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# 数学代写|概率论代考Probability Theory代写|A little point set topology

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## 数学代写|概率论代考Probability Theory代写|A little point set topology

The reader will be familiar with the topology of the real line and, more expansively, finite-dimensional Euclidean space. In this setting she is no doubt familiar, at least at an intuitive level, with the concept of open sets. We begin with the real line.

DEFINITION 1 A subset $\mathbb{A}$ of the real line is open if, for each $x \in \mathbb{A}$, there exists an open interval wholly contained in $\mathbb{A}$ which has $x$ as an interior point, that is, there exists $(a, b) \subseteq \mathbb{A}$ with $a<x<b$. A subset of the line is closed if its complement is open.

The union of any collection-finite, countably infinite, or even uncountably infiniteof open sets is open. This is easy to see as, if $\wedge$ is any index set and $\left{\mathbb{A}\lambda, \lambda \in \Lambda\right}$ is any collection of open sets, then any point $x$ in the union must lie in some set $\mathbb{A}\lambda$ and, as $\mathbb{A}\lambda$ is open, there exists an open interval wholly contained in $\mathbb{A}\lambda$, and hence also in $\bigcup_{\lambda \in \Lambda} \mathbb{A}_\lambda$, that contains $x$ as an interior point. This might suggest that open sets may have a very complicated structure. But a little thought should convince the reader that the elaboration of the structure of open sets on the line that follows is quite intuitive.

We need two elementary notions from the theory of numbers. A subset $\mathbb{A}$ of real numbers is bounded if there are numbers $a$ and $b$ such that $a<x<b$ for every $x \in \mathbb{A}$. The largest of the lower bounds $a$ is the greatest lower bound (or infimum) of $\mathbb{A}$ and is denoted $\inf \mathbb{A}$; the smallest of the upper bounds $\mathrm{b}$ is the least upper bound (or supremum) of $\mathbb{A}$ and is denoted sup $\mathbb{A}$. The key fact that we will need is that every bounded set $\mathbb{A}$ has a greatest lower bound and a least upper bound. (The reader who is not familiar with these notions will find them fleshed out in Section XXI.1 in the Appendix.)

THEOREM 1 Any open set on the line is the union of a countable number of pairwise disjoint open intervals and a fortiori every open set is a Borel set.

PROOF: Suppose $\mathbb{A}$ is an open subset of the real line, $x$ a point in $\mathbb{A}$. Let $\mathbb{I}_x$ be the union of all the open intervals wholly contained in $\mathbb{A}$ for which $x$ is an interior point. (It is clear that $\mathbb{I}_x$ is non-empty as $x$ has to be contained in at least one such interval.) Then $\mathbb{I}_x \subseteq \mathbb{A}$ for each $x \in \mathbb{A}$, whence $\bigcup_x \mathbb{I}_x \subseteq \mathbb{A}$; on the other hand, each $y \in \mathbb{A}$ lies in $\mathbb{I}_y$, hence also in $\bigcup_x \mathbb{I}_x$, and so $\bigcup_x \mathbb{I}_x \supseteq \mathbb{A}$. Thus, $\bigcup_x \mathbb{I}_x=\mathbb{A}$.

We now claim that $\mathbb{I}_x$ is an open interval for each $x \in \mathbb{A}$ and, moreover, if $a=\inf \mathbb{I}_x$ and $\mathrm{b}=\sup \mathbb{I}_x$ then $\mathbb{I}_x=(a, b)$. To see this, consider any point $t$ with $a<t<x$. Then, by definition of infimum, there exists a point $s \in \mathbb{I}_x$ with $a<s<t$. But then $s$ lies in some open interval wholly contained in $\mathbb{A}$ and which contains $x$ as an interior point. It follows that all points in the closed interval $[s, x]$ are contained in this interval and $a$ fortiori so is $t$. This implies that all points $t \in(a, x)$ are contained in $\mathbb{I}_x$. An identical argument using the definition of supremum shows that all points $t \in(x, b)$ are also contained in $\mathbb{I}_x$. And, of course, it is patent that $x \in \mathbb{I}_x$. Thus, ( $\left.a, b\right) \subseteq \mathbb{I}_x$. But it is clear that $\mathbb{I}_x \subseteq(\mathrm{a}, \mathrm{b})$ by the definition of the points $a$ and $b$. It follows that, indeed, $\mathbb{I}_x=(a, b)$. Thus, we may identify $\mathbb{I}_x$ as the largest open interval wholly contained in $\mathbb{A}$ with $x$ as an interior point.

## 数学代写|概率论代考Probability Theory代写|Chance domains with side information

The character of the definition of conditional probability is best illustrated by settings in our common experience.

EXAMPLES: 1) Return to coin tossing. Suppose a (fair) coin is tossed thrice. Identifying the results of the tosses sequentially, the sample space is identified as the set of eight elements
$$\Omega={\mathfrak{H H H}, \mathfrak{H} \mathfrak{H}, \mathfrak{H} \mathfrak{H}, \mathfrak{H} \mathfrak{T}, \mathfrak{T H}, \mathfrak{T} \mathfrak{H}, \mathfrak{T} \mathfrak{T}, \mathfrak{T} \mathfrak{T}}$$
each of which has equal probability $1 / 8$ of occurrence. Let $A$ be the event that the first toss is a head. It is clear that the probability of $A$ is $4 / 8=1 / 2$.

Suppose now that one is informed that the outcome of the experiment was exactly one head. What is the probability of A conditioned upon this information? Let $\mathrm{H}$ be the event that exactly one head occurs. Then $\mathrm{H}$ consists of the sample points $\mathfrak{H} \mathfrak{T} \mathfrak{T}, \mathfrak{H} \mathfrak{T}$, and $\mathfrak{T} \mathfrak{T} \mathfrak{H}$. If exactly one head occurs then the outcome must be one of the three elements of $\mathrm{H}$ each of which perforce is equally likely to have been the observed outcome. The event $A$ can then occur if, and only if, the observed outcome was $\mathfrak{H T}$ T. Consequently, the probability of $A$ given that $\mathrm{H}$ has occurred is now $1 / 3$. Side information about the outcome of the experiment in the form of the occurrence of $\mathrm{H}$ affects the projections of whether the outcomes comprising $A$ could have occurred.

2) Dice. Suppose two six-sided dice are thrown. The probability of the event A that at least one six is recorded is then easily computed to be $1-25 / 36=$ $11 / 36$. If one is informed, however, that the sum of the face values is 8 then the possible outcomes of the experiment reduce from 36 pairs of integers $(i, j)$ with $1 \leq i, j \leq 6$ to the outcomes ${(2,6),(3,5),(4,4),(5,3),(6,2)}$ only, each of which is equally likely to occur. The probability that at least one six is recorded, conditioned on the sum of face values being 8 , is then $2 / 5$.
3) Families. A family has two children. If one of them is known to be a boy, what is the probability that the other is also a boy? Unprepared intuition may suggest a probability of one-half. But consider: listing the genders of the two children, elder first, the sample space for this problem may be considered to be $\Omega={\mathfrak{b} \mathfrak{b}, \mathfrak{b g}, \mathfrak{g} \mathfrak{b}, \mathfrak{g g}}$ with the natural assignment of probability $1 / 4$ to each of the four possibilities. If it is known that one child is a boy, the sample space reduces to the three equally likely possibilities ${\mathfrak{b} \mathfrak{b}, \mathfrak{b} \mathfrak{g}, \mathfrak{g} \mathfrak{b}}$. Only one of the outcomes, $\mathfrak{b} \mathfrak{b}$, in the reduced space is identified with the event that the other child is a boy and so, given that one child is a boy, the probability that the other is also a boy is $1 / 3$. Again, side information about the outcome of the experiment in the form of the occurrence of an auxiliary event affects event probabilities.

# 概率论代写

## 数学代写|概率论代考Probability Theory代写|Chance domains with side information

$$\Omega=\mathfrak{H} \mathfrak{H} \mathfrak{H}, \mathfrak{H}, \mathfrak{H H}, \mathfrak{H} \mathfrak{T}, \mathfrak{T} \mathfrak{H}, \mathfrak{H}, \mathfrak{T} \mathfrak{T}, \mathfrak{T}$$

2）骰子。假设郑出两个六面骰子。事件 $A$ 至少有一个六被记录的概率很容易计算为 $1-25 / 36=11 / 36$. 然 而，如果有人被告知面值之和为 8 ，则实验的可能结果会从 36 对整数减少 $(i, j)$ 和 $1 \leq i, j \leq 6$ 结果 $(2,6),(3,5),(4,4),(5,3),(6,2)$ 只有，其中每一个都同样可能发生。以面值总和为 8 为条件，至少有一个 六被记录的概率是 $2 / 5$.
3）家庭。一个家庭有两个孩子。如果已知其中一个是男孩，那么另一个也是男孩的概率是多少? 毫无准备的直 觉可能表明概率是二分之一。但是考虑: 列出两个孩子的性别，大在前，这个问题的样本空间可以认为是 $\Omega=\mathfrak{b} \mathfrak{b}, \mathfrak{b g}, \mathfrak{g} \mathfrak{b}, \mathfrak{g} \mathfrak{g}$ 随着概率的自然分配 $1 / 4$ 四种可能性中的每一种。如果已知一个孩子是男孩，则样本空间 减少为三个等可能的可能性 $\mathfrak{b} \mathfrak{b}, \mathfrak{b} \mathfrak{g}, \mathfrak{g} \mathfrak{b}$. 结果只有一个， $\mathfrak{b} \mathfrak{b}$ ，在缩小的空间中，被识别为另一个孩子是男孩的 事件，因此，假设一个孩子是男孩，另一个孩子也是男孩的概率是 $1 / 3$. 同样，以辅助事件发生的形式出现的关 于实验结果的辅助信息会影响事件概率。

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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。