### 计算机代写|机器学习代写machine learning代考|Probability Theory

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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

## 计算机代写|机器学习代写machine learning代考|Set Theory

A set describes an ensemble of elements, also referred to as events. An elementary event $x$ refers to a single event among a sampling space (or universe) denoted by the calligraphic letter $\mathcal{S}$. By definition, a sampling space contains all the possible events, $E \subseteq \mathcal{S}$. The special case where an event is equal to the sampling space, $E=\mathcal{S}$, is called a certain event. The opposite, $E=\emptyset$, where an event is an empty set, is called a null event. $E$ refers to the complement of a set, that is, all elements belonging to $\mathcal{S}$ and not to $E$. Figure $3.3$ illustrates these concepts using a Venn diagram.

Let us consider the example, ${ }^{5}$ of the state of a structure following an earthquake, which is described by a sampling space,
\begin{aligned} \mathcal{S} &={\text { no damage, light damage, important damage, collapse }} \ &={N, L, I, C} . \end{aligned}
In that context, an event $E_{1}={\mathbf{N}, \mathrm{L}}$ could contain the no damage and light damage events, and another event $E_{2}={C}$ could contain only the collapsed state. The complements of these events are, respectively, $\overline{E_{1}}={\mathrm{I}, \mathrm{C}}$ and $\overline{E_{2}}={\mathrm{N}, \mathrm{L}, \mathrm{I}}$.
The two main operations for events, union and intersection, are illustrated in figure 3.4. A union is analogous to the “or” operator, where $E_{1} \cup E_{2}$ holds if the event belongs to either $E_{1}, E_{2}$, or both. The intersection is analogous to the “and” operator, where $E_{1} \backslash E_{2} \equiv E_{1} E_{2}$ holds if the event belongs to both $E_{1}$ and $E_{2}$. As a convention, intersection has priority over union. Moreover, both operations are commutative, associative, and distributive.

Given a set of $n$ events $\left{E_{1}, E_{2}, \cdots, E_{n}\right} \in \mathcal{S}, E_{1}, E_{2}, \cdots, E_{n}$,the events are mutually exclusive if $E_{i} E_{j}=\emptyset, \forall i \neq j$, that is, if the intersection for any pair of events is an empty set. Events $E_{1}, E_{2}, \cdots, E_{n}$ are collectively exhaustive if $\cup_{i=1}^{n} E_{i}=\mathcal{S}$, that is, the union of all events is the sampling space. Events $E_{1}, E_{2}, \cdots, E_{n}$ are mutually exclusive and collectively exhaustive if they satisfy both properties simultaneously. Figure $3.5$ presents examples of mutually exclusive (3.5a), collectively exhaustive (3.5b), and mutually exclusive and collectively exhaustive $(3.5 \mathrm{c}-\mathrm{d})$ events. Note that the difference between (b) and (c) is the absence of overlap in the latter.

## 计算机代写|机器学习代写machine learning代考|Probability of Events

$\operatorname{Pr}\left(E_{i}\right)$ denotes the probability of the event $E_{i}$. There are two main interpretations for a probability: the Frequentist and the Bayesian. Frequentists interpret a probability as the number of occurrences of $E_{i}$ relative to the number of samples $s$, as $s$ goes to $\infty$,
$$\operatorname{Pr}\left(E_{i}\right)=\lim {s \rightarrow \infty} \frac{#\left{E{i}\right}}{s} .$$
For Bayesians, a probability measures how likely is $E_{i}$ in comparison with other events in $\mathcal{S}$. This interpretation assumes that the nature of uncertainty is epistemic, that is, it describes our knowledge of a phenomenon. For instance, the probability depends on the available knowledge and can change when new information is obtained. Throughout this book we are adopting this Bayesian interpretation.

By definition, the probability of an event is a number between zero and one, $0 \leq \operatorname{Pr}\left(E_{i}\right) \leq 1$. At the ends of this spectrum, the probability of any event in $\mathcal{S}$ is one, $\operatorname{Pr}(\mathcal{S})=1$, and the probability of an empty set is zero, $\operatorname{Pr}(\emptyset)=0$. If two events $E_{1}$ and $E_{2}$ are mutually exclusive, then the probability of the events’ union is the sum of each event’s probability. Because the union of an event and its complement are the sampling space, $E \cup E=\mathcal{S}$ (see figure $3.5 \mathrm{~d}$ ), and because $\operatorname{Pr}(\mathcal{S})=1$, then the probability of the complement is $\operatorname{Pr}(E)=1-\operatorname{Pr}(E)$.

When events are not mutually exclusive, the general addition rule for the probability of the union of two events is
$$\operatorname{Pr}\left(E_{1} \cup E_{2}\right)=\operatorname{Pr}\left(E_{1}\right)+\operatorname{Pr}\left(E_{2}\right)-\operatorname{Pr}\left(E_{1} E_{2}\right) .$$
This general addition rule is illustrated in figure 3.6, where if we simply add the probability of each event without accounting for the subtraction of $\operatorname{Pr}\left(E_{1} E_{2}\right)$, the probability of the intersection of both events will be counted twice.

## 计算机代写|机器学习代写machine learning代考|The probability of a single e

The probability of a single event is referred to as a maryinal probability. A joint probability designates the probability of the intersection of events. The terms in equation $3.1$ can be rearranged to explicitly show that the joint probability of two events $\left{E_{1}, E_{2}\right}$ is the product of a conditional probability and its associated marginal,
\begin{aligned} \operatorname{Pr}\left(E_{1} E_{2}\right) &=\operatorname{Pr}\left(E_{1} \mid E_{2}\right) \cdot \operatorname{Pr}\left(E_{2}\right) \ &=\operatorname{Pr}\left(E_{2} \mid E_{1}\right) \cdot \operatorname{Pr}\left(E_{1}\right) . \end{aligned}
In cases where $E_{1}$ and $E_{2}$ are statistically independent, $E_{1} \perp E_{2}$,
Note: Statis conditional probabilities are equal to the marginal, tween a pair
$$E_{1} \perp E_{2} \begin{cases}\operatorname{Pr}\left(E_{1} \mid E_{2}\right)=\operatorname{Pr}\left(E_{1}\right) & \text { that learning } \ \operatorname{Pr}\left(E_{2} \mid E_{1}\right)=\operatorname{Pr}\left(E_{2}\right) & \text { other. }\end{cases}$$
In the special case of statistically independent events, the joint probability reduces to the product of the marginals,
$$\operatorname{Pr}\left(E_{1} E_{2}\right)=\operatorname{Pr}\left(E_{1}\right) \cdot \operatorname{Pr}\left(E_{2}\right)$$
The joint probability for $n$ events can be broken down into $n-1$ conditionals and one marginal probability using the chain rule,
\begin{aligned} \operatorname{Pr}\left(E_{1} E_{2} \cdots E_{n}\right) &=\operatorname{Pr}\left(E_{1} \mid E_{2} \cdots E_{n}\right) \operatorname{Pr}\left(E_{2} \cdots E_{n}\right) \ &=\operatorname{Pr}\left(E_{1} \mid E_{2} \cdots E_{n}\right) \operatorname{Pr}\left(E_{2} \mid E_{3} \cdots E_{n}\right) \operatorname{Pr}\left(E_{3} \cdots E_{n}\right) \ &=\operatorname{Pr}\left(E_{1} \mid E_{2} \cdots E_{n}\right) \operatorname{Pr}\left(E_{2} \mid E_{3} \cdots E_{n}\right) \cdots \operatorname{Pr}\left(E_{n-1} \mid E_{n}\right) \operatorname{Pr}\left(E_{n}\right) \end{aligned}
Let us define $\left{E_{1}, E_{2}, E_{3}, \cdots, E_{n}\right} \in \mathcal{S}$, a set of mutually exclusive and collectively exhaustive events, that is, $E_{i} E_{j}=\emptyset, \forall i \neq$ $j, \cup_{i=1}^{n} E_{i}=\mathcal{S}$ – and an event $A$ belonging to the same sampling

space, that is, $A \in \mathcal{S}$. This context is illustrated using a Venn diagram in figure 3.7. The probability of the event $A$ can be obtained by summing the joint probability of $A$ and each event $E_{i}$,
$$\operatorname{Pr}(A)=\sum_{i=1}^{n} \underbrace{\operatorname{Pr}\left(A \mid E_{i}\right) \cdot \operatorname{Pr}\left(E_{i}\right)}{\operatorname{Pr}\left(A E{i}\right)} .$$

## 计算机代写|机器学习代写machine learning代考|Probability of Events

\operatorname{Pr}\left(E_{i}\right)=\lim {s \rightarrow \infty} \frac{#\left{E{i}\right}}{s} 。\operatorname{Pr}\left(E_{i}\right)=\lim {s \rightarrow \infty} \frac{#\left{E{i}\right}}{s} 。

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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。