## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|CS5850

statistics-lab™ 为您的留学生涯保驾护航 在代写计算数学基础Mathematical Foundations of Computing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。我们的专家在代写计算数学基础Mathematical Foundations of Computing方面经验极为丰富，各种代写计算数学基础Mathematical Foundations of Computing相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Random Variables

So far, we have restricted ourselves to studying events, which are collections of outcomes of experiments or observations. However, we are often interested in abstract quantities or outcomes of experiments that are derived from events and observations but are not themselves events or observations. For example, if we throw a fair die, we may want to compute the probability that the square of the face value is smaller than 10. This is random and can be associated with a probability and, moreover, depends on some underlying random events. Yet, it is neither an event nor an observation: It is a random variable. Intuitively, a random variable is a quantity that can assume any one of a set of values, called its domain $\boldsymbol{D}$, and whose value can be stated only probabilistically. In this section, we will study random variables and their distributions.

More formally, a real random variable-the one most commonly encountered in applications having to do with computer networking-is a mapping from events in a sample space $S$ to the domain of real numbers. The probability associated with each value assumed by a real random variable ${ }^{2}$ is the probability of the underlying event in the sample space, as illustrated in Figure 1.1.

A random variable is discrete if the set of values it can assume is finite and countable. The elements of $D$ should be mutually exclusive-that is, the random variable cannot simultaneously take on more than one value-and exhaustive-the random variable cannot assume a value that is not an element of $D$.

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Cumulative Density Function

The domain of a discrete real random variable $X_{d}$ is totally ordered; that is, for any two values $x_{1}$ and $x_{2}$ in the domain, either $x_{1}>x_{2}$ or $x_{2}>x_{1}$. We define the cumulative density function $F\left(X_{d}\right)$ by
$$F(x)=\sum_{i \mid x_{i} \leq x} p\left(x_{i}\right)=p\left(X_{d} \leq x\right)$$
Note the difference between $F\left(X_{d}\right)$, which denotes the cumulative distribution of random variable $X_{d}$, and $F(x)$, which is the value of the cumulative distribution for the value $X_{d}=x$

Similarly, the cumulative density function of a continuous random variable $X_{c}$, denoted $F\left(X_{c}\right)$, is given by
$$F(x)=\int_{-\infty}^{x} f(y) d y=p\left(X_{c} \leq x\right)$$
By definition of probability, in both cases, $0 \leq F\left(X_{d}\right) \leq 1,0 \leq F\left(X_{c}\right) \leq 1$.

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Cumulative Density Function

$$F(x)=\sum_{i \mid x_{i} \leq x} p\left(x_{i}\right)=p\left(X_{d} \leq x\right)$$

$$F(x)=\int_{-\infty}^{x} f(y) d y=p\left(X_{c} \leq x\right)$$

## 广义线性模型代考

statistics-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。

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|CSMAX170

statistics-lab™ 为您的留学生涯保驾护航 在代写计算数学基础Mathematical Foundations of Computing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。我们的专家在代写计算数学基础Mathematical Foundations of Computing方面经验极为丰富，各种代写计算数学基础Mathematical Foundations of Computing相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Joint and Conditional Probability

Thus far, we have defined the terms used in studying probability and considered single events in isolation. Having set this foundation, we now turn our attention to the interesting issues that arise when studying sequences of events. In doing so, it is very important to keep track of the sample space in which the events are defined: A common mistake is to ignore the fact that two events in a sequence may be defined on different sample spaces.

Consider two processes with sample spaces $S_{1}$ and $S_{2}$ that occur one after the other. The two processes can be viewed as a single joint process whose outcomes are the tuples chosen from the product space $S_{1} \times S_{2}$. We refer to the subsets of the product space as joint events. Just as before, we can associate probabilities with outcomes and events in the product space. To keep things straight, in this section, we denote the sample space associated with a probability as a subscript, so that $P_{S_{1}}(E)$ denotes the probability of event $E$ defined over sample space $S_{1}$, and $P_{S_{1} \times S_{2}}(E)$ is an event defined over the product space $S_{1} \times S_{2}$.
EXAMPLE 1.10: JOINT PROCESS AND JOINT EVENTS
Consider sample space $S_{1}={1,2,3}$ and sample space $S_{2}={a, b, c}$. Then, the product space is given by ${(1, a),(1, b),(1, c),(2, a),(2, b),(2, c),(3, a),(3, b)$, $(3, c)}$. If these events are equiprobable, the probability of each tuple is $\frac{1}{9}$. Let $E={1,2}$ be an event in $S_{1}$ and $F={b}$ be an event in $S_{2}$. Then, the event $E F$ is given by the tuples ${(1, b),(2, b)}$ and has probability $\frac{1}{9}+\frac{1}{9}=\frac{2}{9}$.
We will return to the topic of joint processes in Section $1.8$. We now turn our attention to the concept of conditional probability.

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Bayes’s Rule

One of the most widely used rules in the theory of probability is due to an English country minister: Thomas Bayes. Its significance is that it allows us to infer “backwards” from effects to causes rather than from causes to effects. The derivation of his rule is straightforward, though its implications are profound.
We begin with the definition of conditional probability (Equation 1.4):
$$P_{S \times S}(F \mid E)=\frac{P_{S \times S}(E F)}{P_{S}(E)}$$
If the underlying sample spaces can be assumed to be implicitly known, we can rewrite this as
$$P(E F)=P(F \mid E) P(E)$$
We interpret this to mean that the probability that both $E$ and $F$ occur is the product of the probabilities of two events: first, that $E$ occurs; second, that conditional on $E, F$ occurs.

Recall that $P(F \mid E)$ is defined in terms of the event $F$ following event $E$. Now, consider the converse: $F$ is known to have occurred. What is the probability that $E$ occurred? This is similar to the problem: If there is fire, there is smoke, but if we see smoke, what is the probability that it was due to a fire? The probability we want is $P(E \mid F)$. Using the definition of conditional probability, it is given by
$$P(E \mid F)=\frac{P(E F)}{P(F)}$$
Substituting for $P(F)$ from Equation 1.7, we get
$$P(E \mid F)=\frac{P(F \mid E)}{P(F)} P(E)$$
which is Bayes’s rule. One way of interpreting this is that it allows us to compute the degree to which some effect, or posterior $F$, can be attributed to some cause, or prior $E$.

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Joint and Conditional Probability

$(1, a),(1, b),(1, c),(2, a),(2, b),(2, c),(3, a),(3, b) \$, \$(3, c)$. 如果这些事件是等概率的，则每个元组的概 率为 $\frac{1}{9}$. 让 $E=1,2$ 成为一个事件 $S_{1}$ 和 $F=b$ 成为一个事件 $S_{2}$. 那么，事件 $E F$ 由元组给出 $(1, b),(2, b)$ 并且有 概率 $\frac{1}{9}+\frac{1}{9}=\frac{2}{9}$.

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Bayes’s Rule

$$P_{S \times S}(F \mid E)=\frac{P_{S \times S}(E F)}{P_{S}(E)}$$

$$P(E F)=P(F \mid E) P(E)$$

$$P(E \mid F)=\frac{P(E F)}{P(F)}$$

$$P(E \mid F)=\frac{P(F \mid E)}{P(F)} P(E)$$

## 广义线性模型代考

statistics-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。

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|ECE3020

statistics-lab™ 为您的留学生涯保驾护航 在代写计算数学基础Mathematical Foundations of Computing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。我们的专家在代写计算数学基础Mathematical Foundations of Computing方面经验极为丰富，各种代写计算数学基础Mathematical Foundations of Computing相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Axioms of Probability

One of the breakthroughs in modern mathematics was the realization that the theory of probability can be derived from just a handful of intuitively obvious axioms. Several variants of the axioms of probability are known. We present the three axioms as stated by Kolmogorov to emphasize the simplicity and elegance that lie at the heart of probability theory.

1. $0 \leq P(E) \leq 1$; that is, the probability of an event lies between 0 and 1 .
2. $P(S)=1$, that is, it is certain that at least some event in $S$ will occur.
3. Given a potentially infinite set of mutually exclusive events $E_{1}, E_{2}, \ldots$
$$P\left(\bigcup_{i=1}^{\infty} E_{i}\right)=\sum_{i=1}^{\infty} P\left(E_{i}\right)$$
That is, the probability that any one of the events in the set of mutually exclusive events occurs is the sum of their individual probabilities. For any finite set of $n$ mutually exclusive events, we can state the axiom equivalently as
$$P\left(\bigcup_{i=1}^{n} E_{i}\right)=\sum_{i=1}^{n} P\left(E_{i}\right)$$
An alternative form of axiom 3 is:
$$P\left(E_{1} \vee E_{2}\right)=P\left(E_{1}\right)+P\left(E_{2}\right)-P\left(E_{1} \wedge E_{2}\right)$$
This alternative form applies to non-mutually exclusive events.

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Subjective and Objective Probability

The axiomatic approach is indifferent as to how the probability of an event is determined. It turns out that there are two distinct ways in which to determine the probability of an event. In some cases, the probability of an event can be derived from counting arguments. For instance, given the roll of a fair die, we know that only six outcomes are possible and that all outcomes are equally likely, so that the probability of rolling, say, a 1 , is $1 / 6$. This is called its objective probability. Another way of computing objective probabilities is to define the probability of an event as being the limit of a counting process, as the next example shows.

Consider a measurement device that measures the packet header types of every packet that crosses a link. Suppose that during the course of a day, the device samples $1,000,000$ packets, of which 450,000 are UDP packets, 500,000 are TCP packets, and the rest are from other transport protocols. Given the large number of underlying observations, to a first approximation, we can consider the probability that a randomly selected packet uses the UDP protocol to be $450,000 / 1,000,000=0.45$. More precisely, we state
$$P(U D P)=\operatorname{Lim}_{t \rightarrow \infty}(\operatorname{UDPCoun} t(t)) /(\text { TotalPacket } \operatorname{Coun}(t)),$$
where UDPCount(t) is the number of UDP packets seen during a measurement interval of duration $t$, and TotalPacketCount $(t)$ is the total number of packets seen during the same measurement interval. Similarly, $P(T C P)=0.5$.

Note that in reality, the mathematical limit cannot be achieved, because no packet trace is infinite. Worse, over the course of a week or a month, the underlying workload could change, so that the limit may not even exist. Therefore, in practice, we are forced to choose “sufficiently large” packet counts and hope that the ratio thus computed corresponds to a probability. This approach is also called the frequentist approach to probability.

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Axioms of Probability

1. $0 \leq P(E) \leq 1$; 也就是说，事件的概率介于 0 和 1 之间。
2. $P(S)=1$ ，也就是说，可以肯定的是，至少有一些事件发生在 $S$ 会发生。
3. 给定一组可能无限的互斥事件 $E_{1}, E_{2}, \ldots$
$$P\left(\bigcup_{i=1}^{\infty} E_{i}\right)=\sum_{i=1}^{\infty} P\left(E_{i}\right)$$
也就是说，这组互斥事件中任何一个事件发生的概率是它们各自概率的总和。对于任何有限集 $n$ 互斥事件， 我们可以将公理等价地表述为
$$P\left(\bigcup_{i=1}^{n} E_{i}\right)=\sum_{i=1}^{n} P\left(E_{i}\right)$$
公理 3 的另一种形式是:
$$P\left(E_{1} \vee E_{2}\right)=P\left(E_{1}\right)+P\left(E_{2}\right)-P\left(E_{1} \wedge E_{2}\right)$$
这种替代形式适用于非互斥事件。

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Subjective and Objective Probability

$$P(U D P)=\operatorname{Lim}_{t \rightarrow \infty}(\operatorname{UDPCoun} t(t)) /(\text { TotalPacket Coun }(t)),$$

## 广义线性模型代考

statistics-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。