# 统计代写|随机过程代写stochastic process代考|LAW OF TOTAL PROBABILITY

#### Doug I. Jones

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

couryes™为您提供可以保分的包课服务

couryes-lab™ 为您的留学生涯保驾护航 在代写随机过程stochastic process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。我们的专家在代写随机过程stochastic process代写方面经验极为丰富，各种代写随机过程stochastic process相关的作业也就用不着说。

## 统计代写|随机过程代写stochastic process代考|LAW OF TOTAL PROBABILITY

Sometimes it is not possible to calculate directly the probability of the occurrence of an event $A$, but it is possible to find $P(A \mid B)$ and $P\left(A \mid B^c\right)$ for some event $B$. In such cases, the following theorem, which is conceptually rich and has widespread applications, is used. It states that $P(A)$ is the weighted average of the probability of $A$ given that $B$ has occurred and probability of $A$ given that it has not occurred.

Theorem 3.3 (Law of Total Probability) Let $B$ be an event with $P(B)>0$ and $P\left(B^c\right)>0$. Then for any event $A$,
$$P(A)=P(A \mid B) P(B)+P\left(A \mid B^c\right) P\left(B^c\right) .$$
Proof: By Theorem 1.7,
$$P(A)=P(A B)+P\left(A B^c\right) .$$
Now $P(B)>0$ and $P\left(B^c\right)>0$. These imply that $P(A B)=P(A \mid B) P(B)$ and $P\left(A B^c\right)=P\left(A \mid B^c\right) P\left(B^c\right)$. Putting these in (3.7), we have proved the theorem.

## 统计代写|随机过程代写stochastic process代考|BAYES’ FORMULA

To introduce Bayes’ formula, let us first examine the following problem. In a bolt factory, 30 , 50 , and $20 \%$ of production is manufactured by machines I, II, and III, respectively. If 4,5 , and $3 \%$ of the output of these respective machines is defective, what is the probability that a randomly selected bolt that is found to be defective is manufactured by machine III? To solve this problem, let $A$ be the event that a random bolt is defective and $B_3$ be the event that it manufactured by machine III. We are asked to find $P\left(B_3 \mid A\right)$. Now
$$P\left(B_3 \mid A\right)=\frac{P\left(B_3 A\right)}{P(A)} .$$
So, to calculate $P\left(B_3 \mid A\right)$, we need to know the quantities $P\left(B_3 A\right)$ and $P(A)$. But neither of these is given. However, since $P\left(A \mid B_3\right)$ and $P\left(B_3\right)$ are known we use the relation
$$P\left(B_3 A\right)=P\left(A \mid B_3\right) P\left(B_3\right)$$
to find $P\left(B_3 A\right)$. To calculate $P(A)$, we use the law of total probability. Let $B_1$ and $B_2$ be the events that the bolt is manufactured by machines I and II, respectively. Then $\left{B_1, B_2, B_3\right}$ is a partition of the sample space; hence
$$P(A)=P\left(A \mid B_1\right) P\left(B_1\right)+P\left(A \mid B_2\right) P\left(B_2\right)+P\left(A \mid B_3\right) P\left(B_3\right) .$$
Substituting (3.11) and (3.12) in (3.10), we arrive at a formula, called Bayes’ formula, which enables us to calculate $P\left(B_3 \mid A\right)$ readily:
\begin{aligned} P\left(B_3 \mid A\right) & =\frac{P\left(B_3 A\right)}{P(A)} \ & =\frac{P\left(A \mid B_3\right) P\left(B_3\right)}{P\left(A \mid B_1\right) P\left(B_1\right)+P\left(A \mid B_2\right) P\left(B_2\right)+P\left(A \mid B_3\right) P\left(B_3\right)} \ & =\frac{(0.03)(0.20)}{(0.04)(0.30)+(0.05)(0.50)+(0.03)(0.20)} \approx 0.14 . \end{aligned}

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|LAW OF TOTAL PROBABILITY

$$P(A)=P(A \mid B) P(B)+P\left(A \mid B^c\right) P\left(B^c\right) .$$

$$P(A)=P(A B)+P\left(A B^c\right) .$$

## 统计代写|随机过程代写stochastic process代考|BAYES’ FORMULA

$$P\left(B_3 \mid A\right)=\frac{P\left(B_3 A\right)}{P(A)} .$$

$$P\left(B_3 A\right)=P\left(A \mid B_3\right) P\left(B_3\right)$$

$$P(A)=P\left(A \mid B_1\right) P\left(B_1\right)+P\left(A \mid B_2\right) P\left(B_2\right)+P\left(A \mid B_3\right) P\left(B_3\right) .$$

\begin{aligned} P\left(B_3 \mid A\right) & =\frac{P\left(B_3 A\right)}{P(A)} \ & =\frac{P\left(A \mid B_3\right) P\left(B_3\right)}{P\left(A \mid B_1\right) P\left(B_1\right)+P\left(A \mid B_2\right) P\left(B_2\right)+P\left(A \mid B_3\right) P\left(B_3\right)} \ & =\frac{(0.03)(0.20)}{(0.04)(0.30)+(0.05)(0.50)+(0.03)(0.20)} \approx 0.14 . \end{aligned}

## 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 :)