## 统计代写|统计推断代写Statistical inference代考|STAT7604

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference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。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference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|统计推断代写Statistical inference代考|Permutations and combinations

ii. A combination of length $k \leq n$ is an unordered subset of $Q$ containing $k$ elements.

We distinguish between these two cases by using (…) to denote permutation and ${\ldots}$ to denote combination.
Claim 2.3.4 (Number of permutations and number of combinations)
i. If the number of permutations of length $k$ that can be formed from $n$ distinct elements is denoted ${ }^n P_k$, then
$${ }^n P_k=\frac{n !}{(n-k) !} .$$
ii. If the number of combinations of length $k$ that can be formed from $n$ distinct elements is denoted ${ }^n C_k$, then
$${ }^n C_k=\frac{n !}{k !(n-k) !}$$
The number of permutations, ${ }^n P_k=n \times(n-1) \times \ldots \times(n-k+1)$, is a direct consequence of the multiplication rule. Note that one implication of this is that the number of ways of ordering all $n$ elements is ${ }^n P_n=n$ !. The general expression for the number of combinations requires a little more thought.

Suppose that we know the number of combinations of length $k$, that is, we know ${ }^n C_k$. By the above argument, the number of ways of ordering each one of these combinations is $k$ !. The multiplication rule then tells us that ${ }^n P_k=k !{ }^n C_k$. By rearranging we arrive at the general result, ${ }^n C_k={ }^n P_k / k !$.

## 统计代写|统计推断代写Statistical inference代考|Number of combinations and multinomial coefficients

In Claim 2.3.4 we define ${ }^n C_k=n ! /(k !(n-k) !)$ as being the number of combinations of length $k$ from $n$ distinct objects. These numbers arise in a number of different sometimes surprising – contexts and are worthy of consideration in their own right. A common notation for the number of combinations is
$$\left(\begin{array}{l} n \ k \end{array}\right)={ }^n C_k=\frac{n !}{k !(n-k) !} .$$
This quantity is sometimes referred to as ” $n$ choose $k$ “. We start by considering a property that is closely related to our original definition in terms of counting combinations.
Proposition 2.3.6
Consider a collection of $n$ objects, $k$ of which are of type a and $(n-k)$ of which are of type $b$. The number of ways of arranging these objects into sequences of type a and type $b$ is $\left(\begin{array}{l}n \ k\end{array}\right)$.
Proof.
Consider the problem as one of positioning $k$ things of type $a$ into $n$ slots (the remaining slots will be filled with things of type $b$ ). If we label the slots $1, \ldots, n$, the problem is then equivalent to selecting a set of $k$ numbers from ${1, \ldots, n}$; each number we choose will give us a position occupied by something of type $a$, so order is unimportant. By Claim 2.3.4, the number of ways of doing this is $\left(\begin{array}{l}n \ k\end{array}\right)$.

The number of combinations also appears in the expansion of expressions of the form $(a+b)^n$. In order to expand this type of expression we can write it out in full and multiply out the brackets; for example,
\begin{aligned} (a+b) &=a+b \ (a+b)^2 &=(a+b)(a+b)=a^2+a b+b a+b^2=a^2+2 a b+b^2 \ (a+b)^3 &=(a+b)(a+b)(a+b)=\ldots=a^3+3 a^2 b+3 a b^2+b^3 \end{aligned}

## 统计代写|统计推断代写统计推断代考|排列和组合

i。如果可以由$n$个不同元素组成的长度为$k$的排列数表示为${ }^n P_k$，则
$${ }^n P_k=\frac{n !}{(n-k) !} .$$
ii。如果可以由$n$个不同元素组成的长度为$k$的组合的数量表示为${ }^n C_k$，则
$${ }^n C_k=\frac{n !}{k !(n-k) !}$$

## 统计代写|统计推断代写统计推断代考|组合和多项系数的数目

$$\left(\begin{array}{l} n \ k \end{array}\right)={ }^n C_k=\frac{n !}{k !(n-k) !} .$$

\begin{aligned} (a+b) &=a+b \ (a+b)^2 &=(a+b)(a+b)=a^2+a b+b a+b^2=a^2+2 a b+b^2 \ (a+b)^3 &=(a+b)(a+b)(a+b)=\ldots=a^3+3 a^2 b+3 a b^2+b^3 \end{aligned}

## 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|统计推断代写Statistical inference代考|STATS2107

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference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。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference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|统计推断代写Statistical inference代考|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 nsual), 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 $P$ 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 $2.2$.
Example 2.2.7 (Intuitive and not so intuitive probability measures)
Suppose that we have a measurable space $(\Omega, \mathcal{F})$. Two functions that we have encountered before satisfy the properties of probability measures.

1. Equation (2.1) defines a probability measure $\mathrm{P}$ by $\mathrm{P}(A)=|A| /|\Omega|$ for $A \in \mathcal{F}$, where $|A|$ is the number of outcomes in $A$. It is easy to show that this satisfies the conditions for a probability measure.

## 统计代写|统计推断代写Statistical inference代考|Methods for counting outcomes

Suppose that we have a sample space $\Omega$ which is finite, that is, $|\Omega|<\infty$, and a probability measure $\mathrm{P}(A)=|A| /|\Omega|$ for all $A \in \mathcal{F}$. Example 2.1.1 gives a simple illustration of throwing two fair dice, for which $|\Omega|=6 \times 6=36$. In this case the entire sample space is easy to map out and the probabilities of events are readily calculated by counting. In practical experiments, the sample space is usually too large to be written down in its entirety.
Example 2.3.1

1. To play Lotto you select six numbers from ${1, \ldots, 59}$, pay $£ 2$ and receive a ticket with your numbers printed on it. The main Lotto draw on a Saturday night consists of six balls selected at random without replacement from an urn containing 59 balls labelled $1, \ldots, 59$. Initially, we would like to know the probability of winning with a single entry, that is, we would like to know the probability that the six numbers drawn match those on our ticket.
2. I have a lecture in a room with 100 people in it ( 99 students and me). What is the probability that there is at least one pair of people with the same birthday?

In order to tackle problems of the sort given in Example 2.3.1 we need to develop formal methods for counting outcomes. The basis of these methods is the following simple claim.
Claim 2.3.2 (Multiplication rule)
Suppose that $Q_1, \ldots, Q_k$ are experiments and that experiment $Q_i$ has $n_i$ possible outcomes for $i=1, \ldots, k$. The experiment consisting of the ordered sequence $\left(Q_1, Q_2, \ldots, Q_k\right)$ has $n_1 \cdot n_2 \cdot \ldots \cdot n_k=\prod_{i=1}^k n_i$ possible outcomes.
Example 2.3.1 (Revisited I)
We can use the multiplication rule to calculate the size of the sample space in each of our two cases.

1. In Lotto there are 59 ways to choose the first ball. Since this ball is not replaced, there are 58 ways to choose the second ball, and so on. Thus the number of outcomes of the form (ball $1, \ldots$, ball 6 ) is $59 \times 58 \times 57 \times 56 \times 55 \times 54=3.244 \times 10^{10}$ (4 significant figures). We assume, on the basis that the balls are selected at random, that each of these outcomes is equally likely. Of course for the lottery the order in which the numbers appear is unimportant – we will consider this in more detail in section 2.3.1.

.

## 统计代写|统计推断代写统计推断代考|结果计数方法

.统计方法

1. 要玩乐透，你从${1, \ldots, 59}$中选择6个号码，支付$£ 2$，就会收到一张印有你的号码的彩票。周六晚上的乐透主抽奖由6个随机抽取的球组成，没有替换从一个瓮中59个球，标签为$1, \ldots, 59$。首先，我们想知道单次抽奖中奖的概率，也就是说，我们想知道抽到的6个号码与彩票上的号码相匹配的概率。我在一个有100人(99个学生和我)的房间里有一个讲座。至少有一对同一天生日的人的概率是多少?

## 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|统计推断代写Statistical inference代考|MAST30020

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference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。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference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|统计推断代写Statistical inference代考|Intuitive probability

Every year, at the start of the first lecture, we ask students to put their hand up if they do not know what probability is; no-one puts their hand up. We then ask for volunteers to explain probability to their colleagues; no-one volunteers. Probability is something about which we all have some intuitive notions, but these are rather hard to explain. The following simple example is used to illustrate.
Example 2.1.1 (Roll of two fair dice)
We roll two fair dice. What is the probability that the sum of the values on the dice is greater than 10? You should be able to work this out easily. The rest of this section is an attempt to give a thorough account of the reasoning you might have used to arrive at your answer.

The first thing to note is that probabilities are always associated with events. The probability of an event is a number between 0 and 1 (inclusive) providing an indication of how likely the event is; an event with probability 0 will not happen while an event with probability 1 is certain to happen. We can stretch our intuition a bit further. Some informal definitions are helpful at this stage.
Definition 2.1.2 (Experiment, sample space, and events)
i. An experiment is a repeatable procedure that has a well-defined set of possible outcomes.

ii. The sample space, $\Omega$, is the set of all possible outcomes of an experiment. Thus, any sample outcome $\omega$ is a member of the sample space $\Omega$, that is, $\omega \in \Omega$.
iii. An event, $A$, is a set of outcomes that is of interest to us. An event is a subset of the sample space, $A \subseteq \Omega$.
iv. The complement of $A$, denoted $A^c$, is the set of all outcomes not contained in $A$, that is, $A^c={\omega \in \Omega \mid \omega \notin A}$.

If all the outcomes in the sample space are equally likely and the sample space is finite, we can construct an intuitively appealing definition of probability of the event $A$
$$\mathrm{P}(A)=\frac{|A|}{|\Omega|}$$
where $|A|$ is the number of outcomes that are in $A$, and $|\Omega|$ is the total number of possible outcomes. The statement that the sample space is finite means that there is a finite number of possible outcomes of the experiment, that is, $|\Omega|<\infty$.

It is important to remember that probability is a mathematical construct. When we apply probability ideas to real situations we always make assumptions. Thus, probability statements are statements about a mathematical model, not statements about reality.

## 统计代写|统计推断代写Statistical inference代考|Mathematical probability

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.
Definition 2.2.1 ( $\sigma$-algebra)
Let $\Psi$ be a set and let $\mathcal{G}$ be a collection of subsets of $\Psi$. We say that $\mathcal{G}$ is a $\sigma$-algebra defined on $\Psi$ when the following conditions hold:
i. $\varnothing \in \mathcal{G}$,
ii. if $A \in \mathcal{G}$ then $A^c \in \mathcal{G}$,
iii. if $A_1, A_2, \ldots \in \mathcal{G}$ then $\bigcup_{i=1}^{\infty} A_i \in \mathcal{G}$.

We will discuss some of the intuitive reasoning behind these properties in the context of probability in section 2.2.2. The following example shows two $\sigma$-algebras that may be constructed for any set that has a non-trivial subset.
Example 2.2.2 (Small and large $\sigma$-algebras)
Consider a set $\Psi$ together with a non-trivial subset $A \subset \Psi$. Two examples of $\sigma$ algebras defined on $\Psi$ are given below.

1. The smallest non-degenerate $\sigma$-algebra contains 4 elements. $G=\left{\varnothing, A, A^c . \Psi\right}$. where $A \subset \Psi$.
2. The $\sigma$-algebra with the largest number of members is given by including every subset of $\Psi$. We can write this as $G={A: A \subseteq \Psi}$. This is referred to as the power set of $\Psi$, and is sometimes written $\mathcal{P}(\Psi)$ or ${0,1}^{\Psi}$.

The pair consisting of a set and a $\sigma$-algebra defined on that set, $(\Psi, \mathcal{G})$, is referred to as a measurable space. As the name suggests, we define measure on $(\Psi, \mathcal{G})$.
Definition 2.2.3 (Measure)
Given a measurable space $(\Psi, \mathcal{G})$, a measure on $(\Psi, \mathcal{G})$ is a function, $m: \mathcal{G} \rightarrow \mathbb{R}^{+}$, such that,
i. $m(A) \geq 0$ for all $A \in \mathcal{G}$,
ii. $m(\varnothing)=0$,
iii. if $A_1, A_2, \ldots \in \mathcal{G}$ are disjoint then $m\left(\bigcup_{i=1}^{\infty} A_i\right)=\sum_{i=1}^{\infty} m\left(A_i\right)$.

## 统计代写|统计推断代写统计推断代考|直观概率

i。实验是一个可重复的过程，它具有一组定义良好的可能结果。

$$\mathrm{P}(A)=\frac{|A|}{|\Omega|}$$
，其中$|A|$是在$A$中的结果的数量，$|\Omega|$是可能的结果的总数。样本空间是有限的这一说法意味着实验的可能结果数量是有限的，即$|\Omega|<\infty$ .

## 统计代写|统计推断代写统计推断代考|数学概率

i。$\varnothing \in \mathcal{G}$，
ii。如果$A \in \mathcal{G}$，则$A^c \in \mathcal{G}$，
iii。如果$A_1, A_2, \ldots \in \mathcal{G}$那么$\bigcup_{i=1}^{\infty} A_i \in \mathcal{G}$ .

1. 最小的非简并$\sigma$ -algebra包含4个元素。$G=\left{\varnothing, A, A^c . \Psi\right}$。其中$A \subset \Psi$ .
2. $\sigma$ -代数的成员数最大由包含$\Psi$的每个子集给出。我们可以把它写成$G={A: A \subseteq \Psi}$。这被称为$\Psi$的幂集，有时写成$\mathcal{P}(\Psi)$或${0,1}^{\Psi}$ .

i。$m(A) \geq 0$为所有$A \in \mathcal{G}$，
ii。$m(\varnothing)=0$，
iii。如果$A_1, A_2, \ldots \in \mathcal{G}$不连接，则$m\left(\bigcup_{i=1}^{\infty} A_i\right)=\sum_{i=1}^{\infty} m\left(A_i\right)$ .

## 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|统计推断代写Statistical inference代考|MAST20005

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference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。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference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|统计推断代写Statistical inference代考|The Concept of a Statistical Space

The term simple stems from the fact that this represents a particular case of the more general formulation of a statistical space $\left[\left(\mathbf{S}{(n)}, \mathfrak{3}{(n)}, \mathbb{P}_{(n)}\right), \mathcal{G}_n\right]$, where each trial, say $\mathcal{A}_i$, is associated with a different probability space $\left(S_i, \Im_i, \mathbb{P}_i(.)\right.$ ) (i.e., non-ID) and the trials are not necessarily independent. As argued in Chapters 5-8, in many disciplines the IID formulation is inadequate because observational data rarely satisfy such conditions.

A simple statistical space $\left[(S, \Im, \mathbb{P}(.))^n, \mathcal{G}_n^{\text {IID }}\right]$ represents our first formalization of the notion of a random experiment $\mathcal{E}$. This formulation, however, is rather abstract because it involves arbitrary sets and set functions. The main aim of the next chapter is to reduce it to a more appropriate form by mapping this mathematical structure onto the real line where numerical data live.
The story so far in symbols
$$\mathcal{E}:=\left[\begin{array}{c} {[\mathrm{a}]} \ {[\mathrm{b}]} \ {[\mathrm{c}]} \end{array}\right] \Rightarrow\left(\begin{array}{l} S \ (\Im, \mathbb{P}(.)) \ \mathcal{G}_n \end{array}\right) \Longrightarrow\left[(S, \Im, \mathbb{P}(.))^n, \mathcal{G}_n^{\mathrm{IID}}\right]$$
The purpose of this chapter has been to provide an introduction to probability theory using the formalization of a simple chance mechanism we called a random experiment $(\mathcal{E})$ defined by conditions $[\mathrm{a}]-[\mathrm{c}]$. The formalization had a primary objective: to motivate some of the most important concepts of probability theory and define them in a precise mathematical way in the form of a statistical space. The questions addressed along the way include the following:
Why these particular primitive notions $(S, \mathfrak{3}, \mathbb{P}(.))$ ?
The probability space $(S, \Im, \mathbb{P}(.))$ provides an idealized mathematical description of the stochastic mechanism that gives rise to the events in $\Re$.

## 统计代写|统计推断代写Statistical inference代考|The Unfolding Story Ahead

In Chapter 3 the probability space $(S, \Im, \mathbb{P}(.))$ is mapped onto the real line $(\mathbb{R})$ to define a probability model of the form $\Phi={f(x ; \boldsymbol{\theta}), \boldsymbol{\theta} \in \Theta, x \in \mathbb{R}}$. In Chapter 4 the sampling space is transformed into a special type of sampling model we call a random sample: a set of random variables $\mathbf{X}:=\left(X_1, X_2, \ldots, X_n\right)$ which are independent and identically distributed. The unfolding story in symbols:
$$(S, \Im, \mathbb{P}(.)) \rightarrow \Phi={f(x ; \boldsymbol{\theta}), \boldsymbol{\theta} \in \Theta, x \in \mathbb{R}}, \quad \mathcal{G}_n^{\mathrm{IID}} \rightarrow \mathbf{X}:=\left(X_1, X_2, \ldots, X_n\right)$$

Random experiment, outcomes set (sample space), elementary outcomes, events, sure event, impossible event, set-theoretic union, intersection, complementation, partition of a set, empty set, finite set, infinite set, countable set, uncountable set, Venn diagrams, de Morgan’s law, mutually exclusive events, event space, power set, field of events, sigma field of events, Borel field, function, domain and co-domain of a function, range of a function, probability set function, countable additivity, probability space, mathematical deduction, conditional probability, total probability rule, Bayes’ rule, independent events, pairwise independent events, sampling space, independent trials, identically distributed trials, statistical space.

## 统计代写|统计推断代写统计推断代考|统计空间的概念

$$\mathcal{E}:=\left[\begin{array}{c} {[\mathrm{a}]} \ {[\mathrm{b}]} \ {[\mathrm{c}]} \end{array}\right] \Rightarrow\left(\begin{array}{l} S \ (\Im, \mathbb{P}(.)) \ \mathcal{G}_n \end{array}\right) \Longrightarrow\left[(S, \Im, \mathbb{P}(.))^n, \mathcal{G}_n^{\mathrm{IID}}\right]$$

## 统计代写|统计推断代写统计推断代考|The展开的故事在前方

$$(S, \Im, \mathbb{P}(.)) \rightarrow \Phi={f(x ; \boldsymbol{\theta}), \boldsymbol{\theta} \in \Theta, x \in \mathbb{R}}, \quad \mathcal{G}_n^{\mathrm{IID}} \rightarrow \mathbf{X}:=\left(X_1, X_2, \ldots, X_n\right)$$

## 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|统计推断代写Statistical inference代考|STAT3923

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference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。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference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|统计推断代写Statistical inference代考|The Concept of Independence Among Events

The notion of conditioning can be used to determine whether two events $A$ and $B$ are related in the sense that information about the occurrence of one, say $B$, alters the probability of the occurrence of $A$. If knowledge of the occurrence of $B$ does not alter the probability of event $A$, it is natural to say that $A$ and $B$ are independent.
More formally, $A$ and $B$ are independent if
$$\mathbb{P}(A \mid B)=\mathbb{P}(A) \Leftrightarrow \mathbb{P}(B \mid A)=\mathbb{P}(B) .$$
Using the conditional probability formula (2.12), we can deduce that two events $A$ and $B$ are independent if
$$\mathbb{P}(A \cap B)=\mathbb{P}(A) \cdot \mathbb{P}(B) .$$
Note that this notion of independence can be traced back to Cardano in the $1550 \mathrm{~s}$.
Example 2.52 For $A={(H H),(T T)}$ and $B={(T T),(H T)}, A \cap B={(T T)}$ and thus $\mathbb{P}(A \cap B)=1 / 4=\mathbb{P}(A) \cdot \mathbb{P}(B)$, implying that $A$ and $B$ are independent.

It is very important to distinguish between independent and mutually exclusive events; the definition of the latter does not involve probability. Indeed, two independent events with positive probability cannot be mutually exclusive. This is because if $\mathbb{P}(A)>0$ and $\mathbb{P}(B)>0$ and they are independent, then $\mathbb{P}(A \cap B)=\mathbb{P}(A) \cdot \mathbb{P}(B)>0$, but mutual exclusiveness implies that $\mathbb{P}(A \cap B)=0$, since $A \cap B=\varnothing$. The intuition behind this result is that mutually exclusive events are informative about each other because the occurrence of one precludes the occurrence of the other.
Example 2.53 For $A={(H H),(T T)}$ and $B={(H T),(T H)}, A \cap B=\varnothing$ but
$$\mathbb{P}(A \cap B)=0 \neq \frac{1}{4}=\mathbb{P}(A) \cdot \mathbb{P}(B) .$$
Independence can be generalized to more than two events but in the latter case we need to distinguish between pairwise, joint, and mutual independence. For example, in the case of three events $A, B$, and $C$, we say that they are jointly independent if
$$\mathbb{P}(A \cap B \cap C)=\mathbb{P}(A) \cdot \mathbb{P}(B) \cdot \mathbb{P}(C)$$

## 统计代写|统计推断代写Statistical inference代考|The Concept of Random Trials

The first notion we need to formalize pertains to a finite sequence of trials. Let us denote the $n$ trials by $\left{\mathcal{A}1, \mathcal{A}_2, \mathcal{A}_3, \ldots, \mathcal{A}_n\right}$ and associate each trial with a probability space $\left(S_i, \Omega_i, \mathbb{P}_i(.)\right), i=1,2, \ldots, n$, respectively. In order to be able to discuss any relationship between trials, we need to encompass them in an overall probability space; without it, we cannot formalize condition (ii) above. The overall probability space that suggests itself is the product probability space $$\left(S_1, \Im_1, \mathbb{P}_1(.)\right) \times\left(S_2, \Im_2, \mathbb{P}_2(.)\right) \times \cdots \times\left(S_n, \Im_n, \mathbb{P}_n(.)\right),$$ which can be thought of as a triple of the form $$\left(\left[S_1 \times S_2 \times \cdots \times S_n\right],\left[\Im_1 \times \Im_2 \times \cdots \times \Im_n\right],\left[\mathbb{P}_1 \times \mathbb{P}_2 \times \cdots \times \mathbb{P}_n\right]\right):=\left(\mathbf{S}{(n)}, \Im_{(n)}, \mathbb{P}{(n)}\right),$$ in an obvious notation. The technical question that arises is whether $\left(\mathbf{S}{(n)}, \Im_{(n)}, \mathbb{P}{(n)}\right)$ is a proper probability space. To be more precise, the problem is whether $\mathbf{S}{(n)}$ is a proper outcomes set, $\Im_{(n)}$ has the needed structure of a $\sigma$-field, and $\mathbb{P}{(n)}$ defines a set function which satisfies the three axioms. The answer to the first scale of the question is in the affirmative, since the outcomes set can be defined by $$\mathbf{S}{(n)}=\left{\mathbf{s}{(n)}: \mathbf{s}{(n)}:=\left(s_1, s_2, \ldots, s_n\right), s_i \in S_i, \quad i=1,2, \ldots, n\right} .$$
It turns out that indeed $\Omega_{(n)}$ has the needed structure of a $r$-field (for finite $n$ ) and $\mathbb{P}_{(n)}$ defines a set function which satisfies the three axioms; the technical arguments needed to prove these claims are beyond the scope of the present book; see Billingsley (1995).

Having established that the product probability space is a proper probability space, we can proceed to view the sequence of trials $\left{\mathcal{A}1, \mathcal{A}_2, \mathcal{A}_3, \ldots, \mathcal{A}_n\right}$ as an event in $\left(\mathbf{S}{(n)}, \Im_{(n)}, \mathbb{P}_{(n)}\right)$. An event to which we can attach probabilities.

The first component of condition [c] can easily be formalized by ensuring that the probability space $(S, \Im, \mathbb{P}(.))$ remains the same from trial to trial, in the sense
$$\text { [i] }\left(S_i, \Im_i, \mathbb{P}_i(.)\right)=(S, \Im, \mathbb{P}(.)) \text {, for all } i=1,2, \ldots, n \text {. }$$
We refer to this as the identical distribution (ID) condition.

## 统计代写|统计推断代写统计推断代考|事件中的独立性概念

$A={(H H),(T T)}$ 和 $B={(T T),(H T)}, A \cap B={(T T)}$ 因此 $\mathbb{P}(A \cap B)=1 / 4=\mathbb{P}(A) \cdot \mathbb{P}(B)$，暗示 $A$ 和 $B$

$$\mathbb{P}(A \cap B)=0 \neq \frac{1}{4}=\mathbb{P}(A) \cdot \mathbb{P}(B) .$$

$$\mathbb{P}(A \cap B \cap C)=\mathbb{P}(A) \cdot \mathbb{P}(B) \cdot \mathbb{P}(C)$$ ，我们说它们是联合独立的

## 统计代写|统计推断代写统计推断代考|随机试验的概念

$$\text { [i] }\left(S_i, \Im_i, \mathbb{P}_i(.)\right)=(S, \Im, \mathbb{P}(.)) \text {, for all } i=1,2, \ldots, n \text {. }$$

## 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|统计推断代写Statistical inference代考|STAT3013

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference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。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference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|统计推断代写Statistical inference代考|Mathematical Deduction

As a deductive science, mathematics begins with a set of fundamental statements we call axioms (the premises) and ends with other fundamental statements we call theorems, which are derived from the axioms using deductive logical inference. To get some idea of mathematical deduction, let us derive a few such theorems pertaining to probability as specified above.

Accepting the axioms [A1]-[A3] (Table 2.9) as “true,” we can proceed to derive certain corollaries which provide a more complete picture of the mathematical framework.
Theorem 2.1 $\mathbb{P}(\bar{A})=1-\mathbb{P}(A)$, for any $A \in \Im$.
Since $\bar{A} \cup A=S$ and $\bar{A} \cap A=\varnothing$, we can use axioms [A1] and [A3] to deduce that
$$\mathbb{P}(S)=1=\mathbb{P}(\bar{A} \cup A)=\mathbb{P}(\bar{A})+\mathbb{P}(A) .$$
The first equality is axiom [A1], the second follows from the fact that $\bar{A} \cup A=S$, and the third from the fact that $\bar{A} \cap A=\varnothing$ and axiom [A3].

Example 2.44 In the case of tossing a coin twice, let $A={(H H),(H T),(T H)}$. Given that $\bar{A}={(T T)}$, using Theorem $2.1$ we can deduce that $\mathbb{P}(\bar{A})=1 / 4$.

The next result is almost self-evident, but in mathematics we need to ensure that it follows from the axioms. Using Theorem $2.1$ for $A=S$ (and hence $\bar{A}=\varnothing$ ), we deduce:
Theorem 2.2 $\mathbb{P}(\varnothing)=0$.
The next theorem extends axiom $[\mathbf{A 3}]$ to the case where $(A \cap B) \neq \varnothing$.
Theorem 2.3 $\mathbb{P}(A \cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A \cap B)$, for any $A \in \Im, B \in \Im$.
The way to prove this is to define $A \cup B$ in terms of mutually exclusive events and then use [A3]. It is not difficult to see that the events $C={A-(A \cap B)}$ and $B$ are mutually exclusive and $C \cup B=A \cup B$. Hence, by axiom [A3]:
$$\mathbb{P}(A \cup B)=\mathbb{P}(C \cup B)=\mathbb{P}{A-(A \cap B)}+\mathbb{P}(B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A \cap B)$$

## 统计代写|统计推断代写Statistical inference代考|Conditional Probability and Independence

As a prelude to formalizing condition [c] of a Random Experiment $\mathcal{E}$, we need to digress to discuss a very important notion in probability theory, that of conditioning. This notion arises naturally when one has certain additional information relating to the experiment in question that might affect the relevant probabilities.

Example 2.46 In the case of tossing a coin twice, if we (somehow) know that the actual outcome has at least one $T$, then this information will affect the probabilities of certain events. For instance, the outcome $(H H)$ now has zero probability, and thus the outcomes $(H T),(T H)$, and $(T T)$ have probabilities equal to $1 / 3$, not $1 / 4$ as before. Let us formalize this argument in a more systematic fashion by defining the event $B$ “at least one $T “: \quad B={(H T),(T H),(T T)}$.
Without knowing $B$, the outcomes set and the probability distribution are
\begin{aligned} &S_2={(H H),(H T),(T H),(T T)}, \ &\mathbf{r}^{\mathbf{*}}=\left{\mathbb{I}^n(I I I)=\frac{1}{4}, \mathbb{H}^{\sharp}(I T)=\frac{1}{4}, \mathbb{I}^{\mathbb{N}}(T I I)=\frac{1}{4}, \mathbb{I}^n(T T)=\frac{1}{4}\right} . \end{aligned} With the knowledge provided by $B$, these become
\begin{aligned} &S_B={(H T),(T H),(T T)}, \ &\mathbf{P}_B=\left{P_B(H T)=\frac{1}{3}, P_B(T H)=\frac{1}{3}, P_B(T T)=\frac{1}{3}\right} . \end{aligned}
In a sense, the event $B$ has become the new outcomes set and the probabilities are now conditional on $B$ in the sense that
$$P_B(H T)=\mathbb{P}((H T) \mid B)=\frac{1}{3}, \quad P_B(T H)=\mathbb{P}((T H) \mid B)=\frac{1}{3}, \quad P_B(T T)=\mathbb{P}((T T) \mid B)=\frac{1}{3} .$$
A general way to derive these conditional probabilities is the conditional rule
$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}, \text { for } \mathbb{P}(B)>0$$
for any event $A \in \Im$, where $\mathbb{P}(.)$ is the original probability set function defined on $\Im$.
Example 2.47 For $A={(T H)}$ and $A \cap B={(T H)}$, (2.12) implies $\mathbb{P}(A \mid B)=(1 / 4) /(3 / 4)=1 / 3$

## 统计代写|统计推断代写统计推断代考|数学推导

$$\mathbb{P}(S)=1=\mathbb{P}(\bar{A} \cup A)=\mathbb{P}(\bar{A})+\mathbb{P}(A) .$$

$$\mathbb{P}(A \cup B)=\mathbb{P}(C \cup B)=\mathbb{P}{A-(A \cap B)}+\mathbb{P}(B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A \cap B)$$

## 统计代写|统计推断代写统计推断代考|条件概率与独立性

\begin{aligned} &S_2={(H H),(H T),(T H),(T T)}, \ &\mathbf{r}^{\mathbf{*}}=\left{\mathbb{I}^n(I I I)=\frac{1}{4}, \mathbb{H}^{\sharp}(I T)=\frac{1}{4}, \mathbb{I}^{\mathbb{N}}(T I I)=\frac{1}{4}, \mathbb{I}^n(T T)=\frac{1}{4}\right} . \end{aligned} 提供的知识 $B$，它们变成
\begin{aligned} &S_B={(H T),(T H),(T T)}, \ &\mathbf{P}_B=\left{P_B(H T)=\frac{1}{3}, P_B(T H)=\frac{1}{3}, P_B(T T)=\frac{1}{3}\right} . \end{aligned}

$$P_B(H T)=\mathbb{P}((H T) \mid B)=\frac{1}{3}, \quad P_B(T H)=\mathbb{P}((T H) \mid B)=\frac{1}{3}, \quad P_B(T T)=\mathbb{P}((T T) \mid B)=\frac{1}{3} .$$导出这些条件概率的一般方法是条件规则$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}, \text { for } \mathbb{P}(B)>0$$

$A={(T H)}$ 和 $A \cap B={(T H)}$，(2.12)暗示 $\mathbb{P}(A \mid B)=(1 / 4) /(3 / 4)=1 / 3$

## 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|统计推断代写Statistical inference代考|MAST90100

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference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。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference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|统计推断代写Statistical inference代考|Experimental vs. Observational Data

In most sciences, such as physics, chemistry, geology, and biology, the observed data are often generated by the modelers themselves in well-designed experiments. In econometrics the modeler is often faced with observational as opposed to experimental data. This has two important implications for empirical modeling. First, the modeler needs to develop better skills in validating the model assumptions, because random (IID) sample realizations are rare with observational data. Second, the separation of the data collector and the data analyst requires the modeler to examine thoroughly the nature and structure of the data in question.

In economics, along with the constant accumulation of observational data collection grew the demand to analyze these data series with a view to a better understanding of economic phenomena such as inflation, unemployment, exchange rate fluctuations, and the business cycle, as well as improving our ability to forecast economic activity. A first step toward attaining these objectives is to study the available data by being able to answer questions such as:

(i) How were the data collected and compiled?
(ii) What is the subject of measurement and what do the numbers measure?
(iii) What are the measurement units and scale?
(iv) What is the measurement period?
(v) What is the link between the data and any corresponding theoretical concepts?

## 统计代写|统计推断代写Statistical inference代考|Observed Data and the Nature of a Statistical Model

A data set comprising $n$ observations will be denoted by $\mathbf{x}{0}:=\left(x{1}, x_{2}, \ldots, x_{n}\right)$.
REMARK: It is crucial to emphasize the value of mathematical symbolism when one is discussing probability theory. The clarity and concision this symbolism introduces to the discussion is indispensable.
It is common to classify economic data according to the observation units:
(i) Cross-section $\left{x_{k}, k=1,2, \ldots, n\right}, k$ denotes individuals (firms, states, etc.);
(ii) Time series $\left{x_{t}, t=1,2, \ldots, T\right}, t$ denotes time (weeks, months, years, etc.).
For example, observed data on consumption might refer to consumption of different households at the same point in time or aggregate consumption (consumers’ expenditure) over time. The first will constitute cross-section, the second time-series data. By combining these two (e.g. observing the consumption of the same households over time), we can define a third category:
(iii) Panel (longitudinal) $\left{x_{\mathbf{k}}, \mathbf{k}:=(k, t), k=1,2, \ldots, n, t=1,2, \ldots, T\right}$, where $k$ and $t$ denote the index for individuals and time, respectively.
NOTE: In this category the index $\mathbf{k}$ is two-dimensional but $x_{\mathbf{k}}$ is one-dimensional.

## 统计代写|统计推断代写Statistical inference代考|Experimental vs. Observational Data

(i) 数据是如何收集和编制的？
(ii) 测量的主题是什么？这些数字测量的是什么？
(iii) 计量单位和尺度是什么？
(iv) 测量周期是多少？
(v) 数据与任何相应的理论概念之间的联系是什么？

## 统计代写|统计推断代写Statistical inference代考|Observed Data and the Nature of a Statistical Model

(i) 横截面\left{x_{k}, k=1,2, \ldots, n\right}, k\left{x_{k}, k=1,2, \ldots, n\right}, k表示个人（公司、国家等）；
(ii) 时间序列\left{x_{t}, t=1,2, \ldots, T\right}, t\left{x_{t}, t=1,2, \ldots, T\right}, t表示时间（周、月、年等）。

（iii）面板（纵向）\left{x_{\mathbf{k}}, \mathbf{k}:=(k, t), k=1,2, \ldots, n, t=1,2, \ldots, T\right}\left{x_{\mathbf{k}}, \mathbf{k}:=(k, t), k=1,2, \ldots, n, t=1,2, \ldots, T\right}， 在哪里ķ和吨分别表示个人和时间的指数。

## 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|统计推断代写Statistical inference代考|MAST20005

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference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。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference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|统计推断代写Statistical inference代考|Chance Regularity Patterns and Real-World Phenomena

In the case of the experiment of casting two dice, the chance mechanism is explicit and most people will be willing to accept on faith that if this experiment is actually performed properly, then the chance regularity patterns of IID will be present. The question that naturally arises is whether data generated by real-world stochastic phenomena also exhibit such patterns. It is argued that the overwhelming majority of observable phenomena in many disciplines can be viewed as stochastic, and thus amenable to statistical modeling.

Example 1.4 Consider an example from economics where the t-plot of $X=\Delta \ln (E R)$, i.e. log-changes of the Canadian/US dollar exchange rate (ER), for the period 1973-1991 (weekly observations) is shown in Figure 1.6.

What is interesting about the data in Figure $1.6$ is the fact that they exhibit a number of chance regularity patterns very similar to those exhibited by the dice observations in Figure 1.1, but some additional patterns are also discernible. The regularity patterns exhibited by both sets of data are:
(a) the arithmetic average over the ordering (time) appears to be constant;
(b) the band of variation around this average appears to be relatively constant.
In contrast to the data in Figure 1.2, the distributional pattern exhibited by the data in Figure $1.5$ is not a triangular. Instead:
(c) the graph of the relative frequencies (histogram) in Figure $1.7$ exhibits a certain bellshaped symmetry. The Normal density is inserted in order to show that it does not fit well at the tails, in the mid-section, and the top, which is much higher than the Normal curve. As argued in Chapter 5, Student’s $t$ provides a more appropriate distribution for this data; see Figures $3.23$ and 3.24. In addition, the data in Figure $1.6$ exhibit another regularity pattern:
(d) there is a sequence of clusters of small and big changes in succession.
At this stage the reader might not have been convinced that the features noted above are easily discernible from t-plots. An important dimension of modeling in this book is to discuss how to read systematic information in data plots, which will begin in chapter $5 .$

## 统计代写|统计推断代写Statistical inference代考|Chance Regularities and Statistical Models

Motivated by the desire to account for (model) these chance regularities, we look to probability theory to find ways to formalize them in terms of probabilistic concepts. In particular, the stable relative frequencies regularity pattern (Tables $1.3-1.5$ ) will be formalized using the concept of a probability distribution (see Chapter 5). The unpredictability pattern will be related to the concept of Independence ([2]), and the approximate “sameness” pattern to the Homogeneity (ID) concept ([3]). To render statistical model specification easier, the probabilistic concepts aiming to “model” the chance regularities can be viewed as belonging to three broad categories:

These broad categories can be seen as defining the basic components of a statistical model in the sense that every statistical model is a blend of components from all three categories. The first recommendation to keep in mind in empirical modeling is:

1. A statistical model is simply a set of (internally) consistent probabilistic assumptions from the three broad categories (D), $(M)$, and $(\mathrm{H})$ defining a stochastic generating mechanism that could have given rise to the particular data.

The statistical model is chosen to represent a description of a chance mechanism that accounts for the systematic information (the chance regularities) in the data. The distinguishing feature of a statistical model is that it specifies a situation, a mechanism, or a process in terms of a certain probabilistic structure. The main objective of Chapters 2-8 is to introduce numerous probabilistic concepts and ideas that render the choice of an appropriate statistical model an educated guess and not a hit-or-miss selection.

## 统计代写|统计推断代写Statistical inference代考|Chance Regularity Patterns and Real-World Phenomena

（a）排序（时间）上的算术平均值似乎是恒定的；
(b) 围绕这个平均值的变化带似乎是相对恒定的。

（c）图中的相对频率图（直方图）1.7呈现出一定的钟形对称性。插入法线密度是为了表明它在尾部、中间部分和顶部不能很好地拟合，这比法线曲线高得多。如第 5 章所述，学生的吨为这些数据提供更合适的分布；看图3.23和 3.24。此外，图中的数据1.6表现出另一种规律性模式：
(d) 有一系列大小连续变化的簇。

## 统计代写|统计推断代写Statistical inference代考|Chance Regularities and Statistical Models

1. 统计模型只是来自三大类 (D) 的一组（内部）一致的概率假设，(米)， 和(H)定义可能产生特定数据的随机生成机制。

## 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|统计推断代写Statistical inference代考|STAT3923

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference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。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference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|统计推断代写Statistical inference代考|Chance Regularity Patterns

The chance regularities denote patterns that are usually revealed using a variety of graphical techniques and careful preliminary data analysis. The essence of chance regularity, as suggested by the term itself, comes in the form of two entwined features:
chance an inherent uncertainty relating to the occurrence of particular outcomes; regularity discernible regularities associated with an aggregate of many outcomes.
TERMINOLOGY: The term “chance regularity” is used in order to avoid possible confusion with the more commonly used term “randomness.”

At first sight these two attributes might appear to be contradictory, since “chance” is often understood as the absence of order and “regularity” denotes the presence of order. However, there is no contradiction because the “disorder” exists at the level of individual outcomes and the order at the aggregate level. The two attributes should be viewed as inseparable for the notion of chance regularity to make sense.

A glance at Table $1.1$ suggests that the observed data constitute integers between 2 and 12 , but no real patterns are apparent, at least at first sight. To bring out any chance regularity patterns we use a graph as shown in Figure 1.1, t-plot: $\left{\left(t, x_{t}\right), t=1,2, \ldots, n\right}$.

The first distinction to be drawn is that between chance regularity patterns and deterministic regularities that is easy to detect.

## 统计代写|统计推断代写Statistical inference代考|From Chance Regularities to Probabilities

The question that naturally arises is whether the available substantive information pertaining to the mechanism that gave rise to the data in Figure $1.1$ would affect the choice of a statistical model. Common sense suggests that it should, but it is not clear what its role should be. Let us discuss that issue in more detail.

The actual data-generating mechanism (DGM). It turns out that the data in Table $1.1$ were generated by a sequence of $n=100$ trials of casting two dice and adding the dots of the two sides facing up. This game of chance was very popular in medieval times and a favorite pastime of soldiers waiting for weeks on end outside the walls of European cities they had under siege, looking for the right opportunity to assail them. After thousands of trials these illiterate soldiers learned empirically (folk knowledge) that the number 7 occurs more often than any other number and that 6 occurs less often than 7 but more often than $5 ; 2$ and 12 would occur the least number of times. One can argue that these soldiers had an instinctive understanding of the empirical relative frequencies summarized by the histogram in Figure 1.3.

In this subsection we will attempt to reconstruct how this intuition was developed into something more systematic using mathematization tools that eventually led to probability theory. Historically, the initial step from the observed regularities to their probabilistic formalization was very slow in the making, taking centuries to materialize; see Chapter $2 .$
The first crucial feature of the generating mechanism is its stochastic nature: at each trial (the casting of two dice), the outcome (the sum of the dots of the sides) cannot be predicted with any certainty. The only thing one can say with certainty is that the result of each trial will be one of the numbers ${2,3,4,5,6,7,8,9,10,11,12}$. It is also known that these numbers do not occur equally often in this game of chance.

## 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|统计推断代写Statistical inference代考|STAT3923

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference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。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference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|统计推断代写Statistical inference代考|Penalized ℓ1 recovery

Penalized $\ell_{1}$ recovery of signal $x$ from its observation (1.1) is
$$\widehat{x}{\text {pen }}(y) \in \underset{u}{\operatorname{Argmin}}\left{|u|{1}+\lambda\left|H^{T}(A u-y)\right|\right},$$
where $H \in \mathbf{R}^{m \times N}$, a norm $|\cdot|$ on $\mathbf{R}^{N}$, and a positive real $\lambda$ are parameters of the construction.

Theorem 1.5. Given $A$, positive integer s, and $q \in[1, \infty]$, assume that $(H,|\cdot|)$ satisfies the conditions $\mathbf{Q}{q}(s, \kappa)$ and $\mathbf{Q}{1}(s, \varkappa)$ with $\varkappa<1 / 2$ and $\kappa \geq \varkappa$. Then (i) Let $\lambda \geq 2$. Then for all $x \in \mathbf{R}^{n}, y \in \mathbf{R}^{m}$ it holds: $$\left|\widehat{x}{\text {pen }}(y)-x\right|{p} \leq \frac{4 \lambda^{\frac{1}{p}}}{1-2 \varkappa}\left[1+\frac{\kappa \lambda}{2 s}-\varkappa\right]^{\frac{q(p-1)}{p(q-1)}}\left[\left|H^{T}(A x-y)\right|+\frac{\left|x-x^{}\right|_{1}}{2 s}\right], 1 \leq p \leq q .$$ In particular, with $\lambda=2 s$ we have: $$\left|\widehat{x}{\text {pen }}(y)-x\right|{p} \leq \frac{4(2 s)^{\frac{1}{p}}}{1-2 x}[1+\kappa-x]^{\frac{q(p-1)}{p(q-1)}}\left[\left|H^{T}(A x-y)\right|+\frac{\left|x-x^{2}\right|_{1}}{2 s}\right], 1 \leq p \leq q .$$ (ii) Let $\rho \geq 0$, and let $\Xi_{\rho}$ be given by (1.14). Then for all $x \in \mathbf{R}^{n}$ and all $\eta \in \Xi_{\rho}$ one has: $\lambda \geq 2 s \quad \Rightarrow$ $\left|\widehat{x}{\text {pen }}(A x+\eta)-x\right|{p} \leq \frac{4 \lambda^{\frac{1}{p}}}{1-2 \varkappa}\left[1+\frac{\kappa \lambda}{2 s}-\varkappa\right]^{\frac{q(p-1)}{p(q-1)}}\left[\rho+\frac{\left|x-x^{}\right|_{1}}{2 s}\right], 1 \leq p \leq q ;$
$\lambda=2 s \quad \Rightarrow$
$\left|\widehat{x}{\text {pen }}(A x+\eta)-x\right|{p} \leq \frac{4(2 s)^{\frac{1}{p}}}{1-2 \varkappa}[1+\kappa-\varkappa]^{\frac{q(p-1)}{p(q-1)}}\left[\rho+\frac{\left|x-x^{x}\right|_{1}}{2 s}\right], 1 \leq p \leq q .$
For proof, see Section 1.5.2.

## 统计代写|统计推断代写Statistical inference代考|VERIFIABILITY AND TRACTABILITY ISSUES

The good news about $\ell_{1}$ recovery stated in Theorems $1.3,1.4$, and $1.5$ is “conditional” – we assume that we are smart enough to point out a pair $(H,|\cdot|)$ satisfying condition $\mathbf{Q}{1}(s, \varkappa)$ with $\varkappa<1 / 2$ (and condition $\mathbf{Q}{q}(s, \kappa)$ with a “moderate” $\varkappa^{8}$ ). The related issues are twofold:

1. First, we do not know in which range of $s, m$, and $n$ these conditions, or even the weaker than $\mathbf{Q}{1}(s, \varkappa), \varkappa<1 / 2$, nullspace property can be satisfied; and without the nullspace property, $\ell{1}$ minimization becomes useless, at least when we want to guarantee its validity whatever be the $s$-sparse signal we want to recover;
2. Second, it is unclear how to verify whether a given sensing matrix $A$ satisfies the nullspace property for a given $s$, or a given pair $(H,|\cdot|)$ satisfics the condition $\mathbf{Q}_{q}(s, \kappa)$ with given parameters.
What is known about these crucial issues can be outlined as follows.
3. It is known that for given $m, n$ with $m \ll n$ (say, $m / n \leq 1 / 2$ ), there exist $m \times n$ sensing matrices which are $s$-good for the values of $s$ “nearly as large as $m$,” specifically, for $s \leq O(1) \frac{m}{\ln (n / m)} \cdot{ }^{9}$ Moreover, there are natural families of matrices where this level of goodness “is a rule.” E.g., when drawing an $m \times n$ matrix at random from Gaussian or Rademacher distributions (i.e., when filling the matrix with independent realizations of a random variable which is either a standard (zero mean, unit variance) Gaussian one, or takes values $\pm 1$ with probabilities $0.5$ ), the result will be $s$-good, for the outlined value of $s$, with prohahility approashing 1 as $m$ and $n$ grow. All this remains true when instead of speaking about matrices $A$ satisfying “plain” nullspace properties, we are speaking about matrices $A$ for which it is easy to point out a pair $(H,|\cdot|)$ satisfying the condition $\mathbf{Q}_{2}(s, \varkappa)$ with, say, $\varkappa=1 / 4$.

The above results can be considered as a good news. A bad news is that we do not know how to check efficiently, given an $s$ and a sensing matrix $A$, that the matrix is s-good, just as we do not know how to check that $A$ admits good (i.e., satisfying $\mathbf{Q}_{1}(s, \varkappa)$ with $\left.\varkappa<1 / 2\right)$ pairs $(H,|\cdot|)$. Even worse: we do not know an efficient recipe allowing us to build, given $m$, an $m \times 2 m$ matrix $A^{m}$ which is provably $s$-good for $s$ larger than $O(1) \sqrt{m}$, which is a much smaller “level of goodness” than the one promised by theory for randomly generated matrices. ${ }^{10}$ The “common life” analogy of this situation would be as follows: you know that $90 \%$ of bricks in your wall are made of gold, and at the same time, you do not know how to tell a golden brick from a usual one.

## 统计代写|统计推断代写Statistical inference代考|Penalized ℓ1 recovery

$$|\widehat{x} \operatorname{pen}(y)-x| p \leq \frac{4 \lambda^{\frac{1}{p}}}{1-2 \varkappa}\left[1+\frac{\kappa \lambda}{2 s}-\varkappa\right]^{\frac{q(p-1)}{p(q-1)}}\left[\left|H^{T}(A x-y)\right|+\frac{|x-x|{1}}{2 s}\right], 1 \leq p \leq q .$$ 特别是，与 $\lambda=2 s$ 我们有: $$|\widehat{x} \operatorname{pen}(y)-x| p \leq \frac{4(2 s)^{\frac{1}{p}}}{1-2 x}[1+\kappa-x]^{\frac{q(p-1)}{p(q-1)}}\left[\left|H^{T}(A x-y)\right|+\frac{\left|x-x^{2}\right|{1}}{2 s}\right], 1 \leq p \leq q .$$
(ii) 让 $\rho \geq 0$ ，然后让 $\Xi_{\rho}$ 由 (1.14) 给出。那么对于所有人 $x \in \mathbf{R}^{n}$ 和所有 $\eta \in \Xi_{\rho}$ 一个有: $\lambda \geq 2 s \Rightarrow$ $|\widehat{x} \operatorname{pen}(A x+\eta)-x| p \leq \frac{4 \lambda^{\frac{1}{p}}}{1-2 \varkappa}\left[1+\frac{\kappa \lambda}{2 s}-\varkappa\right]^{\frac{q(p-1)}{p(q-1)}}\left[\rho+\frac{|x-x|{1}}{2 s}\right], 1 \leq p \leq q$ $\lambda=2 s \quad \Rightarrow$ $|\widehat{x} \operatorname{pen}(A x+\eta)-x| p \leq \frac{4(2 s)^{\frac{1}{p}}}{1-2 \varkappa}[1+\kappa-\varkappa]^{\frac{q(p-1)}{p(q-1)}}\left[\rho+\frac{\left|x-x^{x}\right|{1}}{2 s}\right], 1 \leq p \leq q$.

## 统计代写|统计推断代写Statistical inference代考|VERIFIABILITY AND TRACTABILITY ISSUES

1. 首先，我们不知道在哪个范围内 $s, m$ ，和 $n$ 这些条件，甚至弱于 $\mathbf{Q} 1(s, \varkappa), \varkappa<1 / 2$ ，可以满足零空间性 质；并且没有 nullspace 属性， $\ell$ 1最小化变得无用，至少当我们想要保证它的有效性时 $s$ – 我们想要恢复的 稀疏信号;
2. 二、不清楚如何验证给定的传感矩阵是否 $A$ 满足给定的零空间属性 $s$ ，或给定的一对 $(H,|\cdot|)$ 满足条件 $\mathbf{Q}_{q}(s, \kappa)$ 给定参数。
对这些关键问题的了解可以概括如下。
3. 众所周知，对于给定 $m, n$ 和 $m \ll n$ (说， $m / n \leq 1 / 2$ )， 存在 $m \times n$ 传感矩阵是 $s$ – 有利于价值观 $s^{\text {“差 }}$ 不多大 $m$,”具体来说，对于 $s \leq O(1) \frac{m}{\ln (n / m)} \cdot{ }^{9}$ 此外，在某些自然矩阵族中，这种良好程度“是一种规
则。例如，当绘制一个 $m \times n$ 从高斯或 Rademacher 分布中随机生成矩阵（即，当用随机变量的独立实 现填充矩阵时，该随机变量要么是标准 (零均值，单位方差) 高斯变量，要么取值 $\pm 1$ 有概率 $0.5)$ ，结果将 是 $s$-好，对于概述的价值 $s$, 概率接近 1 为 $m$ 和 $n$ 生长。当不谈论矩阵时，所有这些都是正确的 $A$ 满足“普通” 零空间属性，我们正在谈论矩阵 $A$ 很容易指出一对 $(H,|\cdot|)$ 满足条件 $\mathbf{Q}{2}(s, \varkappa)$ 与，说， $\varkappa=1 / 4$. 上述结果可以认为是一个好消息。一个坏消息是我们不知道如何有效地检查，给定一个 $s$ 和传感矩阵 $A$ ，矩阵是 s-good，就像我们不知道如何检查 $A$ 承认好（即满足 $\mathbf{Q}{1}(s, \varkappa)$ 和 $\left.\varkappa<1 / 2\right)$ 对 $(H,|\cdot|)$. 更糟糕的是：我们不知 道一个有效的配方允许我们构建，给定 $m ， 一$ 个 $m \times 2 m$ 矩阵 $A^{m}$ 这是可证明的 $s$ – 适合 $s$ 比大 $O(1) \sqrt{m}$ ，这是 一个比理论所承诺的随机生成矩阵小得多的“善良水平”。 ${ }^{10}$ 这种情况的“普通生活”类比如下：你知道 $90 \%$ 你墙上 的砖是金做的，同时，你不知道如何区分金砖和普通砖。

## 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。