19th Ave New York, NY 95822, USA

# 电气工程代写|模拟和数字通信代写Analogue and Digital Communications代考|EE360K Statistical Averages

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

## 电气工程代写|模拟和数字通信代写Analogue and Digital Communications代考|Statistical Averages

We know from simple reasoning that in a board examination of national or international importance, where $N$ ( $N$ very large) students appear, then we may regard the number of students who obtain $i$ marks as a random variable. If the marks obtained are rounded off to the nearest whole number, and if there are $n_0$ students with 0 marks, $n_1$ students with 1 mark, .., $n_i$ students with $i$ marks, and so on, then the average in the board examination is
$$\text { Average marks }=\frac{\sum_{i=0}^{i=100} i n_i}{\sum_{i=0}^{i=100} n_i}=\frac{\sum_{i=0}^{i=100} i n_i}{N}$$
If we regard $n_i / N$ as the relative frequency of occurrence, (the probability) of obtaining $i$ marks, then
$$\text { Average marks }=\sum_{i=0}^{i=100} \underbrace{i}{” r v^{\prime \prime}} \underbrace{\frac{n_i}{N}}{\text {“Probability” }}$$
from this result, we can proceed to calculate the average value of a random variable, $x$.

The probability of occurrence of an event, $X$, in the interval $d x,(x \leq X \leq x+d x)$ is
$$f_X(x) d x$$
then (using the earlier reasoning) the first step in finding the mean is
therefore we ‘sum’ (integrate) this expression to give the statistical average of the event
$$\text { Mean } x=E[X]=m_X=\int_{-\infty}^{\infty} x f_X(x) d x$$

## 电气工程代写|模拟和数字通信代写Analogue and Digital Communications代考|Moments

Another set of averages which are often used are the $n$th moments and $n$th central moments of a random variable. In general, the $n$th moment of a random variable, $X$ is given by
$$E\left[X^n\right]=\int_{-\infty}^{\infty} x^n f_X(x) d x$$
where $E[\cdots]$ is the mean or expectation. The $n$th central moment of a random variable is
$$E\left[\left(X-m_X\right)^n\right]=\int_{-\infty}^{\infty}\left(x-m_X\right)^n f_X(x) d x$$
the most used moments are the ones with $n=2$. For example in Eq. (3.46), for $n=2$,
$$E\left[X^2\right]=\int_{-\infty}^{\infty} x^2 f_X(x) d x$$
gives us the mean of the square of the random variable, which may be connected to the ‘power’ of the random variable. In Eq. (3.47), when $n=2$,
$$\sigma_X^2=E\left[\left(X-m_X\right)^2\right]$$
is called the variance of the random variable. Looking at the variance in more detail,
\begin{aligned} \sigma_X^2 &=E\left[\left(X-m_X\right)^2\right] \ &=\int_{-\infty}^{\infty}\left(x-m_X\right)^2 f_X(x) d x \ &=\int_{-\infty}^{\infty}\left[x^2-2 x m_X+m_X^2\right] f_X(x) d x \ &=E\left[x^2\right]-m_X^2 \end{aligned}
$\sigma_X$ is also called the standard deviation of the random variable. In this way we can apply the average to any function $g(X)$,
$$E[g(X)]=\int_{-\infty}^{\infty} g(x) f_X(x) d x$$

## 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。