Posted on Categories:Probability theory, 数学代写, 概率论

# 数学代写|概率论代考Probability Theory代写|Basic definition

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## 数学代写|概率论代考Probability Theory代写|Basic definition

We define a probability triple or (probability) measure space or probability space to be a triple $(\Omega, \mathcal{F}, \mathbf{P})$, where:

• the sample space $\Omega$ is any non-empty set (e.g. $\Omega=[0,1]$ for the uniform distribution considered above);
• the $\sigma$-algebra (read “sigma-algebra”) or $\sigma$-field (read “sigma-field”) $\mathcal{F}$ is a collection of subsets of $\Omega$, containing $\Omega$ itself and the empty set $\emptyset$, and closed under the formation of complements ${ }^*$ and countable unions and countable intersections (e.g. for the uniform distribution considered above, $\mathcal{F}$ would certainly contain all the intervals $[a, b]$, but would contain many more subsets besides);
• the probability measure $\mathbf{P}$ is a mapping from $\mathcal{F}$ to $[0,1]$, with $\mathbf{P}(\emptyset)=0$ and $\mathbf{P}(\Omega)=1$, such that $\mathbf{P}$ is countably additive as in (1.2.3).

This definition will be in constant use throughout the text. Furthermore it contains a number of subtle points. Thus, we pause to make a few additional observations.

The $\sigma$-algebra $\mathcal{F}$ is the collection of all events or measurable sets. These are the subsets $A \subseteq \Omega$ for which $\mathbf{P}(A)$ is well-defined. We know from Proposition 1.2.6 that in general $\mathcal{F}$ might not contain all subsets of $\Omega$, though we still expect it to contain most of the subsets that come up naturally.

To say that $\mathcal{F}$ is closed under the formation of complements and countable unions and countable intersections means, more precisely, that
(i) For any subset $A \subseteq \Omega$, if $A \in \mathcal{F}$, then $A^C \in \mathcal{F}$;
(ii) For any countable (or finite) collection of subsets $A_1, A_2, A_3, \ldots \subseteq \Omega$, if $A_i \in \mathcal{F}$ for each $i$, then the union $A_1 \cup A_2 \cup A_3 \cup \ldots \in \mathcal{F}$
(iii) For any countable (or finite) collection of subsets $A_1, A_2, A_3, \ldots \subseteq \Omega$, if $A_i \in \mathcal{F}$ for each $i$, then the intersection $A_1 \cap A_2 \cap A_3 \cap \ldots \in \mathcal{F}$.

## 数学代写|概率论代考Probability Theory代写|Constructing probability triples

We clarify the definition of Subsection 2.1 with a simple example. Let us again consider the Poisson(5) distribution considered in Subsection 1.1. In this case, the sample space $\Omega$ would consist of all the non-negative integers:

$\Omega={0,1,2, \ldots}$. Also, the $\sigma$-algebra $\mathcal{F}$ would consist of all subsets of $\Omega$. Finally, the probability measure $\mathbf{P}$ would be defined, for any $A \in \mathcal{F}$, by
$$\mathbf{P}(A)=\sum_{k \in A} e^{-5} 5^k / k ! .$$
It is straightforward to check that $\mathcal{F}$ is indeed a $\sigma$-algebra (it contains all subsets of $\Omega$, so it’s closed under any set operations), and that $\mathbf{P}$ is a probability measure defined on $\mathcal{F}$ (the additivity following since if $A$ and $B$ are disjoint, then $\sum_{k \in A \cup B}$ is the same as $\sum_{k \in A}+\sum_{k \in B}$ ).

So in the case of Poisson(5), we see that it is entirely straightforward to construct an appropriate probability triple. The construction is similarly straightforward for any discrete probability space, i.e. any space for which the sample space $\Omega$ is finite or countable. We record this as follows.

# 概率论代写

## 数学代写|概率论代考Probability Theory代写|Basic definition

$\sigma$ -algebra(读作“sigma-algebra”)或$\sigma$ -field(读作“sigma-field”)$\mathcal{F}$是$\Omega$的子集的集合，包含$\Omega$本身和空集$\emptyset$，并且封闭于补数${ }^*$和可数并集和可数交集(例如，对于上面考虑的均匀分布，$\mathcal{F}$肯定包含所有区间$[a, b]$)。但会包含更多的子集);

$\sigma$ -代数$\mathcal{F}$是所有事件或可测量集的集合。这些是为其定义了$\mathbf{P}(A)$的子集$A \subseteq \Omega$。我们从命题1.2.6中知道，一般来说$\mathcal{F}$可能不包含$\Omega$的所有子集，尽管我们仍然期望它包含大多数自然出现的子集。

(i)对于任意子集$A \subseteq \Omega$，若$A \in \mathcal{F}$，则$A^C \in \mathcal{F}$;
(ii)对于任意可数(或有限)子集$A_1, A_2, A_3, \ldots \subseteq \Omega$的集合，如果对每个$i$都有$A_i \in \mathcal{F}$，则有并集$A_1 \cup A_2 \cup A_3 \cup \ldots \in \mathcal{F}$
(iii)对于任意可数(或有限)子集$A_1, A_2, A_3, \ldots \subseteq \Omega$的集合，如果对每个$i$都有$A_i \in \mathcal{F}$，则相交$A_1 \cap A_2 \cap A_3 \cap \ldots \in \mathcal{F}$。

## 数学代写|概率论代考Probability Theory代写|Constructing probability triples

$\Omega={0,1,2, \ldots}$． 同样，$\sigma$ -代数$\mathcal{F}$将由$\Omega$的所有子集组成。最后，对于任何$A \in \mathcal{F}$，概率测度$\mathbf{P}$将由
$$\mathbf{P}(A)=\sum_{k \in A} e^{-5} 5^k / k ! .$$

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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。