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• 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

## 数学网课代修|概率统计代写Probability and Statistics代考|Discrete Random Variables

In our dice example, the random variable $X$ could take on six values in the set ${1,2,3,4,5,6}$. We say that the support of $X$ is ${1,2,3,4,5,6}$, meaning the list of the values the random variable can take on. This is a finite set here.

In the ALOHA example, Section $1.6, X_{1}$ and $X_{2}$ each have support ${0,1,2}$, again a finite set.

Now think of another experiment, in which we toss a coin until we get a head. Let $N$ be the number of tosses needed. Then the support of $N$ is the set ${1,2,3, \ldots}$ This is a countably infinite set. ${ }^{1}$

Now think of one more experiment, in which we throw a dart at the interval $(0,1)$, and assume that the place that is hit, $R$, can take on any of the values between 0 and 1 . Here the support is an uncountably infinite set.

We say that $X, X_{1}, X_{2}$ and $N$ are discrete random variables, while $\mathrm{R}$ is continuous. We’ll discuss continuous random variables in Chapter 6 .

Note that discrete random variables are not necessarily integer-valued. Consider the random variable $X$ above (number of dots showing on a die). Define $W=0.1 X$. $W$ still takes on values in a finite set $(0,0.1, \ldots, 0.6)$, so it too is discrete.

## 数学网课代修|概率统计代写Probability and Statistics代考|Independent Random Variables

We already have a definition for the independence of events; what about independence of random variables? The answer is that we say two random variables are independent if events corresponding to them are independent.
In the dice example above, it is intuitively clear that the random variables $X$ and $Y$ “do not affect” each other. If I know, say, that $X=6$, that knowledge won’t help me guess $Y$ at all. For instance, the probahility that $Y=2$, knowing $X$, is still $1 / 6$. Writing this mathematically, we have
$$P(Y=2 \mid X=6)=P(Y=2)$$

which in turn implies
$$P(Y=2 \text { and } X=6)=P(Y=2) P(X=6)$$
In other words, the events ${X=6}$ and ${Y=2}$ are independent, and similarly the events ${X=i}$ and ${Y=j}$ are independent for any $i$ and $j$. This leads to our formal definition of independence:

Definition 4 Random variables $X$ and $Y$ are said to be independent if for any sets $I$ and $J$, the corresponding events ${X$ is in $I}$ and ${Y$ is in $J}$ are independent, i.e.,
$P(X$ is in $I$ and $Y$ is in $J)=P(X$ is in $I) \cdot P(Y$ is in $J)$
So the concept simply means that $X$ doesn’t affect $Y$ and vice versa, in the sense that knowledge of one does not affect probabilities involving the other. The definition extends in the obvious way to sets of more than two random variables.

The notion of independent random variables is absolutely central to the field of probability and statistics, and will pervade this entire book.

## 数学网课代修|概率统计代写Probability and Statistics代考|Independent Random Variables

$$P(Y=2 \mid X=6)=P(Y=2)$$

$$P(Y=2 \text { and } X=6)=P(Y=2) P(X=6)$$

## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师