# 统计代写|概率论代写Probability theory代考|MATHS2103

#### Doug I. Jones

Lorem ipsum dolor sit amet, cons the all tetur adiscing elit

couryes-lab™ 为您的留学生涯保驾护航 在代写概率论Probability theory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Probability theory代写方面经验极为丰富，各种代写概率论Probability theory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
couryes™为您提供可以保分的包课服务

## 统计代写|概率论代写Probability theory代考|Elementary Conditional Probabilities

Example 8.1 We throw a die and consider the events
$A:={$ the face shows an odd number $}$,
$B:={$ the face shows three or smaller $}$.
Clearly, $\mathbf{P}[A]=\frac{1}{2}$ and $\mathbf{P}[B]=\frac{1}{2}$. However, what is the probability that $A$ occurs if we already know that $B$ occurs?

We model the experiment on the probability space $(\Omega, \mathcal{A}, \mathbf{P})$, where $\Omega=$ ${1, \ldots, 6}, \mathcal{A}=2^{\Omega}$ and $\mathbf{P}$ is the uniform distribution on $\Omega$. Then
$$A={1,3,5} \quad \text { and } B={1,2,3} .$$
If we know that $B$ has occurred, it is plausible to assume the uniform distribution on the remaining possible outcomes; that is, on ${1,2,3}$. Thus we define a new probability measure $\mathbf{P}_B$ on $\left(B, 2^B\right)$ by
$$\mathbf{P}_B[C]=\frac{# C}{# B} \quad \text { for } C \subset B \text {. }$$

By assigning the points in $\Omega \backslash B$ probability zero (since they are impossible if $B$ has occurred), we can extend $\mathbf{P}_B$ to a measure on $\Omega$ :
$$\mathbf{P}[C \mid B]:=\mathbf{P}_B[C \cap B]=\frac{#(C \cap B)}{# B} \text { for } C \subset \Omega \text {. }$$
In this way, we get $\mathbf{P}[A \mid B]=\frac{#{1,3}}{#{1,2,3}}=\frac{2}{3} . \diamond$
Motivated by this example, we make the following definition.
Definition $8.2$ (Conditional probability) Let $(\Omega, \mathcal{A}, \mathbf{P})$ be a probability space and $B \in \mathcal{A}$. We define the conditional probability given $B$ for any $A \in \mathcal{A}$ by
\mathbf{P}[A \mid B]=\left{\begin{aligned} \frac{\mathbf{P}[A \cap B]}{\mathbf{P}[B]}, & \text { if } \mathbf{P}[B]>0, \ 0, & \text { otherwise. } \end{aligned}\right.
Remark $8.3$ The specification in (8.1) for the case $\mathbf{P}[B]=0$ is arbitrary and is of no importance. $\diamond$

## 统计代写|概率论代写Probability theory代考|Conditional Expectations

Let $X$ be a random variable that is uniformly distributed on $[0,1]$. Assume that if we know the value $X=x$, the random variables $Y_1, \ldots, Y_n$ are independent and $\operatorname{Ber}_x$-distributed. So far, with our machinery we can only deal with conditional probabilities of the type $\mathbf{P}[\cdot \mid X \in[a, b]], a<b$ (since $X \in[a, b]$ has positive probability). How about $\mathbf{P}\left[Y_1=\ldots=Y_n=1 \mid X=x\right]$ ? Intuitively, this should be $x^n$. We thus need a notion of conditional probabilities that allows us to deal with conditioning on events with probability zero and that is consistent with our intuition. In the next section, we will see that in the current example this can be done using transition kernels. First, however, we have to consider a more general situation.
In the following, $\mathcal{F} \subset \mathcal{A}$ will be a sub- $\sigma$-algebra and $X \in \mathcal{L}^1(\Omega, \mathcal{A}, \mathbf{P})$. In analogy with Lemma 8.10, we make the following definition.

Definition 8.11 (Conditional expectation) A random variable $Y$ is called a conditional expectation of $X$ given $\mathcal{F}$, symbolically $\mathbf{E}[X \mid \mathcal{F}]:=Y$, if:
(i) $Y$ is $\mathcal{F}$-measurable.
(ii) For any $A \in \mathcal{F}$, we have $\mathbf{E}\left[X \mathbb{1}_A\right]=\mathbf{E}\left[Y \mathbb{1}_A\right]$.
For $B \in \mathcal{A}, \mathbf{P}[B \mid \mathcal{F}]:=\mathbf{E}\left[\mathbb{1}_B \mid \mathcal{F}\right]$ is called a conditional probability of $B$ given the $\sigma$-algebra $\mathcal{F}$.
Theorem 8.12 $\mathbf{E}[X \mid \mathcal{F}]$ exists and is unique (up to equality almost surely).
Since conditional expectations are defined only up to equality a.s., all equalities with conditional expectations are understood as equalities a.s., even if we do not say so explicitly.

# 概率论代考

## 统计代写|概率论代写概率论代考|基本条件概率

$A:={$脸上显示一个奇数$}$，
$B:={$脸上显示三个或更小的$}$ .

## 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。

Days
Hours
Minutes
Seconds

# 15% OFF

## On All Tickets

Don’t hesitate and buy tickets today – All tickets are at a special price until 15.08.2021. Hope to see you there :)