a<x<a+b,
$$and the corresponding ASD-II has PDF$$
f(x ; a, b)=1 /\left[b \pi \sqrt{1-((x-a) / b)^{2}}\right] \text { for } a-b<x<a+b .
$$This is a location-and-scale distribution that is symmetric around a and is U-shaped. It reduces to the SASD by the transformation Y=(X-a) / b. Differentiate w.r.t. x, and equate to zero to get the minimum at x=a, with minimum value f(a)=1 /(b \pi). In terms of the minimum value, the PDF (5.6) can be written as$$
f(x ; a, b)=f(a) / \sqrt{1-((x-a) / b)^{2}} \text { for } a-b<x<a+b .
$$Next, consider the PDF$$
f(x ; a, b)=1 /[\pi \sqrt{(x-a)(b-x)}]=\left[\pi^{2}(x-a)(b-x)\right]^{-1 / 2} \quad \text { for } \quad a<x<b
$$## 统计代写|工程统计代写engineering statistics代考|RELATION TO OTHER DISTRIBUTIONS This is a special case of the beta distribution (Chapter 4) when a=1 / 2, b=1 / 2. Hence, all properties of beta distribution are applicable to SASD-I as well. In particular, X and 1-X are identically distributed. As the range of SASD-I is (0,1), the transformation Y=-\log (X) results in log-arcsine distribution discussed in page 60. If X_{k}^{\prime} s are IID Beta-I( \left.\mathrm{I} \frac{2 k-1}{2 n}, \frac{1}{2 n}\right) random variables, the distribution of the geometric mean (GM) of them Y=\left(\prod_{k=1}^{n} X_{k}\right)^{1 / n} is SASD-I distributed for n \geq 2 ([90], [22]). This has the interpretation that the log-arcsine law is decomposable into a sum (or average) of independent log-beta random variables (so that arcsine distribution is not additively decomposable or infinite divisible). If U is \operatorname{CUNI}(0,1) then Y=-\cos (\pi U / 2) is arcsine distributed. Conversely, if X has an arcsine distribution, U=(2 / \pi) \arcsin (\sqrt{x}) has the U(0,1) distribution. Differentiate w.r.t. u to get |\partial y / \partial u|= (\pi / 2) \sin (\pi u / 2), so that |\partial u / \partial y|=(2 / \pi) / \sqrt{1-\cos ^{2}(\pi u / 2)}=(2 / \pi) / \sqrt{1-y^{2}}. An alternate way to state this is as follows. If U \sim \operatorname{CUNI}(-\pi, \pi), the distribution of Y=\cos (u) is SASD-I (see below). Similarly, the SASD-II is related to the U(0,1) distribution as X=\cos (\pi u), because |\partial x / \partial u|=\pi \sin (\pi u)=\pi \sqrt{1-\cos ^{2}(\pi u)}=\pi \sqrt{1-x^{2}}. The transformation Y=2 X-1 and Y=X^{2} when applied to SASD-II results in SASD-I. Similarly, if X \sim SASD-I, then Y=\sqrt{X} is SASD-II. If \Phi 0 denotes the CDF of a normal distribution, \Phi^{-1}(F(x)) \sim \mathrm{N}(0,1) where F(x) denotes the CDF of ASD. Problem 5.2 If X \sim SASD-I prove that Y=1 / X has PDF f(y)=1 /[\pi y \sqrt{y-1}] for y>1. Example 5.3 Distribution of \cos (\mathrm{X}) If X \sim \operatorname{CUNI}(-\pi, \pi), find the distribution of Y= \cos (X) Solution 5.4 As X \sim \operatorname{CUNI}(-\pi, \pi), F(x)=1 / 2 \pi. From y=\cos (x), we get |d y / d x|= \sin (x)=\sqrt{1-\cos ^{2}(x)}=\sqrt{1-y^{2}}, so that f(y)=(1 / 2 \pi)\left(1 / \sqrt{1-y^{2}}\right). Since the equation y=\cos (x) has two solutions in -\pi, \pi as x_{1}=\cos ^{-1}(y) and x_{2}=2 \pi-x_{1}, the PDF becomes f(y)=1 /\left(\pi \sqrt{1-y^{2}}\right), which is SASD-II. ## 统计代写|工程统计代写engineering statistics代考|PROPERTIES OF ARCSINE DISTRIBUTION The SASD-I is a special case of beta type-I distribution. It is symmetric around the mean ( 1 / 2) and is concave upward (the probability decreases and then increases), but satisfies the log-convex property. As x \rightarrow 0 or x \rightarrow 1 the PDF f(x) \rightarrow \infty. Put Y=X-\frac{1}{2} to get$$
f(y)=\frac{1}{\pi \sqrt{(y+1 / 2)(1 / 2-y)}}, \quad-1 / 2<y<1 / 2 .
$$As (1 / 2+y)(1 / 2-y)=\left(1 / 4-y^{2}\right), the PDF becomes f(y)=(2 / \pi) / \sqrt{\left(1-4 y^{2}\right)}, for -1 / 2<y<1 / 2. The mean is 0.5 and variance is 0.125 for the standard arcsine distribution (see below). As the distribution is symmetric, coefficient of skewness is zero. The kurtosis coefficient is \beta_{2}=3 / 2. Thus, it is always platykurtic. Note that the density is maximum when x is near 0 or 1 with the center as a cusp (U-shaped), and minimum at x=0.5 with minimum value 2 / \pi. Hence, there are two modes (bimodal) that are symmetrically placed in the tails. This is the reason why it is platykurtic. Arcsine distribution is the exact antithesis of bell-shaped laws because (i) the mean coincides with the minimum (whereas mean coincides with maximum for bell-shaped laws), (ii) lower and upper limits correspond to asymptotes (density rises up to \infty ) (whereas bellshaped laws tail off to zero), (iii) bimodal (bell-shaped laws are unimodal), and (iv) statistical measures are more prone to outliers as the peaks are away from the mean (samples from bellshaped distributions have lesser chance of outliers). Due to these peculiarities, the convergence of central limit theorem to normality is slow. The hazard function is given by$$
1 / h(x ; a, b)=b \sqrt{1-[(x-a) / b]^{2}} \arccos ((x-a) / b)

## 统计代写|工程统计代写engineering statistics代考|PROPERTIES OF ARCSINE DISTRIBUTION

SASD-I 是 beta I 型分布的一个特例。它围绕均值对称（1/2)并且向上凹（概率先减小后增大），但满足对数凸性质。作为X→0或者X→1PDF格式F(X)→∞. 放是=X−12要得到

F(是)=1圆周率(是+1/2)(1/2−是),−1/2<是<1/2.

1/H(X;一个,b)=b1−[(X−一个)/b]2阿尔科斯⁡((X−一个)/b)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|工程统计代写engineering statistics代考|ENGRD 2700

statistics-lab™ 为您的留学生涯保驾护航 在代写工程统计engineering statistics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写工程统计engineering statistics代写方面经验极为丰富，各种代写工程统计engineering statistics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|GENERAL BETA DISTRIBUTION

General three-parameter beta distribution is given by
$$f_{x}(a, b, c)=(x / c)^{a-1}(1-x / c)^{b-1} / c B(a, b) .$$
The four-parameter beta distribution follows from (4.1) using $y=(x-a) /(b-a)$ as
$$f(x ; a, b, c, d)=\frac{\Gamma(c+d)}{\Gamma(c) \Gamma(d)(b-a)^{c+d-1}}(x-a)^{c-1}(b-x)^{d-1}$$
This could also be written as
$$f(x ; a, b, c, d)=\frac{\Gamma(c+d)}{\Gamma(c) \Gamma(d)(b-a)}[(x-a) /(b-a)]^{c-1}[1-(x-a) /(b-a)]^{d-1},$$

which can be transformed to Beta-I using $y=(x-a) /(b-a)$. This has mean $(a d+b c) /(c+$ $d)$, and variance $\sigma^{2}=c d(b-a)^{2} /\left[(c+d+1)(c+d)^{2}\right]$. The location parameters are “a”, “b” and scale parameters are $\mathrm{c}$ and $\mathrm{d}$. Coefficient of skewness is $2 c d(d-c) /\left[(c+d)^{2}(c+\right.$ $\left.d)^{(3)}\left[c d /\left((c+d)(c+d)^{(2)}\right)\right]\right]$ where $(c+d)^{(k)}$ is raising Pochhammer notation with $(c+$ $d)^{(3)}=(c+d)(c+d+1)(c+d+2)$. The mode is $\frac{a(d-1)+b(c-1)}{(c+d-2)}$ for $c$ not 1 and $d$ not 1 . The beta-geometric (discrete) distribution is defined in terms of CBF as
$$f(x ; a, b)=B(a+1, x+b-1) / B(a, b), \quad \text { for } \mathrm{x}=1,2,3, \ldots$$
This satisfies the recurrence relation $(a+b+x-1) p_{x}(a, b)=(x+b-2) p_{x-1}(a, b)$ with $p_{0}=B(a+1, b-1) / B(a, b)$. A change of origin transformation $Y=X-1$ results in the PMF $f(x ; a, b)=B(a+1, x+b) / B(a, b)$, for $\mathrm{x}=0,1,2, \ldots$

## 统计代写|工程统计代写engineering statistics代考|GEOTECHNICAL ENGINEERING

The shear strength parameters in geotechnical engineering (cohesive force $c$, and internal friction angle $\phi$ ) are crucial in accurate reliability analysis. The risk assessment accuracy can then be modeled using a joint distribution of $c$ and $\phi$. Data scarcity may lead to inaccurate estimates of the probability of failure. Either a truncated normal, half-normal, truncated lognormal ${ }^{4}$ or a beta distribution (with range $[a, b]$ ) is assumed for the above parameters. As there are multiple parameters (like cohesive force, internal friction angle, unit weight of soils) involved, one approach is to approximate the joint distribution by a univariate distribution. This is called the “copula-approach,” or “copula modeling technique.” As the shear strength parameter is more important to achieve high accuracy, marginal distribution of it using the beta law can improve the accuracy of reliability analysis. Restricting attention to only the shear-strength (c) and internal friction angle $(\phi)$, the bivariate CDF $F(c, \phi)$ can be expressed in terms of individual marginal distributions and a copula function as
$$F(c, \phi)=C\left(F_{1}(c), F_{2}(\phi) ; \theta\right),$$
where $C O$ denotes the copula. Take partial derivative $\partial^{2} / \partial c \partial \phi$ to get
$$f(c, \phi)=c\left(F_{1}(c), F_{2}(\phi) ; \theta\right) f_{1}(c) f_{2}(\phi),$$
where $f_{1}(c)$ and $f_{2}(\phi)$ are the marginal PDFs. Some geotechnical processes are multi-modal (exhibit two or more distinct peaks) in which case linear combination of appropriate uni-modal distributions are used in reliability analysis under uncertainties.

## 统计代写|工程统计代写engineering statistics代考|BETA DISTRIBUTION IN PERT

The program (or project) evaluation and review technique (PERT) is a diagrammatic tool used in project management. It was first introduced in 1957 for the U.S. Navy’s Polariz nuclear submarine design and construction scheduling project. The project must be comprised of tasks (called
${ }^{4}$ As the soil properties are strictly non-negative, the lognormal is preferred over normal distribution.

activities) with a dependency among them. Each activity is uniquely identified using a start and end dates (or times in micro-projects), and represented by an arrow. Isolated activities that do not have dependency among other activities are excluded from PERT. This implies that the PERT graph is always a directed acyclic graph (DAG) with the project start-date as the start-node (or source), and project finish-date as the end-node (sink) with predecessor and successor events for all intermediate activities. ${ }^{5}$ Its primary purpose is to analyze various activities so as to provide a best and worst estimates on project completion time and costs. In other words, uncertainty is incorporated in a controlled manner so that projects can be scheduled without knowing the precise details and durations of all the activities involved. The information on early-start (ES), early-finish (EF), late-start (LS), late-finish (LF), and expected duration can be obtained for internal nodes (and sink node) so that management can schedule activities in an optimal way (manpower, materials, machines, etc.) to complete a project within constraints. A critical path (which is the path with the longest time to complete) is identified from the source to the sink which identifies all activities with slack. Even internal nodes can be analyzed to understand each completed phase of a complex project, so that management can periodically review the progress within scheduled time and cost expenditures. A similar tool called critical path method (CPM) is also popular in project management. Although PERT and CPM are complementary tools, CPM uses one time and one cost estimation for each activity, so that PERT is more versatile for analysis of milestones in big projects.

PERT uses four types of time estimates to accomplish an activity. An optimistic-estimate (o) is the minimum possible time required, a pessimistic-estimate (p) is the maximum possible time required, a most-likely time $(\mathrm{m})$ is the best estimate of the time required (mode), and an expected time $(\mathrm{o}+4 \mathrm{~m}+\mathrm{p}) / 6$ is the average (arithmetic mean) time required, with variance $(p-$ $o)^{2} / 36$. Activity duration in PERT networks (used in project planning and implementations) are assumed to follow the beta distribution, in which case more precise estimates are available for expected time as $(2 \mathrm{o}+9 \mathrm{~m}+2 \mathrm{p}) / 13$. It may also be associated with any particular set of PERT estimates. The four-parameter beta distribution is typically used in PERT modeling (especially to model earth-moving activities in construction projects). The PDF is given by
$$f(x ; a, b, p, q)=(x-c)^{a-1}(d-x)^{b-1} /\left[(d-c)^{a+b-1} B(a, b)\right],$$
where $c$ (most optimistic completion time) is the lower and $d$ (most pessimistic completion time) is the upper limit on activity duration.

## 统计代写|工程统计代写engineering statistics代考|GENERAL BETA DISTRIBUTION

FX(一个,b,C)=(X/C)一个−1(1−X/C)b−1/C乙(一个,b).

F(X;一个,b,C,d)=Γ(C+d)Γ(C)Γ(d)(b−一个)C+d−1(X−一个)C−1(b−X)d−1

F(X;一个,b,C,d)=Γ(C+d)Γ(C)Γ(d)(b−一个)[(X−一个)/(b−一个)]C−1[1−(X−一个)/(b−一个)]d−1,

F(X;一个,b)=乙(一个+1,X+b−1)/乙(一个,b), 为了 X=1,2,3,…

## 统计代写|工程统计代写engineering statistics代考|GEOTECHNICAL ENGINEERING

F(C,φ)=C(F1(C),F2(φ);θ),

F(C,φ)=C(F1(C),F2(φ);θ)F1(C)F2(φ),

## 统计代写|工程统计代写engineering statistics代考|BETA DISTRIBUTION IN PERT

4由于土壤性质严格非负，因此对数正态分布优于正态分布。

PERT 使用四种类型的时间估计来完成一项活动。乐观估计 (o) 是所需的最小可能时间，悲观估计 (p) 是所需的最大可能时间，最可能的时间(米)是所需时间（模式）的最佳估计，以及预期时间(○+4 米+p)/6是所需的平均（算术平均）时间，有方差(p− ○)2/36. 假设 PERT 网络中的活动持续时间（用于项目规划和实施）遵循 beta 分布，在这种情况下，可以对预期时间进行更精确的估计，因为(2○+9 米+2p)/13. 它也可能与任何特定的PERT估计集相关联。四参数 beta 分布通常用于 PERT 建模（尤其是用于建模建筑项目中的土方活动）。PDF由下给出

F(X;一个,b,p,q)=(X−C)一个−1(d−X)b−1/[(d−C)一个+b−1乙(一个,b)],

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|工程统计代写engineering statistics代考|ENGG 202

statistics-lab™ 为您的留学生涯保驾护航 在代写工程统计engineering statistics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写工程统计engineering statistics代写方面经验极为丰富，各种代写工程统计engineering statistics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|TYPE-II BETA DISTRIBUTION

Beta distribution of the second kind (also called type-II beta distribution, beta-prime distribution, or inverted beta distribution (IBD)) is obtained from the above by the transformation $Y=X /(1-X)$ or equivalently $X=Y /(1+Y)$. When $x \rightarrow 0, y \rightarrow 0$, and when $x \rightarrow 1$, $y \rightarrow \infty$. Hence, the range of $Y$ is from 0 to $\infty$. The PDF is given by
$$f(y ; a, b)=y^{a-1} /\left[B(a, b)(1+y)^{a+b}\right], \quad y>0, a, b>0 .$$
The Beta-I distribution is used to model random experiments or occurrences that vary between two finite limits, that are mapped to the $(0,1)$ range, while Beta-II is used when upper limit is infinite. It is also used in risk analysis in finance and marketing, etc.

Put $a=b=1$ to get Beta( $(1,1)$, which is identical to $\mathrm{U}(0,1)$ distribution. If $X$ is Beta$\mathrm{I}(a, b)$ then $(1-X) / X$ is $\operatorname{Beta}-\mathrm{II}(b, a)$, and $X /(1-X)$ is $\operatorname{Beta-II}(a, b)$. If $X$ and $Y$ are independent gamma random variables GAMMA $(a, \lambda)$ and GAMMA $(b, \lambda)$, then $X /(X+Y)$ is $\operatorname{Beta}(a, b)$. As gamma and $\chi^{2}$ are related, this result can also be stated in terms of normal variates as follows. If $X$ and $Y$ are independent normal variates, then $Z=X^{2} /\left(X^{2}+Y^{2}\right)$ is Beta-I distributed. In addition, if $X_{1}, X_{2}, \ldots, X_{k}$ are IID $N(0,1)$ and $Z_{1}=X_{1}^{2} /\left(X_{1}^{2}+X_{2}^{2}\right)$, $Z_{2}=\left(X_{1}^{2}+X_{2}^{2}\right) /\left(X_{1}^{2}+X_{2}^{2}+X_{3}^{2}\right)$, and so on, $Z_{j}=\sum_{i=1}^{j} X_{i}^{2} / \sum_{i=1}^{j+1} X_{i}^{2}$, then each of them are Beta-I distributed, as also the product of any consecutive set of $Z_{j}^{\prime}$ s are beta distributed. The logistic distribution and type II beta distribution are related as $Y=-\ln (\mathrm{X})$. If $X$ is $\mathrm{Beta}-\mathrm{I}(a, b)$ then $Y=\ln (X /(1-X))$ has a generalized logistic distribution. Dirichlet distribution is a gencralization of beta distribution. Order statistic from uniform distribution is beta distributed. In general, $j^{t h}$ highest order statistic from a uniform distribution is $\operatorname{Beta}-\mathrm{I}(j, n-j+1)$.

## 统计代写|工程统计代写engineering statistics代考|MOMENTS AND GENERATING FUNCTIONS OF TYPE-II BETA

The mean and variance are $\mu=a /(b-1)$ and $\sigma^{2}=a(a+b-1) /\left[(b-1)^{2}(b-2)\right]$ for $b>2$. Consider $\mathrm{E}\left(Y^{k}\right)$
$$\int_{0}^{\infty} y^{k} f_{y}(a, b) d y=\int_{0}^{\infty} y^{a+k-1} /\left[B(a, b)(1+y)^{a+b}\right] d y$$
Put $x=y /(1+y)$ so that $y=x /(1-x),(1+y)=1 /(1-x)$ and $d y / d x=[(1-x)-$ $x(-1)] /(1-x)^{2}$. This simplifies to $1 /(1-x)^{2}$. The range of $X$ is $[0,1]$. Hence, $(4.22)$ becomes
$$(1 / B(a, b)) \int_{0}^{\infty} y^{a+k-1} /(1+y)^{a+b} d y=(1 / B(a, b)) \int_{0}^{1} x^{a+k-1}(1-x)^{b-k-1} d x .$$
This is $B(a+k, b-k) / B(a, b)$. Put $k=1$ to get the mean as $\Gamma(a+1) \Gamma(b-1) \Gamma(a+$ b) $/[\Gamma(a) \Gamma(b) \Gamma(a+b)]$. Write $\Gamma(a+1)=a \Gamma(a)$ in the numerator, and $\Gamma(b)=(b-1) \Gamma(b-$ 1) in the denominator and cancel out common factors to get $\mu=a /(b-1)$. Put $k=2$ to get the second moment as $B(a+2, b-2) / B(a, b)=\Gamma(a+2) \Gamma(b-2) \Gamma(a+b) /[\Gamma(a) \Gamma(b) \Gamma(a+$ $b)]=a(a+1) /[(b-1)(b-2)]$. From this the variance is obtained as $a(a+1) /[(b-1)(b-$ 2)] $-a^{2} /(b-1)^{2}$. Take $\mu=a /(b-1)$ as a common factor. This can now be written as $\mu\left(\frac{a+1}{b-2}-\mu\right)$. Substitute for $\mu$ inside the bracket and take $(b-1)(b-2)$ as common denominator. The numerator simplifies to $b-a+2 a-1=(a+b-1)$. Hence, the variance becomes $\sigma^{2}=a(a+b-1) /\left[(b-1)^{2}(b-2)\right]$. As $(a+1) /(b-2)-\mu=(a+b) /[(b-1)(b-2)]$, this expression is valid for $b>2$. Unlike the Beta-I distribution whose variance is always bounded, the variance of Beta-II can be increased arbitrarily by keeping b constant (say near $\left.2^{+}\right)$and letting $a \rightarrow \infty$. It can also be decreased arbitrarily when $(a+1) /(b-2)$ tends to $\mu=a /(b-1)$. The expectation of $[X /(1-X)]^{k}$ is easy to compute in terms of complete gamma function as $\mathrm{E}[X /(1-X)]^{k}=\frac{\Gamma(a+k) \Gamma(b-k)}{\Gamma(a) \Gamma(b)}$. See Table $4.2$ for further properties.

Example 4.14 The mode of Beta-II distribution Prove that the mode of Beta-II distribution is $(a-1) /(b+1)$.
Solution 4.15 Differentiate the PDF (without constant multiplier) w.r.t. $y$ to get
$$f^{\prime}(y)=\left[(1+y)^{a+b}(a-1) y^{a-2}-y^{a-1}(a+b)(1+y)^{a+b-1}\right] /(1+y)^{2(a+b)}$$

Equate the numerator to zero and solve for $y$ to get $y[a+b-a+1]=(a-1)$, or $y=(a-$ 1) $/(b+1)$As the Beta-I random variable takes values in $[0,1]$, any CDF can be substituted for $x$ to get a variety of new distributions (Chattamvelli (2012) [36]). For instance, put $x=\Phi(x)$, the CDF of a normal variate to get the beta-normal distribution with PDF
$$f(x ; a, b)=(1 / B[a, b]) \phi(x)[\Phi(x)]^{a-1}[1-\Phi(x)]^{b-1}$$

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

Tail areas of several statistical distributions are related to the beta CDF, as discussed below. The survival function of a binomial distribution $\operatorname{BINO}(n, p)$ is related to the left tail areas of Beta-I distribution as:
$$\sum_{x=a}^{n}\left(\begin{array}{l} n \ x \end{array}\right) p^{x} q^{n-x}=\mathrm{I}{p}(a, n-a+1)$$ Using the symmetry relationship, the CDF becomes $$\sum{x=0}^{a-1}\left(\begin{array}{l} n \ x \end{array}\right) p^{x} q^{n-x}=\mathrm{I}_{q}(n-a+1, a) .$$

When both $a$ and $b$ are integers, this has a compact representation as
$$\mathrm{I}{x}(a, b)=1-\sum{k=0}^{a-1}\left(\begin{array}{c} a+b-1 \ k \end{array}\right) x^{k}(1-x)^{a+b-1-k} .$$
The survival function of negative binomial distribution is related as follows:
$$\sum_{x=a}^{n}\left(\begin{array}{c} n+x-1 \ x \end{array}\right) p^{n} q^{x}=\mathrm{I}{q}(a, n)=1-\mathrm{I}{p}(n, a)$$
The relationship between the CDF of central $F$ distribution and the IBF is
$$\mathrm{F}{m, n}(x)=\mathrm{I}{y}(m / 2, n / 2),$$
where $(m, n)$ are the numerator and denominator $\mathrm{DoF}$ and $y=m x /(n+m x)$. Similarly, Student’s $t$ CDF is evaluated as
$$\mathrm{T}{n}(t)=(1 / 2)\left(1+\operatorname{sign}(\mathrm{t}) \mathrm{I}{x}(1 / 2, n / 2)\right)=(1 / 2)\left(1+\operatorname{sign}(\mathrm{t})\left[1-\mathrm{I}_{y}(n / 2,1 / 2)\right]\right),$$
where $x=t^{2} /\left(n+t^{2}\right), y=1-x=n /\left(n+t^{2}\right), \operatorname{sign}(\mathrm{t})=+1$ if $\mathrm{t}>0,-1$ if $\mathrm{t}<0$ and is zero for $t=0$.

The IBF is related to the tail areas of binomial, negative binomial, Student’s $t$, central $F$ distributions. It is also related to the confluent hypergeometric function, generalized logistic distribution, the distribution of order statistics from uniform populations, and the Hotelling’s $\mathrm{T}^{2}$ statistic. The hypergeometric function can be approximated using the IBF also [145]. The Dirichlet (and its inverse) distribution can be expressed in terms of IBF [140]. It is related to the CDF of noncentral distributions. For instance, the CDF of singly noncentral beta (Seber (1963) [121]), singly type-II noncentral beta, and doubly noncentral beta (Chattamvelli (1995) [31]), noncentral T (Chattamvelli (2012) [36], Craig (1941) [48]), noncentral F (Chattamvelli (1996) [33], Patnaik (1949) [107]), and the sample multiple correlation coefficient (Ding and Bargmann (1991) [53], Ding (1996) [52]) could all be evaluated as infinite mixtures of IBF. It is used in string theory to calculate and reproduce the scattering amplitude in terms of Regge trajectories, and to model properties of strong nuclear force.

## 统计代写|工程统计代写engineering statistics代考|TYPE-II BETA DISTRIBUTION

F(是;一个,b)=是一个−1/[乙(一个,b)(1+是)一个+b],是>0,一个,b>0.
Beta-I 分布用于模拟在两个有限限制之间变化的随机实验或事件，这些限制映射到(0,1)范围，而 Beta-II 用于上限为无限时。它还用于金融和营销等领域的风险分析。

## 统计代写|工程统计代写engineering statistics代考|MOMENTS AND GENERATING FUNCTIONS OF TYPE-II BETA

∫0∞是ķF是(一个,b)d是=∫0∞是一个+ķ−1/[乙(一个,b)(1+是)一个+b]d是

(1/乙(一个,b))∫0∞是一个+ķ−1/(1+是)一个+bd是=(1/乙(一个,b))∫01X一个+ķ−1(1−X)b−ķ−1dX.

F′(是)=[(1+是)一个+b(一个−1)是一个−2−是一个−1(一个+b)(1+是)一个+b−1]/(1+是)2(一个+b)

F(X;一个,b)=(1/乙[一个,b])φ(X)[披(X)]一个−1[1−披(X)]b−1

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

∑X=一个n(n X)pXqn−X=我p(一个,n−一个+1)使用对称关系，CDF 变为

∑X=0一个−1(n X)pXqn−X=我q(n−一个+1,一个).

∑X=一个n(n+X−1 X)pnqX=我q(一个,n)=1−我p(n,一个)

F米,n(X)=我是(米/2,n/2),

IBF 与二项式、负二项式、Student’s 的尾部区域有关吨, 中央F分布。它还与汇合的超几何函数、广义逻辑分布、均匀总体的顺序统计分布以及 Hotelling 的吨2统计。超几何函数也可以使用 IBF 来近似 [145]。Dirichlet（及其逆）分布可以用 IBF [140] 来表示。它与非中心分布的 CDF 有关。例如，单非中心 beta (Seber (1963) [121])、单 II 型非中心 beta 和双重非中心 beta (Chattamveli (1995) [31])、非中心 T (Chattamveli (2012) [36] 的 CDF , Craig (1941) [48]), noncentral F (Chattamveli (1996) [33], Patnaik (1949) [107]) 和样本多重相关系数 (Ding and Bargmann (1991) [53], Ding (1996) ) [52]) 都可以被评估为 IBF 的无限混合。它在弦理论中用于根据雷格轨迹计算和再现散射幅度，并模拟强核力的特性。

## 有限元方法代写

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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。

## 统计代写|工程统计代写engineering statistics代考|STAT2110

statistics-lab™ 为您的留学生涯保驾护航 在代写工程统计engineering statistics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写工程统计engineering statistics代写方面经验极为丰富，各种代写工程统计engineering statistics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|INTRODUCTION

The beta distribution has a long history that can be traced back to the year 1676 in a letter from Issac Newton to Henry Oldenbeg (see Dutka (1981) [55]). It is widely used in civil, geotechnical, earthquake, and metallurgical engineering due to its close relationship with other continuous distributions. The PDF of Beta-I $(a, b)$ is given by ${ }^{1}$
$$f(x ; a, b)=x^{a-1}(1-x)^{b-1} / B(a, b)$$
where $00$ and $b>0$ results in a variety of distributional shapes. The Beta-I distribution is a proper choice in risk-modeling because the risks in many applications can be lower and upper bounded, and scaled to any desired range (say $(0,1)$ range) [78]. Events constrained to happen within a finite interval can be modeled due to the wide variety of shapes assumed by this distribution.

It is also used in Bayesian models with unknown probabilities, in order-statistics and reliability analysis. In Bayesian analysis, the prior distribution is assumed to be Beta-I for binomial proportions. It is used to model the proportion of fat (by weight) in processed or canned food, percentage of impurities in some manufactured products like food items, cosmetics, laboratory chemicals, etc. Data in the form of proportions arise in many applied fields like marketing, toxicology, bioinformatics, genomics, etc. Beta distribution is the preferred choice when these quantities exhibit extra variation than expected. Important distributions belonging to the beta family are discussed below. These include type I and type-II beta distributions. We will use the respective notations $\operatorname{Beta}-\mathrm{I}(a, b)$, and $\operatorname{Beta}-\mathrm{II}(a, b) .{ }^{2}$ Beta distributions with three or more parameters are also briefly mentioned.

## 统计代写|工程统计代写engineering statistics代考|ALTERNATE REPRESENTATIONS

Write $c=a-1$ and $d=b-1$ to get the alternate form
$$f(x ; c, d)=x^{c}(1-x)^{d} / B(c+1, d+1) .$$ Put $x=\sin ^{2}(\theta)$ in (4.2) to get
$$f(\theta ; c, d)=\sin ^{2 c}(\theta) \cos ^{2 d}(\theta) / B(c+1, d+1) \text { for } 0<\theta<\pi / 2$$
Some applications use $a$ and $n-a+1$ as parameters resulting in
$$f(x ; a, b, n)=x^{a-1}(1-x)^{n-a} / B(a, n-a+1) .$$
A symmetric beta distribution results when $a=b$ with PDF
$$f(x ; a)=x^{a-1}(1-x)^{a-1} / B(a, a)=[x(1-x)]^{a-1} \Gamma(2 a) /[\Gamma(a)]^{2} .$$
Beta distributions defined on $(-1,+1)$ are encountered in some applications. Using the transformation $Y=2 X-1$ we get $f(y)=f(x) / 2=f((y+1) / 2) / 2$. This results in the PDF
$$f(y ; a, b)=[(y+1) / 2]^{a-1}[(1-y) / 2]^{b-1} /[2 B(a, b)] .$$
This simplifies to
$\left.f(y ; a, b)=C(1+y)^{a-1}(1-y)\right]^{b-1}$ where $-1<y<1$, and $C=1 /\left[2^{a+b-1} B(a, b)\right] .$
This also can be generalized to 4-parameters as
$$\left.f(x ; a, b, c, d)=C(1+x / c)^{a-1}(1-x / d)\right]^{b-1}$$
and to the 6-parameters as
$$\left.f(x ; a, b, c, d, p, q)=C(1+(x-p) / c)^{a-1}(1-(x-q) / d)\right]^{b-1},$$
where $C$ is the normalizing constant, which is found using the well-known integral
$$\int_{a}^{b}(x-a)^{a-1}(b-x)^{b-1} d x=(b-a)^{a+b-1} B(a, b) .$$
These are related to the Berstein-type basis functions $Y_{k}^{n}(x ; a, b, m)=\left(\begin{array}{c}m \ k\end{array}\right)(x-a)^{k}(b-$ $x)^{n-k} /(b-a)^{m}[147]$. Truncated and size-biased versions of them are used in several engineering fields.

## 统计代写|工程统计代写engineering statistics代考|RELATION TO OTHER DISTRIBUTIONS

It is a special case of gamma distribution with $m=1$. It reduces to uniform (rectangular) distribution $\mathrm{U}(0,1)$ for $a=b=1$. A triangular-shaped distribution results for $a=1$ and $b=2$, or vice versa. When $a=b=1 / 2$, this distribution reduces to the arcsine distribution of first kind (Chapter 5). If $b=1$ and $a \neq 1$, it reduces to power-series distribution $f(x ; a)=a x^{a-1}$ using the result $\Gamma(a+1)=a * \Gamma(a)$. A J-shaped distribution is obtained when $a$ or $b$ is less than one. If $X$ and $Y$ are IID gamma distributed with parameters $a$ and $b$, the ratio $Z=X /(X+Y)$ is Beta-I $(a, b)$ distributed. If $X_{k}^{\prime}$ s are IID Beta-I $\left(\frac{2 k-1}{2 n}, \frac{1}{2 n}\right)$ random variables, the distribution of the geometric mean (GM) of them $Y=\left(\prod_{k=1}^{n} X_{k}\right)^{17 n}$ is SASD-I distributed (Chapter 5) for $n \geq 2([90],[22])$. This has the interpretation that the GM of Beta-I $\left(\frac{2 k-1}{2 n}, \frac{1}{2 n}\right)$ random variables converges to arcsine law whereas the AM tends to the normal law (central limit theorem). As $\chi^{2}$ distribution is a special case of gamma distribution, a similar result follows as $Z=\chi_{m}^{2} /\left(\chi_{m}^{2}+\chi_{n}^{2}\right) \sim \operatorname{Beta}-\mathrm{I}(m / 2, n / 2)$

As $\left(\chi_{m}^{2}+\chi_{n}^{2}\right)$ is independent of $Z$, the above result can be generalized as follows: If $X_{1}, X_{2}, \ldots X_{n}$ are IID normal variates with zero means and variance $\sigma_{k}^{2}$, then $Z_{1}=X_{1}^{2} /\left(X_{1}^{2}+\right.$ $\left.X_{2}^{2}\right), Z_{2}=\left(X_{1}^{2}+X_{2}^{2}\right) /\left(X_{1}^{2}+X_{2}^{2}+X_{3}^{2}\right)$, and so on are mutually independent beta random variables. If $X$ has an $F(m, n)$ distribution, then $Y=(m / n) X /[1+(m / n) X]$ is beta distributed. The beta distribution is also related to the Student’s $t$ distribution under the transformation $x=1 /\left(1+t^{2} / n\right)$. Similarly, $y=-\log (x)$ has PDF
$$f(y ; a, b)=\exp (-a y)(1-\exp (-y))^{b-1}$$
and $y=x /(1-x)$ results in beta-prime distribution (page 48). The positive eigenvalue of Roy’s $\theta_{\max }$-criterion used in MANOVA has a Beta-I distribution when $s=\max \left(p, n_{h}\right)=1$, where $p$ is the dimensionality and $n_{h}$ is the DoF of the hypothesis.

Problem 4.1 Prove that $B(a+1, b)=[a /(a+b)] B(a, b)$ where $B(a, b)$ denotes the CBF. What is the value of $B(.5, .5)$ ?

Problem 4.2 If $X \sim \operatorname{Beta}-\mathrm{I}(a, b)$, find the distribution of $Y=(1-X) / X$, and obtain its mean and variance. Find the ordinary moments.

Problem 4.3 If $X$ and $Y$ are independent gamma random variables GAMMA $(a, \lambda)$ and $\operatorname{GAMMA}(b, \lambda)$, then prove that $X /(X+Y)$ is $\operatorname{Beta}(a, b)$

Problem 4.4 Verify whether $f(x ; c, d)=(1+x)^{c-1}(1-x)^{d-1} /\left[2^{c+d-1} \mathrm{~B}(c, d)\right]$ is a PDF for $-1<x<1$, where $B(c, d)$ is the complete beta function.

## 统计代写|工程统计代写engineering statistics代考|INTRODUCTION

Beta 分布有很长的历史，可以追溯到 1676 年 Issac Newton 给 Henry Oldenbeg 的一封信（参见 Dutka (1981) [55]）。由于与其他连续分布关系密切，它被广泛应用于土木、岩土、地震和冶金工程。Beta-I 的 PDF(一个,b)是（谁）给的1

F(X;一个,b)=X一个−1(1−X)b−1/乙(一个,b)

## 统计代写|工程统计代写engineering statistics代考|ALTERNATE REPRESENTATIONS

F(X;C,d)=XC(1−X)d/乙(C+1,d+1).放X=罪2⁡(θ)在 (4.2) 中得到

F(θ;C,d)=罪2C⁡(θ)因2d⁡(θ)/乙(C+1,d+1) 为了 0<θ<圆周率/2

F(X;一个,b,n)=X一个−1(1−X)n−一个/乙(一个,n−一个+1).

F(X;一个)=X一个−1(1−X)一个−1/乙(一个,一个)=[X(1−X)]一个−1Γ(2一个)/[Γ(一个)]2.
Beta 分布定义于(−1,+1)在某些应用程序中遇到。使用转换是=2X−1我们得到F(是)=F(X)/2=F((是+1)/2)/2. 这导致PDF

F(是;一个,b)=[(是+1)/2]一个−1[(1−是)/2]b−1/[2乙(一个,b)].

F(是;一个,b)=C(1+是)一个−1(1−是)]b−1在哪里−1<是<1， 和C=1/[2一个+b−1乙(一个,b)].

F(X;一个,b,C,d)=C(1+X/C)一个−1(1−X/d)]b−1

F(X;一个,b,C,d,p,q)=C(1+(X−p)/C)一个−1(1−(X−q)/d)]b−1,

∫一个b(X−一个)一个−1(b−X)b−1dX=(b−一个)一个+b−1乙(一个,b).

## 统计代写|工程统计代写engineering statistics代考|RELATION TO OTHER DISTRIBUTIONS

F(是;一个,b)=经验⁡(−一个是)(1−经验⁡(−是))b−1

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|工程统计代写engineering statistics代考|STAT 2201

statistics-lab™ 为您的留学生涯保驾护航 在代写工程统计engineering statistics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写工程统计engineering statistics代写方面经验极为丰富，各种代写工程统计engineering statistics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|RELATION TO OTHER DISTRIBUTIONS

It is a special case of gamma distribution with $m=1$ (Chapter 6), and Weibull distribution. The Rosin-Rammler-Bennett (RRB) distribution used in mineral engineering is a special case of one-parameter exponential distribution. If $X \sim \operatorname{EXP}(\lambda)$, and $b$ is a constant, then $Y=X^{1 / b} \sim \operatorname{WEIB}(\lambda, b)$. The sum of $n$ IID exponential variates with the same parameter is gamma (also called Erlang) distributed, and with different parameters has a hyper-exponential distribution. Similarly, if $X_{1}, X_{2}, \ldots, X_{n}$ are IID $\operatorname{EXP}(\lambda)$ and $S_{n}=X_{1}+X_{2}+\cdots+X_{n}$, then $\operatorname{Pr}\left[S_{n}<t<S_{n+1}\right]$ has a Poisson distribution with parameter $\lambda t$. It is also related to the $\mathrm{U}(0,1)$ distribution, and power-law distribution, which is another discrete analogue of this distribution [34]. The extreme value distribution can be considered as a nonlinear generalization of $\operatorname{EXP}(\lambda)$. Its relationship with zero-truncated Poisson (ZTP) distribution is used to generate random numbers [98]. If $X_{1}$ and $X_{2}$ are IID $\operatorname{EXP}(\lambda)$, then $Y=X_{1} /\left(X_{1}+X_{2}\right) \sim \mathrm{U}(0,1)$. This has the implication that “if two random numbers between 0 and 1 are chosen from $\mathrm{U}(0,1)$, the ratio of one of them to their sum is more likely to be close to half; but when the two numbers are chosen from an exponential distribution, the same ratio is uniform between 0 and 1 .” Putting $Y=1 / X$ results in the inverse exponential distribution (Figure 3.2) with PDF
$$f(y ; \lambda)=\left(\lambda / y^{2}\right) \exp (-\lambda / y)$$
If $X_{1}$ and $X_{2}$ are IID EXP(1) random variables, $Y=X_{1}-X_{2}$ has standard Laplace distribution. A mixture of exponential distributions with gamma mixing weights gives rise to Lomax distribution.

## 统计代写|工程统计代写engineering statistics代考|PROPERTIES OF EXPONENTIAL DISTRIBUTION

This distribution has a single parameter, which is positive. It is a reverse-J shaped distribution which is always positively skewed (see Figure 3.1; page 28). Variance of this distribution is the square of the mean, as shown below. This means that when $\lambda \rightarrow 0$, the variance and kurtosis increases without limit (see Figure 3.1; page 28). The CDF is given by
$$F(x ; \lambda)=\left{\begin{array}{cl} 1-\exp (-\lambda x), x \geq 0 & \ 0 & \text { otherwise. } \end{array}\right.$$
The SF is $S(x ; \lambda)=1-\mathrm{CDF}=\exp (-\lambda x)$, so that $f(x ; \lambda)=\lambda S(x ; \lambda)$. From this the hazard function is obtained as
$$h(x)=f(x) /[1-F(x)]=f(x) / S(x)=\lambda,$$

which is constant. When a device or an equipment approximately exhibits constant hazard rate, it is an indication that the $\operatorname{EXP}(\lambda)$ may be a good choice to model the lifetime. If $x_{\alpha}$ is the $\alpha^{\text {th }}$ percentage point, $\alpha=1-\exp \left(-\lambda x_{\alpha}\right)$ from which $x_{\alpha}=-\log (1-\alpha) / \lambda$

Problem 3.1 A left-truncated exponential distribution with truncation point $c$ has PDF $f(x ; \lambda)=\lambda e^{-\lambda x} /\left[1-e^{-c \lambda}\right]$ for $x>c$. Obtain the mean and variance.

Problem 3.2 Prove that the exponential distribution is the continuous-time analog of the geometric distribution.

Solution 3.3 Let $T$ denote the lifetime of a non-repairable item (like light-bulbs, transistors, micro-batteries used in watches, etc.) that wears out over time. Divide $T$ into discrete time units of equal duration $($ say $c)$. This duration may be counted in hours for light bulbs, days for transistors, and so on. Thus, the time-clicks are counted in unit multiples of $c$ (say 1200 hours for light bulbs). Let $N$ denote the number of time-clicks until the item fails, so that $T=N c$. This equation connects the continuous lifetime with discrete time-clicks. Assume that $N$ has a geometric distribution $\mathrm{GEO}(c \lambda)$ where $c \lambda$ denotes the failure probability for each time-click. Then $\operatorname{Pr}[N=n]=(1-c \lambda)^{n-1}(c \lambda)$, where $(1-c \lambda)^{n-1}$ denotes the probability that the unit did not fail during the first $(n-1)$ time-clicks. As the $\mathrm{SF}$ of $T$ is $\operatorname{Pr}[T>k]=\operatorname{Pr}[N c>k]=$ $\operatorname{Pr}[N>\lfloor k / c\rfloor]=(1-c \lambda)^{\lfloor k / c\rfloor}$, where the integer part is taken because $N$ is discrete. This is of the form $(1-\lambda / m)^{m}$, which as $m \rightarrow \infty$ tends to $\exp (-\lambda)$ with $m=1 / c$. As $c \rightarrow 0$, the SF approaches $\exp (-k \lambda)$, which is the SF of exponential distribution.

Problem 3.4 If $X \sim E X P(\lambda)$, find (i) $\operatorname{Pr}[1 \leq X \leq 2]$ and (ii) $\operatorname{Pr}[X \geq x]$ when $\lambda=\ln (2)$.
Problem 3.5 If $\mathrm{X} \sim \mathrm{EXP}(\lambda)$ and $\mathrm{Y} \sim \operatorname{EXP}(\mu)$ prove that $\operatorname{Pr}(X<Y)=\lambda /(\lambda+\mu)$.
This distribution represents the time for a continuous process to change state (from working to non-working, from infected to recovery, detected to non-detected or vice versa). For example, the time between detection of radioactivity by a Geiger counter (absence to presence) is approximately exponentially distributed.

## 统计代写|工程统计代写engineering statistics代考|RANDOM NUMBERS

The easiest way to generate random numbers is using the inverse CDF method. As the quantiles of the distribution are given by $u=F(x)=1-\exp (-\lambda x)$, we get $x=F^{-1}(u)=-\log (1-$ $u) / \lambda$. This becomes $-\theta \log (1-u)$ for the alternate form $f(x, \theta)=(1 / \theta) \exp (-x / \theta)$. As $U(0,1)$ and $1-U$ are identically distributed (page 20 ), we could generate a random number in $[0,1$ ) and obtain $u=-\log (u) / \lambda$ as the required random number (Marsaglia (1961) [98]).

Problem $3.41$ If $X, Y$ are IID $\operatorname{EXP}(1 / 2)$, prove that $Z=(X-Y) / 2$ is Laplace distributed.
Problem 3.42 Find $k$ for the PDF $f(x)=k x^{-p} \exp (-c / x), \quad 00, p>1$.
Show that the $r^{t h}$ moment is $E\left(x^{r}\right)=c^{r} \Gamma(p-r+1) / \Gamma(p-1)$ for $r \leq(p+1)$.
Example 3.43 Mean deviation of exponential distribution Find the mean deviation of the exponential distribution $\mathrm{f}(\mathrm{x}, \lambda)=\lambda e^{-\lambda x}$.
Solution 3.44 We know that the CDF is $1-e^{-\lambda x}$. Thus, the MD is given by
$$\mathrm{MD}=2 \int_{0}^{1 / \lambda}\left(1-e^{-\lambda x}\right) d x$$
Split this into two integrals and evaluate each to get
$$\mathrm{MD}=2[1 / \lambda+(1 / \lambda) \exp (-1)-(1 / \lambda)]=2 /(e \lambda)=2 \mu_{2} * f_{m}$$
where $f_{m}=\lambda e^{-1}=\lambda / e$. Alternatively, use the $\mathrm{SF}()$ version as the exponential distribution tails off to the upper limit.

## 统计代写|工程统计代写engineering statistics代考|RELATION TO OTHER DISTRIBUTIONS

F(是;λ)=(λ/是2)经验⁡(−λ/是)

## 统计代写|工程统计代写engineering statistics代考|PROPERTIES OF EXPONENTIAL DISTRIBUTION

$$F(x ; \lambda)=\left{给出 1−经验⁡(−λX),X≥0 0 否则。 \正确的。 吨H和小号F一世s小号(X;λ)=1−CDF=经验⁡(−λX),s○吨H一个吨F(X;λ)=λ小号(X;λ).Fr○米吨H一世s吨H和H一个和一个rdF在nC吨一世○n一世s○b吨一个一世n和d一个s h(x)=f(x) /[1-F(x)]=f(x) / S(x)=\lambda,$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|工程统计代写engineering statistics代考|STATS 7053

statistics-lab™ 为您的留学生涯保驾护航 在代写工程统计engineering statistics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写工程统计engineering statistics代写方面经验极为丰富，各种代写工程统计engineering statistics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|MOMENTS AND GENERATING FUNCTIONS

The moments are easy to find using the MGF. The mean is directly obtained as $\mu=[1 /(b-$ $a)] \int_{a}^{b} x d x=[1 /(b-a)] \frac{x^{2}}{2} |_{a}^{b}=\left(b^{2}-a^{2}\right) /[2(b-a)]=(a+b) / 2$. Higher-order moments are found using the MGF. Thus,
$$E\left(X^{n}\right)=(1 /(n+1)) \sum_{k=0}^{n} a^{k} b^{n-k}$$
The MGF is
$$M_{x}(t)=E\left(e^{t x}\right)=\int_{x=a}^{b}[1 /(b-a)] e^{t x} d x=[1 /(b-a)] e^{t x} /\left.t\right|{a} ^{b}=\left(e^{b t}-e^{a t}\right) /[(b-a) t]$$ The characteristic function $(\mathrm{ChF})$ is $$\phi{x}(t)=(\exp (i b t)-\exp (i a t)) /[(b-a) i t] \quad \text { for } \quad t \neq 0 .$$
This reduces to $\sinh (a t) / a t$ for $\operatorname{CUNI}(-a,+a)$.
Moments can be found from the MGF as follows. Consider $e^{b t} / t=1 / t+b+b^{2} t / 2 !+$ $\cdots+b^{k} t^{k-1} / k !+\cdots .$ As $(1 / t)$ is common in both $e^{b t} / t$ and $e^{a t} / t$, it cancels out. The second term is $(b-a) /(b-a)=1$. Thus,
$$\left(e^{b t}-e^{a t}\right) /[(b-a) t]=1+\frac{1}{b-a}\left(\sum_{k=2}^{\infty}\left[\left(b^{k}-a^{k}\right) /(b-a)\right] t^{k-1} / k !\right) \text {. }$$
If we differentiate $(2.13)(k-1)$ times w.r.t. $t$, all terms below the $(k-1)^{t h}$ term will vanish (as they are derivatives of constants independent of $t$ ‘s) and all terms beyond the $k^{t h}$ term will contain powers of $t$. Only the $(k-1)^{t h}$ term is a constant with a $(k-1)$ ! in the numerator, which cancels out with the $k !$ giving a $k$ in the denominator. By taking the limit as $t \rightarrow 0$, we get
$$\mu_{k-1}^{\prime}=\left.\left(\partial^{k-1} / \partial t^{k-1}\right) M_{x}(t)\right|_{t=0}=\left(b^{k}-a^{k}\right) /[(b-a) k]$$

## 统计代写|工程统计代写engineering statistics代考|TRUNCATED UNIFORM DISTRIBUTIONS

Truncation in the left-tail or right-tail results in the same distribution with a reduced range. Suppose $X \sim \operatorname{CUNI}(a, b)$. If truncation occurs in the left-tail at $x=c$ where $a<c<b$, the PDF is given by
$$g(x ; a, b, c)=f(x ; a, b) /(1-F(c))=(1 /(b-a))[1 /(1-(c-a) /(b-a))]=1 /(b-c) .$$
If truncation occurs at $c$ in the left-tail and $d$ in the right-tail, the PDF is given by
$$g(x ; a, b, c, d)=f(x ; a, b) /(F(d)-F(c))=1 /(d-c)$$
This shows that truncation results in rectangular distributions.
Example 2.34 Even moments of rectangular distribution Prove that the $k^{\text {th }}$ central moment is zero for $k$ odd, and is given by $\mu_{k}=(b-a)^{k} /\left[2^{k}(k+1)\right]$ for $k$ even.

Solution 2.35 By definition, $\mu_{k}=\frac{1}{b-a} \int_{a}^{b}\left(x-\frac{a+b}{2}\right)^{k} d x$. Make the change of variable $y=$ $x-(a+b) / 2$. For $x=a$, we get $y=a-(a+b) / 2=(a-b) / 2=-(b-a) / 2$. Similarly for $x=b$, we get $y=b-(a+b) / 2=(b-a) / 2$. As the Jacobian is $\partial y / \partial x=1$, the integral becomes $\mu_{k}=\frac{1}{b-a} \int_{-(b-a) / 2}^{(b-a) / 2} y^{k} d y$. When $k$ is odd, this is an integral of an odd function in symmetric range, which is identically zero. For $k$ even, we have $\mu_{k}=\frac{2}{b-a} \int_{0}^{(b-a) / 2} y^{k} d y$ $=\left.\frac{2}{b-a}\left[y^{k+1} /(k+1)\right]\right|_{0} ^{(b-a) / 2}=(b-a)^{k} /\left[2^{k}(k+1)\right]$, as the constant $2 /(b-a)$ cancels out.
Example 2.36 Ratio of independent uniform distributions If $X$ and $Y$ are IID CUNI$(0, b)$ variates, find the distribution of $U=X / Y$.

Solution 2.37 Let $U=X / Y, V=Y$ so that the inverse mapping is $Y=V, X=U V$. The Jacobian is $|J|=v$. The joint PDF is $f(x, y)=1 / b^{2}$. Hence, $f(u, v)=v / b^{2}$. The PDF of $u$ is obtained by integrating out $v$. The region of interest is a rectangle of sides $1 \times b$ at the left,and a curve $u v=b$ to its right. Integrating out $v$, we obtain $f(u)=\int_{0}^{b} \frac{v}{b^{2}} d v$ for $0<u \leq 1$, and $f(u)=\int_{0}^{b / u} v / b^{2} d v=1 /\left(2 u^{2}\right)$ for $1<u<\infty .$
$$f(u)=\left{\begin{array}{rll} 1 / 2 & \text { for } \quad 0<u<1 \ 1 /\left(2 u^{2}\right) & \text { for } \quad 1<u<\infty \end{array}\right.$$
which is independent of the parameter $b$.

## 统计代写|工程统计代写engineering statistics代考|APPLICATIONS

This distribution finds applications in many fields. For instance, heat conductivity, thermal diffusion, and voltage fluctuations in a short time period (temporal dimension) are assumed to be uniform distributed. Some of these properties can also be extended to spatial dimensions (areas that are unit distance away from the source). It is used in nonparametric tests like KolmogorovSmirnov test. The rounding errors resulting from grouping data into classes uses a $U(0,1)$ to obtain a correction factor known as Sheppard’s correction. Quantization errors in audio-coding (compression) use this distribution. It is also used in stratified sampling, non-random clustering, etc. Random numbers from other distributions are easy to generate using $\mathrm{U}[0,1]$. These are discussed in subsequent chapters.

Example 2.38 Estimating proportions A jar contains a mixture of two liquids, $\mathrm{L}{1}$ and $\mathrm{L}{2}$, that mixes well in each other (as water and wine, or acid and water). All that is known is that “there is at most three times as much of one as the other.” Find the probability that (i) $\mathrm{L}{1} / \mathrm{L}{2} \leq 2$ and (ii) $\mathrm{L}{1} / \mathrm{L}{2} \geq 1$.

Solution $2.39$ The given condition is $\frac{1}{3} \leq \mathrm{L}{1} / \mathrm{L}{2} \leq 3$. Let $\mathrm{U}=\mathrm{L}{1} / \mathrm{L}{2}$. Assume that $U$ is uniformly distributed in $[1 / 3,3]$. As $3-1 / 3=8 / 3$, we take the density function as $\mathrm{f}(\mathrm{x})=3 / 8$, $\frac{1}{3} \leq x \leq 3$. The required answer for (i) is $\mathrm{P}[\mathrm{U} \leq 2]=\int_{1 / 3}^{2} f(x) d x=\left.(3 / 8) * x\right|{1 / 3} ^{2}=(3 / 8) *$ $(2-1 / 3)=5 / 8$; and (ii) $\mathrm{L}{1} / \mathrm{L}{2} \geq 1=\int{1}^{3} \mathrm{f}(\mathrm{x}) \mathrm{dx}=\left.(3 / 8) * x\right|_{1} ^{3}=6 / 8=0.75$

## 统计代写|工程统计代写engineering statistics代考|MOMENTS AND GENERATING FUNCTIONS

MGF 是

φX(吨)=(经验⁡(一世b吨)−经验⁡(一世一个吨))/[(b−一个)一世吨] 为了 吨≠0.

(和b吨−和一个吨)/[(b−一个)吨]=1+1b−一个(∑ķ=2∞[(bķ−一个ķ)/(b−一个)]吨ķ−1/ķ!).

μķ−1′=(∂ķ−1/∂吨ķ−1)米X(吨)|吨=0=(bķ−一个ķ)/[(b−一个)ķ]

## 统计代写|工程统计代写engineering statistics代考|TRUNCATED UNIFORM DISTRIBUTIONS

G(X;一个,b,C)=F(X;一个,b)/(1−F(C))=(1/(b−一个))[1/(1−(C−一个)/(b−一个))]=1/(b−C).

G(X;一个,b,C,d)=F(X;一个,b)/(F(d)−F(C))=1/(d−C)

$$f(u)=\左{ 1/2 为了 0<在<1 1/(2在2) 为了 1<在<∞\正确的。$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|工程统计代写engineering statistics代考|Rectangular Distribution

statistics-lab™ 为您的留学生涯保驾护航 在代写工程统计engineering statistics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写工程统计engineering statistics代写方面经验极为丰富，各种代写工程统计engineering statistics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|ALTERNATE REPRESENTATIONS

The range of the distribution is symmetric around the origin (say $-a$ to $+a$ ), or centered around a fixed constant (say $\theta-1 / 2, \theta+1 / 2$ ) in several practical applications:
$$f(x ; a, \theta)=\left{\begin{array}{rll} 1 / \theta & \text { for } & a \leq x \leq a+\theta \ 1 /(2 \theta) & \text { for } & a-\theta \leq x \leq a+\theta \ 0 & & \text { otherwise. } \end{array}\right.$$
For $a=1$ we get $f(x ; a, b)=1 / 2$ for $-1<x<1$; and for $a=1 / 2$ we get $f(u ; a, b)=1$ for $-1 / 2<u<1 / 2$

The mean and variance are $\mu=(a+b) / 2$, and $\sigma^{2}=(b-a)^{2} / 12$, as shown on page 21 . Write $\mu=(a+b) / 2$ and $\sigma=(b-a) /(2 \sqrt{3})$. Cross multiply to get $(a+b)=2 \mu$, and $(b-$ $a)=(2 \sqrt{3}) \sigma$. Add them to get $b=\mu+\sqrt{3} \sigma$. Subtracting gives $a=\mu-\sqrt{3} \sigma$, from which $(b-a)=(2 \sqrt{3}) \sigma$. Thus, the PDF becomes
$$f(x ; \mu, \sigma)=1 /(2 \sqrt{3} \sigma), \mu-\sqrt{3} \sigma \leq x \leq \mu+\sqrt{3} \sigma$$
The CDF is
F(x ; a, b)=\left{\begin{aligned} 0 & \text { for } x<\mu-\sqrt{3} \sigma \ 0.50(1+(x-\mu) /(\sqrt{3} \sigma)) & \text { for } \mu-\sqrt{3} \sigma \leq x \leq \mu+\sqrt{3} \sigma \ 1 & \text { for } x>\mu+\sqrt{3} \sigma \end{aligned}\right.
and inverse CDF is $F^{-1}(p)=\mu+\sqrt{3} \sigma(2 p-1)$ for $0<p<1$. Put $\sigma=1$ to get the standardized $\operatorname{CUNI}(-\sqrt{3},+\sqrt{3})$ that has mean zero and variance unity.

Some applications in engineering, theoretical computer science, and number theory use the uniform distribution modulo $k$. This allows the distribution to be extended to the entire real line (because the “mod $\mathrm{k}$ ” maps all such real numbers to $(0, k)$ range), and are more applicable to discrete uniform distribution. The uniform distribution on a circle has PDF $f(x)=1 /(2 \pi)$, for $0<x \leq 2 \pi$.

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

Due to its relationship with many other distributions, it is extensively used in computer generation of random variables. $U(0,1)$ is a special case of Beta-I $(a, b)$ when $a=b=1$. If $X \sim U(0,1)$ then $Y=-\log (X) \sim \mathrm{SED}$ (i.e., $\operatorname{EXP}(1))$, and $Y=-2 \log (X)$ has a $\chi_{2}^{2}$ distribution. If $x_{1}, x_{2}, \ldots, x_{k}$ are independent samples from possibly $k$ different $\mathrm{U}(0,1)$ populations,

$P_{k}=\sum_{j=1}^{k}-2 \ln \left(x_{j}\right)$ being the sum of $k$ IID $\chi_{2}^{2}$ variates has $\chi_{2 k}^{2}$ distribution. This is called Pearson’s statistic in tests of significance [114]. A simple change of variable transformation $Y=(X-a) /(b-a)$ in the general PDF results in the SUD (i.e., $\mathrm{U}(0,1)) . \mathrm{U}(0,1)$ is also related to arcsine distribution as $Y=-\cos (\pi U / 2)$ (Chapter 5). If $X$ is any continuous random variable with CDF $F(x)$, then $U=F(x) \sim U[0,1]$.
Example 2.1 Distribution of $F(x)$
If $X$ is a continuous variate, find the distribution of $U=F(x)$.
Solution 2.2 Consider
$$\mathrm{F}(\mathrm{u})=\operatorname{Pr}(\mathrm{U} \leq u)=\operatorname{Pr}(\mathrm{F}(\mathrm{x}) \leq u)=\operatorname{Pr}\left(\mathrm{x} \leq \mathrm{F}^{-1}(u)\right)=\mathrm{F}\left[\mathrm{F}^{-1}(u)\right]=\mathrm{u}$$
The CDF of a rectangular distribution $\operatorname{CUNI}(a, b)$ is $(x-a) /(b-a)$. Put $a=0, b=1$ to get $F(x)=x$. Equation (2.6) then shows that $U$ is an SUD.

This property can be used to generate random numbers from a distribution if the expression for its CDF (or SF) involves simple or invertible arithmetic or transcendental functions. For example, the CDF of an exponential distribution is $F(x)=1-\exp (-\lambda x)$. Equating to a random number $u$ in the range $[0,1]$ and solving for $x$, we get $1-e^{-\lambda x}=u$ or $x=-\log (1-u) / \lambda$, using which random numbers from exponential distributions can be generated.

Problem 2.3 If a doctor and a nurse arrive a hospital independently and uniformly during $8 \mathrm{AM}$ and $9 \mathrm{AM}$, find the probability that the first patient to arrive has to wait longer than 10 $\mathrm{min}$, if consultation is possible only when both the doctor and nurse are in office.

Problem 2.4 If three random variables $X, Y, Z$ are independently and uniformly distributed over $(0,1)$, show that $\operatorname{Pr}(X>Y Z)=3 / 4$

## 统计代写|工程统计代写engineering statistics代考|PROPERTIES OF RECTANGULAR DISTRIBUTION

This distribution has a special type of symmetry called flat-symmetry. Hence, all odd central moments $\mu_{2 r+1}$ except the first one are zeros. The median always coincides with the mean, and the mode can be any value within the range. As the probability is constant throughout the interval, the range is always finite (and quite often small). As $F(x)=(x-a) /(b-a)$, its inverse is
$$F^{-1}(p)=a+p(b-a), \quad \text { for } 0<p<1 .$$
A uniform distribution defined in an interval $(c, c+\theta)$ has PDF
$$f(x ; \theta)=1 / \theta \text { for } c \leq x \leq c+\theta .$$

Take $c=0$ to get the standard form $f(x ; \theta)=1 / \theta, 0<x<\theta$. This is the analogue of the $\operatorname{DUNI}(\mathrm{N})$ with probability function $f(x ; N)=1 / N$, for $x=0,1,2, \ldots, N-1$ discussed in Chapter 3 of Chattamvelli and Shanmugam (2020) [42]. The transformation $Y=(b-a)-X$ results in the same distribution. In particular, if $X \sim U(0,1)$ then $Y=1-X \sim U(0,1)$. This property of $U(0,1)$ is used in generating random samples from other distributions like the exponential distribution (page 35). Only the extremes of a sample $x_{(1)}$ and $x_{(n)}$ are sufficient to fit this distribution.
Problem 2.18 If $X \sim \mathrm{U}[0,1]$, find the distribution of $Y=\exp (X)$, and its variance.
Problem $2.19$ If $X \sim U(0,1)$, find the distribution of $Y=X /(1-X)$.

## 统计代写|工程统计代写engineering statistics代考|ALTERNATE REPRESENTATIONS

$$f(x ; a, \theta)=\left{ 1/θ 为了 一个≤X≤一个+θ 1/(2θ) 为了 一个−θ≤X≤一个+θ 0 否则。 \正确的。$$

F(X;μ,σ)=1/(23σ),μ−3σ≤X≤μ+3σ
CDF 是
$$F(x ; a, b)=\left{ 0 为了 X<μ−3σ 0.50(1+(X−μ)/(3σ)) 为了 μ−3σ≤X≤μ+3σ 1 为了 X>μ+3σ\正确的。$$

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

F(在)=公关⁡(在≤在)=公关⁡(F(X)≤在)=公关⁡(X≤F−1(在))=F[F−1(在)]=在

## 统计代写|工程统计代写engineering statistics代考|PROPERTIES OF RECTANGULAR DISTRIBUTION

F−1(p)=一个+p(b−一个), 为了 0<p<1.

F(X;θ)=1/θ 为了 C≤X≤C+θ.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|工程统计代写engineering statistics代考|MOMENTS AND CUMULANTS

statistics-lab™ 为您的留学生涯保驾护航 在代写工程统计engineering statistics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写工程统计engineering statistics代写方面经验极为丰富，各种代写工程统计engineering statistics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|MOMENTS AND CUMULANTS

The concept of moments is used in several applied fields like mechanics, particle physics, kinetic gas theory, etc. Moments and cumulants are important characteristics of a statistical distribution and plays an important role in understanding respective distributions. The population moments (also called raw moments or moments around zero) are mathematical expectations of powers of the random variable. They are denoted by Greek letters as $\mu_{r}^{\prime}=\mathrm{E}\left(X^{r}\right)$, and the corresponding sample moment by $m_{r}^{\prime}$. The zeroth moment being the total probability is obviously one. The first moment is the population mean (i.e., expected value of $X, \mu=E(X)$ ). There exist an alternative expectation formula for non-negative continuous distributions as $E(X)=\int_{l l}^{u l}(1-F(x)) d x$, where $l l$ is the lower and $u l$ is the upper limit. This takes the simple and more familiar form $E(X)=\int_{0}^{\infty}(1-F(x)) d x=\int_{0}^{\infty} S(x) d x$ when the range is $(0, \infty)$. Positive moments are obtained when $r$ is a positive integer, negative moments when $r$ is a negative integer, and fractional moments when $r$ is a real number. Alternate expectation formulas exist for higher-order moments as well (see [60]):
$$E\left(X^{r}\right)=\int_{0}^{\infty} r x^{r-1}(1-F(x)) d x=\int_{0}^{\infty} r x^{r-1} S(x) d x ; \quad \text { for } \quad r \geq 1 .$$
Central moments are moments around the mean, denoted by $\mu_{r}=\mathrm{E}\left((X-\mu)^{r}\right)$. As $E(X)=\mu$, the first central moment is always zero, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis.
There are several measures of dispersion available. Examples are mean absolute deviation, variance (and standard deviation (SD)), range, and coefficient of variation (CV) (the ratio of SD and the mean $\left(s / \bar{x}{n}\right)$ for a sample, and $\sigma / \mu$ for a population). A property of the variance is that the variance of a linear combination is independent of the constant term, if any. Mathematically, $\operatorname{Var}(c+b X)=|b| \operatorname{Var}(X)$ (which is devoid of the constant c). Similarly, the variance of a linear combination is the sum of the variances if the variables are independent $(\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y))$. The $\mathrm{SD}$ is the positive square root of variance, and is called volatility in finance and econometrics. It is used for data normalization as $z{i}=\left(x_{i}-\bar{x}\right) / s$ where $s$ is the $\mathrm{SD}$ of a sample $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$. Data normalized to the same scale or frequency can be combined. This technique is used in several applied fields like spectroscopy, thermodynamics, machine learning, etc. The CV quantifies relative variation within a sample or population. Very low CV values indicate relatively little variation within the groups, and very large values do not provide much useful information. It is used in bioinformatics to filter genes, which are usually combined in a serial manner. It also finds applications in manufacturing engineering, education, and psychology to compare variations among heterogeneous groups as it captures the level of variation relative to the mean.

The inverse moments (also called reciprocal moments) are mathematical expectations of negative powers of the random variable. A necessary condition for the existence of inverse moments is that $f(0)=0$, which is true for $\chi^{2}, F$, beta, Weibull, Pareto, Rayleigh, and Maxwell distributions. More specifically, $E(1 / X)$ exists for a non-negative random variable $X$ iff $\int_{0}^{\delta}(f(x) / x) d x$ converges for some small $\delta>0$. Although factorial moments can be defined in terms of Stirling numbers, they are not popular for continuous distributions. The absolute moments for random variables that take both negative and positive values are defined as $v_{k}=E\left(|X|^{k}\right)=\int_{-\infty}^{\infty}|x|^{k} f(x) d x=\int_{-\infty}^{\infty}|x|^{k} d F(x)$

## 统计代写|工程统计代写engineering statistics代考|SIZE-BIASED DISTRIBUTIONS

Any statistical distribution with finite mean can be extended by multiplying the PDF or PMF by $C(1+k x)$, and choosing $C$ such that the total probability becomes one ( $k$ is a user-chosen nonzero constant). This reduces to the original distribution for $k=0$ (in which case $C=1$ ). The unknown $C$ is found by summing over the range for discrete, and by integrating for continuous and mixed distributions. This is called size-biased distribution (SBD), which is a special case of weighted distributions. Consider the continuous uniform distribution with PDF $f(x ; a, b)=1 /(b-a)$ for $a<x<b$, denoted by CUNI $(a, b)$. The size-biased distribution is $g(y ; a, b, k)=[C /(b-a)](1+k y)$. This means that any discrete, continuous, or mixed distribution for which $\mu=E(X)$ exists can provide a size-biased distribution. As shown in Chapter $2, \operatorname{CUNI}(a, b)$ has mean $\mu=(a+b) / 2$. Integrate the above from $a$ to $b$, and use the above result to get $[C /(b-a)] \int_{a}^{b}(1+k y) d y=1$, from which $C=2 /[2+k(a+b)]=1 /(1+k \mu)$. Similarly, the exponential SBD is $g(y ; k, \lambda)=C \lambda(1+k y) \exp (-\lambda y)$ where $C=\lambda^{2} /(k+\lambda)$. As another example, the well-known Rayleigh and Maxwell distributions (discussed in Part II) are not actually new distributions, but simply size-biased Gaussian distributions $\mathrm{N}\left(0, a^{2}\right)$ with biasing term $x$, and $\mathrm{N}(0, k T / m)$ with biasing term $x^{2}$, respectively. Other SBDs are discussed in respective chapters.

We used expectation of a linear function $(1+k x)$ in the above formulation. This technique can be extended to higher-order polynomials acting as weights $\left(\right.$ e.g., $\left.\left(1+b x+c x^{2}\right)\right)$, as also first- or higher-order inverse moments, if the respective moments exist. Thus, if $E(1 /(a+$ $b x)$ ) exists for a distribution with PDF $f(x)$, we could form a new SBD as $g(x ; a, b, C)=$ $C f(x) /(a+b x)$ by choosing $C$ so as to make the total probability unity. This concept was introduced by Fisher (1934) [61] to model ascertainment bias in the estimation of frequencies, and extended by Rao $(1965,1984)[113,115]$. More generally, if $w(x)$ is a non-negative weight function with finite expectation $(E(w(x))<\infty)$, then $w(x) f(x) / E(w(x))$ is the PDF of weighted distribution (in the continuous case; or PMF in discrete case). It is sometimes called lengthbiased distribution when $w(x)=x$, with PDF $g(x)=x f(x) / \mu$ (because the weight $x$ acts as a length from some fixed point of reference). As a special case, we could weigh using $E\left(x^{k}\right)$ and $E\left(x^{-k}\right)$ when the respective moments of order $k$ exists with PDF $x^{\pm k} f(x) / \mu_{\pm k}^{\prime}$. This results in either distributions belonging to the same family (as in $\chi^{2}$, gamma, Pareto, Weibull, F, beta, and power laws), different known families (size-biasing exponential distribution by $E\left(x^{k}\right)$ results in gamma law, and uniform distribution in power law), or entirely new distributions (Student’s T, Laplace, Inverse Gaussian, etc.). Absolute-moment-based weighted distributions can be defined when ordinary moments do not exist as in the case of Cauchy distribution (see Section 9.4, page 124). Fractional powers can also be used to get new distributions for positive random variables. Other functions like logarithmic (for $x \geq 0$ ) and exponential can be used to get nonlinearly weighted distributions. The concept of SBD is applicable to classical (discrete and continuous) distributions as well as to truncated, transmuted, exponentiated, skewed, mixed, and other extended distributions.

## 统计代写|工程统计代写engineering statistics代考|LOCATION-AND-SCALE DISTRIBUTIONS

The LaS distributions are those in which the location information (central tendency) is captured in one parameter, and scale (spread and skewness) is captured in another.

Definition 1.11 A parameter $\theta$ is called a location parameter if the PDF is of the form $f(x \mp$ $\theta)$, and a scale parameter if the PDF is of the form $(1 / \theta) f(x / \theta)$.

Most of the LaS distributions are of the continuous type. Examples are the general normal, Cauchy, and double-exponential (Laplace) distributions. If $\mu$ is the mean and $\sigma$ is the standard deviation of a univariate random variable $X$, and $Z=(X-\mu) / \sigma$ results in a standard distribution (devoid of parameters; see Section 1.1.2 in page 2), we say that $X$ belongs to the LaS family. This definition can easily be extended to the multivariate case where $\mu$ is a vector and $\Sigma$ is a matrix so that $Z=(X-\mu)^{\prime} \Sigma^{-1}(X-\mu)$ is in standard form. Sample data values are standardized using the transformation $z_{k}=\left(x_{k}-\bar{x}\right) / s_{x}$, where $s_{x}$ is the sample standard deviation, which can be applied to samples from any population including $\mathrm{LaS}$ distributions. The resulting values are called $z$-values or $z$-scores.

Write the above in univariate case as $X=\mu+\sigma Z$. As $\sigma$ is positive, a linear transformation with positive slope of any standard distribution results in a location-scale family for the underlying distribution. When $\sigma=1$, we get the one-parameter location family, and when $\mu=0$, we get the scale family. The exponential, gamma, Maxwell, Pareto, Rayleigh, Weibull, and halfnormal are scale-family distributions. The CDF of $X$ and $Z$ are related as $F((x-\mu) / \sigma)=G(x)$, and the quantile functions of $X$ and $Z$ are related as $F^{-1}(p)=\mu+\sigma G^{-1}(p)$. For $X$ continuous, the densities are related as $g(x)=(1 / \sigma) f((x-\mu) / \sigma)$. Maximum likelihood estimates (MLE) of the parameters of LaS distributions have some desirable properties. They are also easy to fit using available data. Extensions of this include $\log$-location-scale (LLS) distributions, nonlinearly transformed $\mathrm{LaS}$ distributions (like trigonometric, transcendental, and other functions of it, etc.) (Jones (2015) [80], Jones and Angela (2015) [81]).

## 统计代写|工程统计代写engineering statistics代考|LOCATION-AND-SCALE DISTRIBUTIONS

LaS 分布是在一个参数中捕获位置信息（集中趋势）而在另一个参数中捕获尺度（分布和偏度）的分布。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写工程统计engineering statistics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写工程统计engineering statistics代写方面经验极为丰富，各种代写工程统计engineering statistics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。