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

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• 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

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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。