## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|MATH2216

statistics-lab™ 为您的留学生涯保驾护航 在代写概率论与统计Probability Theory and Statistics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。我们的专家在代写概率论与统计Probability Theory and Statistics代写方面经验极为丰富，各种代写概率论与统计Probability Theory and Statistics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

## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|Elements of combinatorics

Let’s consider some finite sets $A$ and $B$ which consist of $n$ and $m$ elements $(|A|=n<\infty,|B|=m<\infty)$
$$A=\left{a_1, a_2, \ldots, a_n\right}, \quad B=\left{b_1, b_2, \ldots, b_m\right} .$$
We define a new set (the Cartesian product) $A \times B$ as follows:
$$A \times B=\left{\left(a_i, b_j\right): a_i \in A, b_j \in B\right}$$
Then the number of elements of a set (Cartesian product) is $|A \times B|=|A| \cdot|B|=n \cdot m$, because all elements of this set can be arranged in $n$ rows of $m$ elements in each in the following way:
\begin{aligned} & \left(a_1, b_1\right),\left(a_1, b_2\right), \ldots,\left(a_1, b_m\right), \ & \left(a_2, b_1\right),\left(a_2, b_2\right), \ldots,\left(a_2, b_m\right), \ & \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ & \left(a_n, b_1\right),\left(a_n, b_2\right), \ldots,\left(a_n, b_m\right) \end{aligned}
This statement can be generalized in the following sense. Theorem 1. Let some finite sets be given:
$$\begin{gathered} A_1=\left{a_{11}, a_{12}, \ldots, a_{1 n_1}\right}, A_2=\left{a_{21}, a_{22}, \ldots, a_{2 n_2}\right}, \ldots, A_m=\left{a_{m 1}, a_{m 2}, \ldots, a_{m n_m}\right} \ \left(\left|A_k\right|=n_k<\infty, k=1,2, \ldots, m\right) . \end{gathered}$$

## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|Distribution of balls in boxes

Let there be $r$ balls and $n$ boxes, which are numerated by the numbers $i=1,2, \ldots, n$. Denote the set of boxes by $\Omega_0={1,2, \ldots, n}$.

Let us first consider the case of distinguishable (i.e., having some differences from each other – number, color, etc.) balls.

Denote by $\Omega$ a sample space, corresponding to a random distribution of $r$ balls into $n$ boxes (here and further «random distribution of balls in boxes» means that any ball can get into any box with the same probability). If we denote by $i_j \quad(j=1,2, \ldots, r)$ the number of box into which the ball No. $j$ got, then the sample space corresponding to the given experiment can be described as follows:
$$\Omega=\left{\left(i_1, i_2, \ldots, i_r\right): i_j \in \Omega_0, j=1,2, \ldots, r\right}=\underbrace{\Omega_0 \times \Omega_0 \times \ldots \times \Omega_0}_r$$
From this we see that the experiment consisting in placing $r$ distinguishable balls into $n$ distinguishable boxes and the experiment corresponding to the choice of a random sample of size $r$ from the general population of size $n$ are described by the same sample space (see the previous paragraph 1.1).

Remark. Above we used the figurative language of «balls» and «boxes», but the sample space, constructed earlier for this scheme, allows a large number of interpretations.

For the convenience of further references, we present now a number of schemes that are visually very different but essentially equivalent to the abstract arrangement of $r$ balls in $n$ boxes in the sense that the corresponding outcomes differ only in their verbal description. In this case, the probabilities attributed to elementary events can be different in different examples.

Example 5. a) Birthdays. The distribution of birthdays of $r$ students corresponds to the distribution of $r$ balls into $n=365$ boxes (it is assumed that there are 365 days in a year).
b) When firing at targets, the bullet corresponds to the balls, and the targets to the boxes.
c) In experiments with cosmic rays, particles that fall into Geiger counters play the role of balls, and the counters themselves are boxes.
d) The elevator leaves (rises) with $r$ people and stops on $n$ floors. Then the distribution of people into groups, depending on the floor on which they exit, corresponds to the distribution of $r$ balls in $n$ boxes.
e) The experiment consisting in throwing $r$ dice corresponds to the distribution of $r$ balls in $n=6$ boxes. If the experiment consists in throwing $r$ symmetrical coins, then $n=2$.

From the above formula (17), according to Theorem 2 of the preceding section, it follows that $|\Omega|=n^r$. The latter means that $r$ distinguishable balls can be distributed over $n$ distinguishable boxes in $n^r$ ways.

# 概率论与统计代考

## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|Elements of combinatorics

$A=$ lleft{a_1, a_2, Vdots, a_nlright}, lquad B=॥left{b_1, b_2, Vdots, b_m bight} 。

$$\left(a_1, b_1\right),\left(a_1, b_2\right), \ldots,\left(a_1, b_m\right), \quad\left(a_2, b_1\right),\left(a_2, b_2\right), \ldots,\left(a_2, b_m\right), \ldots \ldots \ldots \ldots \ldots .$$

## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|Distribution of balls in boxes

b)射击目标时，子弹对应球，目标对应盒子。
c) 在宇宙射线实验中，落入盖革计数器的粒子扮演球的角色，而计数器本身就是盅子。
d) 电梯离开（上升） $r$ 人和停止 $n$ 地板。然后，根据他们离开的楼层，人们分组的分布对应于 $r$ 球进 $n$ 盒 子。
e) 投郑实验 $r \ln ^2$ 子对应的分布 $r$ 球进 $n=6$ 盒子。如果实验包括投郑r对称硬币，那么 $n=2$.

## 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。

## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|STAT7610

statistics-lab™ 为您的留学生涯保驾护航 在代写概率论与统计Probability Theory and Statistics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。我们的专家在代写概率论与统计Probability Theory and Statistics代写方面经验极为丰富，各种代写概率论与统计Probability Theory and Statistics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

## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|Classical definition of probability

Consider a finite probability space $(\Omega, \mathcal{A}, P)$.
Let
$$\Omega=\left{\omega_1, \omega_2, \ldots, \omega_n\right},|\Omega|=n<\infty,$$
and all the elementary events are equiprobable:
$$p=P\left(\omega_i\right), i=1, \ldots, n .$$

$$1=\sum_{i=1}^n p=n p, \quad p=\frac{1}{n},$$
and for any event $A \subseteq \Omega$ by the formula $\left(2^{\prime}\right)$
$$P(A)=\sum_{\dot{\tau} \Theta_i \in A} \frac{1}{n}=\frac{|A|}{n}=\frac{|A|}{|\Omega|} .$$
Definition of probability of event $A \subseteq \Omega(|\Omega|=n<\infty)$ by the formula (3) is called a classical definition of probability (the term «uniform discrete distribution» is also often used).

Thus, the classical definition of probability is used in those cases when all the elementary events (outcomes) of the experiment under consideration are equally likely (equiprobable), i.e. in determining conditions of this experiment, no elementary event has any advantages over others.

The classical definition of probability can be formulated differently as follows: The probability of an event $A$ is the ratio of the number of cases (elementary events) which are favorable for $A$ (i.e., leading to the occurrence of event A) to the total number of cases (elementary events); The probability of the event $A$ is equal to the ratio of the number $|A|$ of elements of the event $A$ to the number $|\Omega|=n$ of elements of $\Omega$.

## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|Events. Operations on events

The concepts, definitions, and properties of the probability introduced below apply not only to the case of a discrete probability space, but they also remain valid for all probabilistic spaces considered in what follows.

Certain event. I.e. $\Omega \subseteq \Omega$, so $\Omega={\omega}$ is an event and this event will necessarily happen as a result of experiment. Such an event is called a certain event.

Thus, a certain event (designation: $\Omega$ ) is an event that will necessarily occur as a result of experiment.

Impossible event is an event that will never happen as a result of experiment. An impossible event is denoted by $\varnothing$ (an empty set).

Sum (union) of events. Sum (union) of events $A$ and $B$ (designation: $A \backslash B$ ) is an event consisting of elementary events belonging to at least one of the events $A$ and $B$ :
$$A \cup B={\omega \in \Omega: \omega \in A \text { or } \omega \in B} .$$
Thus, the sum $A \cup B$ of events $A$ and $B$ is an event, which will occur if and only if at least one of them occurs.

Product of events. A product (intersection) of events $A$ and $B$ (designation: $A \cap B$ or $A B$ ) is an event which consists of elementary events belonging to $A$ and $B$ :
$$A \cap B={\omega \in \Omega: \omega \in A, \omega \in B} .$$
So, a product $A \cap B$ of events $A$ and $B$ is an event, which occurs if and only if events $A$ and $B$ occur simultaneously.

Difference of events. A difference of events $A$ and $B$ (designation: $A \backslash B$ ) is an event, which consists of elementary events belonging to $A$ but not belonging to $B$ :
$$A \backslash B={\omega \in \Omega: \omega \in A, \omega \notin B}$$
So, a difference $A \backslash B$ of events $A$ and $B$ is an event, which occurs if and only if an event $A$ occurs and B doesn’t occur.

# 概率论与统计代考

## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|Classical definition of probability

\begin{aligned} & p=P\left(\omega_i\right), i=1, \ldots, n \ & 1=\sum_{i=1}^n p=n p, \quad p=\frac{1}{n} \end{aligned}

$$P(A)=\sum_{\dot{\tau} \Theta_i \in A} \frac{1}{n}=\frac{|A|}{n}=\frac{|A|}{|\Omega|} .$$

## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|Events. Operations on events

$$A \cup B=\omega \in \Omega: \omega \in A \text { or } \omega \in B .$$

$$A \cap B=\omega \in \Omega: \omega \in A, \omega \in B .$$

$$A \backslash B=\omega \in \Omega: \omega \in A, \omega \notin B$$

## 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。

## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|MAТН3603

statistics-lab™ 为您的留学生涯保驾护航 在代写概率论与统计Probability Theory and Statistics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。我们的专家在代写概率论与统计Probability Theory and Statistics代写方面经验极为丰富，各种代写概率论与统计Probability Theory and Statistics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

## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|Sample space. Classical definition of probability

From a historical point of view, the classical definition of probability (classical probability) is the very first definition of probability. The classical definition is based on the notion of equal probability (equiprobability) of the outcomes of the random phenomenon under study. The property of equal probability (equiprobability) is a formally undefined primary concept. Let’s give some examples on the explanation of the meaning of this property.

1. A single toss of a fair coin. It is clear that the possible outcomes of this experiment are: the occurrence of «Head», the occurrence of «Tail». In addition, the coin, possibly, will stand on the edge, roll away somewhere, etc. It is possible to list a number of mutually exclusive events that can occur with a real coin. In the mathematical description of this experiment, it is natural to abstract from a number of insignificant (practically impossible) outcomes and confine ourselves to only two (only possible) outcomes: the occurrence of «Head» (we denote this by «H»), the occurrence of «Tail» (we denote this by «T»). It is clear that if the coin is symmetric, then these two outcomes have no advantages over each other, and these two outcomes are equally likely, in other words, equally probable.
2. A single toss of a dice. A dice is a regular cube made of a homogeneous material, with faces numbered from 1 to 6 . When throwing such a cube, only one of six outcomes can actually be realized: one point falling out, two points falling out, …, six points falling out, and all these outcomes are equally likely (equally probable).
As we see, in the above examples, individual outcomes have no advantages over each other – they are equally possible outcomes.

The outcomes $« \mathrm{H} »$ and $« \mathrm{~T} »$ when throwing a coin are mutually exclusive $-$ they cannot occur simultaneously. The following events (outcomes) are also mutually exclusive: $A_i={i}, i=\overline{1,6},\left(A_i\right.$ means the occurrence of $i$ points $i=\overline{1,6}$ : when $i \neq j$ the events $A_i$ and $A_j$ cannot occur simultaneously. Such only possible and mutually exclusive (simultaneously non-appearing) outcomes of the experiment will be called elementary events (in the future elementary events will be denoted by $\omega$ («omega small»)). In this sense, for example, $A={$ an even number fell out with one dice throwing $}$ is a composite (complex) event: $A={2,4,6}$. This event will occur if and only if at least one of the elementary events ${2},{4},{6}$ occurs.

The set of all possible elementary events $\omega$ is called a sample space and is denoted by $\Omega$ ( «omega large»): $\Omega={\omega}$. Thus, sample space $\Omega$ is a set of all the only possible mutually exclusive outcomes of the experiment (elementary events) such that each experimental result of interest to us can be uniquely described with the help of the elements of this set.

## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|Discrete probability space

The definition of probability $\left(2^{\prime}\right)$ is correct (the series in the right-hand side absolutely converges).

We note here that in the probability theory, we will not be interested in specific numerical values of the function $P$ (this is only a question of the practical value of one or another model). It is clear that in the coin toss model one must assume $P(\mathrm{~T})=P(\mathrm{H})=1 / 2$; in the case of a symmetric dice $P(1)=P(2)=\ldots=P(6)=1 / 6$. In the experiment with coin tossing before the first Tail dropping out one must assume $P\left(\omega_n\right)=2^{-n}, n=1,2, \ldots$. Since $\sum_{n=1}^{\infty} 2^{-n}=1$, then the function $P$, which is given on the outcomes of types $\omega_n=H H \ldots H T$, will define the probability distribution on $\Omega={\mathrm{T}, \mathrm{HT}, \mathrm{HHT}, \ldots, \mathrm{HH} . . \mathrm{HT}, \ldots}$. To calculate within the given probability space, for example, the probability of the event $A={$ the experiment will end at an even step $}=$ $=\left{\omega_2, \omega_4, \ldots\right}$, it is necessary to calculate the sum of the corresponding probabilities: $P(A)=1 / 3$.

Definition. Let $\Omega=\left{\omega_1, \omega_2, \ldots,\right}$ be a discrete sample space, $A \in \mathcal{A}={A: A \subseteq \Omega}$ – a collection of all events (subsets) of $\Omega$.

Then a numerical function $P$, defined on $\mathcal{A}$ and satisfying properties (2), ( $\left.2^{\prime}\right)$, is called a probability or a probability function on $(\Omega, \mathcal{A})$, and the triple $(\Omega, \mathcal{A}, P)$ is called a discrete probability space. (If $\Omega$ is a finite set, then $(\Omega, \mathcal{A}, P)$ is called a finite probability space).

Remark. Everywhere in the future, by $|A|$ we will denote the number of elements of a finite set $A$. Thus, a finite probability space is a triple $(\Omega, \mathcal{A}, P)$ with $|\Omega|<\infty$.

# 概率论与统计代考

## 计算机代写|概率论与统计代写Probability Theory and Statistics代考|Sample space. Classical definition of probability

1.一次抛一枚公平的硬币。很明显，这个实验的可能结果是：出现 «Head»，出现 «Tail»。此外， 硬币可能会站在边缘、滚到某处等。可以列出一些真实硬币可能发生的相互排斥的事件。在这个 实验的数学描述中，很自然地从一些微不足道的（实际上不可能的）结果中抽象出来，并将我们 自己局限于两个 (唯一可能的) 结果：«Head»的出现 (我们用 «H» 表示)，«Tail»的出现（我 们用 «T» 表示) 。很明显，如果硬币是对称的，那么这两个结果就没有优势，而且这两个结果是 等概率的，换句话说，是等概率的。

1. 一次掷骰子。骰子是由均质材料制成的规则立方体，其面编号为 1 到 6 。抛出这样一个立方体， 实际上只能实现六种结果中的一种: 一分掉，两分掉，……，六分掉，所有这些结果都是等概率 的 (等概率)。
正如我们所见，在上述示例中，各个结果之间没有优势―一它们是同样可能的结果。

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。