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## 统计代写|统计推断代考Statistical Inference代写|Ramsey

In order to develop a more rigorous framework for probability, we take a brief detour into an area of mathematics known as measure theory. The ideas here may seem a bit esoteric at first. Later we will see how they relate to our intuition about how probability should behave.

Measure
Consider a set, $\Psi$, and a subset, $A \subseteq \Psi$. We want to get some idea of the size of $A$. If $A$ is finite, one obvious way to do this is just to count the number of elements in $A$. Measures are functions acting on subsets that give us an idea of their size and generalise the notion of counting elements. Since a measure acts on subsets of the sample space, the domain for a measure will be a collection of subsets. In order to ensure that the measure can be defined sensibly, we need this collection to have certain properties.

## 统计代写|统计推断代考Statistical Inference代写|Probability measure

In this section we will show how the framework of section $2.2 .1$ allows us to develop a rigorous definition of probability. Measure gives us a sense of the size of a set. Probability tells us how likely an event is. We will put these two ideas together to define probability as a measure.

To define a measure we need a measurable space, that is, a set and a $\sigma$-algebra defined on the set. Our intuitive description of probability in section $2.1$ introduces the idea of a sample space, $\Omega$, the set of all possible outcomes of our experiment. We also define events as subsets of $\Omega$ containing outcomes that are of interest. From this setup we can generate a measurable space, $(\Omega, \mathcal{F})$, where $\mathcal{F}$ is a $\sigma$-algebra defined on $\Omega$. Here $\mathcal{F}$ is a collection of subsets of $\Omega$ (as usual), and we interpret the elements of $\mathcal{F}$ as being events. Thus, if $A \in \mathcal{F}$ then $A$ is an event. Remember that probability is always associated with events so $\mathcal{F}$ will be the domain for probability measure.
Definition 2.2.6 (Probability measure)
Given a measurable space $(\Omega, \mathcal{F})$, a probability measure on $(\Omega, \mathcal{F})$ is a measure $\mathrm{P}: \mathcal{F} \rightarrow[0,1]$ with the property that $\mathrm{P}(\Omega)=1$.

Note that, as we might expect, the definition restricts the codomain of $\mathrm{P}$ to be the unit interval, $[0,1]$. The triple consisting of a sample space, a collection of events (forming a $\sigma$-algebra on the sample space), and a probability measure, $(\Omega, \mathcal{F}, \mathrm{P})$, is referred to as a probability space.

We give two examples of functions which satisfy the conditions for probability measures. Showing that these functions are probability measures is part of Exercise $2.2$

# 统计推断代写

Measure

## 统计代写|统计推断代考统计推断代写|概率度量

. 在本节中，我们将展示$2.2 .1$节的框架如何允许我们对概率进行严格的定义。测量使我们对一组的大小有一种感觉。概率告诉我们一个事件发生的可能性。我们将把这两个概念结合起来，把概率定义为一种度量 要定义一个度量，我们需要一个可度量空间，即一个集合和一个在集合上定义的$\sigma$ -代数。我们在$2.1$部分对概率的直观描述引入了样本空间$\Omega$的概念，是我们实验的所有可能结果的集合。我们还将事件定义为$\Omega$的子集，其中包含感兴趣的结果。通过这个设置，我们可以生成一个可度量的空间$(\Omega, \mathcal{F})$，其中$\mathcal{F}$是在$\Omega$上定义的$\sigma$ -代数。这里$\mathcal{F}$是$\Omega$(和往常一样)的子集的集合，我们将$\mathcal{F}$的元素解释为事件。因此，如果$A \in \mathcal{F}$则$A$是一个事件。记住，概率总是与事件相关，因此$\mathcal{F}$将是概率度量的域。定义2.2.6(概率度量)

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。

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## 统计代写|统计推断代考Statistical Inference代写|Ramsey

Ramsey, a brilliant student of Russell and Wittgenstein, could fairly be called the Galois of probability theory: He died in 1930, of jaundice, ${ }^{10}$ at the age of 26 . His work was published posthumously as a collection of essays entitled The Foundations of Mathematics (1931). Ramsey, like Borel, got started on subjectivism from critically contemplating Keynes’ theory; Keynes himself, as was noted earlier, appears ultimately to have been persuaded that Ramsey was on the right track. Ramsey, for his part, questioned whether there is, in fact, any such thing as Keynes’ “probability relations”:
All we appear to know about them are certain general propositions, the laws of addition and multiplication; it is as if everyone knew the laws of geometry but no one could tell whether any given object were round or square; and I find it hard to imagine how so large a body of general knowledge can be combined with so slender a stock of particular facts. $(1931$, p. 162)

## 统计代写|统计推断代考Statistical Inference代写|De Finetti

The personalization of probability was carried through most thoroughly by de Finetti, who saw probability as arising from uncertainty, whatever its source. Hence a string of digits in the decimal expansion of $\pi$-say, the 2001st to the 3000 thqualifies, as well as a string of digits in a random number table, as an object of probability theory. Indeed “random,” in his theory, means “unknown to You,” regardless of whether the event is determined or known to anyone else. It is a consequence of de Finetti’s definition that a probability cannot be unknown: Probability characterizes uncertainty, and there is no second-order uncertainty about uncertainty. He compares the concept of unknown probability with “thinking that in a statistical survey it makes sense to indicate, in addition to those whose sex is unknown, those for whom one does not even know ‘whether the sex is unknown or not'” (de Finetti, 1974, p. 84).

In using betting to measure probabilities, de Finetti is of course involved in the comparisons of expectations. Indeed, like Borel (1924), he introduces probability and expectation through the explicitly monetary concept of prices. If $X$ is a “random gain,”
We might ask an individual, e.g. You, to specify the certain gain which is considered equivalent to $X$. This we might call the price (for You) of $X$ (we denote it by $P(X)$ ) in the sense that, on your scale of preference, the random gain $X$ is, or is not, preferred to a certain gain $x$ according as $x$ is less than or greater than $P(x)$. $(1974$, p. 73 )

# 统计推断代写

## 统计代写|统计推断代考Statistical Inference代写|De Finetti

de Finetti 对概率的个性化进行得最为彻底，他认为概率源于不确定性，无论其来源如何。因此，十进制扩展中的一串数字圆周率- 比如说，第 2001 到第 3000 以及随机数表中的一串数字都可以作为概率论的对象。事实上，在他的理论中，“随机”意味着“你不知道”，无论该事件是确定的还是其他人知道的。概率不可能是未知的，这是德菲内蒂定义的结果：概率表征不确定性，并且不存在关于不确定性的二阶不确定性。他将未知概率的概念与“认为在统计调查中，除了那些不知道性别的人之外，指出那些甚至不知道‘性别是否未知’的人是有意义的”（de菲内蒂，1974 年，第 84 页）。

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。

Posted on Categories:Statistical inference, 统计代写, 统计推断

## 统计代写|统计推断代写STATISTICAL INFERENCE代写|ST502 Shewhart Control Charts

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## 统计代写|统计推断代写STATISTICAL INFERENCE代写|Shewhart Control Charts for Monitoring Process Mean

For monitoring the mean of a process, we typically use the sample mean $(\bar{X})$. An example follows.
Example 3.1 A Parametric Shewhart $\bar{X}$ Control Chart
Shewhart charts are typically applied with subgroup data. Column (a) of Table $3.4$ presents some simulated data from a normal distribution, which represent measurements taken from 25 independent samples, each of size 5 $(n=5)$ on a type of wafer. Suppose that, from engineering considerations, the IC mean dimension of the wafers $\mu_{0}$ is specified to be $1.5 \mathrm{~cm}$ and the IC process standard deviation $\sigma_{0}$ is known to be equal to $0.1 \mathrm{~cm}$. The mean of each sample is shown in Column (b) of Table 3.4. To illustrate the calculations, consider sample number 1 . The first charting statistic, the mean of sample 1 , is calculated as follows
$$\bar{X}{1}=\frac{1.3235+1.4128+1.6744+1.4573+1.6914}{5}=\frac{7.5594}{5}=1.5119$$ To apply the formulas for the control limits in Equation 3.1, note that the expected value of $\bar{X}$ (i.e., $\mu{\bar{X}}$ ) is simply the specified process mean $\mu_{0}$, whereas the standard deviation of $\bar{X}$, namely, $\sigma_{\bar{X}}$, is equal to $\frac{\sigma_{0}}{\sqrt{n}}$. Thus, the $C L$ and the $k$-sigma control limits for a Shewhart $\bar{X}$ control chart are given by
\begin{aligned} U C L &=\mu_{0}+k \frac{\sigma_{0}}{\sqrt{n}} \ C L &=\mu_{0} \ L C L &=\mu_{0}-k \frac{\sigma_{0}}{\sqrt{n}} \end{aligned}

## 统计代写|统计推断代写STATISTICAL INFERENCE代写|Shewhart Control Charts for Monitoring Process Variation

Variation is an important aspect of any analysis and thus it is necessary to monitor the process variation or spread and ensure that it is IC. Moreover, as we see in Equation 3.1, the Shewhart control limits for the process mean depend on the process standard deviation. Thus, unless the standard deviation remains IC, the control chart for the mean will not be very informative. So, we need to monitor the variance or the standard deviation using a control chart.

There are several possible statistics that can be used to monitor variation. The most popular choices are the sample range $(R)$, the sample standard deviation $(S)$, and the sample variance $\left(S^{2}\right)$.

Typically, we use a control chart to monitor the process mean together with a control chart to monitor the process variation. If the variation is IC, we go ahead and examine the control chart for the mean. For example, a Shewhart $\bar{X}$ chart for the mean is often used together with a Shewhart $R$ chart for the spread. Note that, for illustration, we consider the Shewhart $R$ chart even though recent literature recommends using a different spread chart, such as the Shewhart $S$ chart; see, for instance, Mahmoud et al. (2010). We do this because the Shewhart $R$ chart is simple and continues to be used in the industry.

In Case $\mathrm{K}$, the values of $\mu$ and $\sigma$ are known or are specified so that they can be used to construct the respective control charts. We illustrate the Shewhart $R$ and $S$ charts for the known standard deviation $\sigma_{0}$.

## 统计代写|统计推断代写STATISTICAL INFERENCE代写|Shewhart Control Charts for Monitoring Process Mean

Shewhart 图通常与子组数据一起应用。表 (a) 栏 $3.4$ 呈现一些来自正态分布的模拟数据，这些数据代表从 25 个独立样本中获取 的测量值，每个样本大小为 $5(n=5)$ 在一种晶圆上。假设，从工程考虑，晶片的 IC 平均尺寸 $\mu_{0}$ 被指定为 $1.5 \mathrm{~cm}$ 和 IC 工艺标准 差 $\sigma_{0}$ 已知等于 $0.1 \mathrm{~cm}$. 每个样本的平均值显示在表 $3.4$ 的 (b) 栏中。为了说明计算，请考虑样本编号 1 。第一个图表统计量，即 样本 1 的平均值，计算如下
$$\bar{X} 1=\frac{1.3235+1.4128+1.6744+1.4573+1.6914}{5}=\frac{7.5594}{5}=1.5119$$

$$U C L=\mu_{0}+k \frac{\sigma_{0}}{\sqrt{n}} C L \quad=\mu_{0} L C L=\mu_{0}-k \frac{\sigma_{0}}{\sqrt{n}}$$

## 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。