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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

电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Binomial Distribution

Consider a series of $n$ Bernoulli experiments where the result of each experiment is independent of the others. We would naturally like to know the number of successes in these $n$ trials. This can be represented by a discrete random variable $X$ with parameters $(n, a)$ and is called a binomial random variable. The probability mass function of a binomial random variable with parameters $(n, a)$ is given by
$$p(i)=\left(\begin{array}{c} n \ i \end{array}\right) a^{i}(1-a)^{n-i}$$
If we set $b=1-a$, then these are just the terms of the expansion $(a+b)^{n}$. The expected value of a variable that is binomially distributed with parameters $(n, a)$ is $n a$.

Consider a local area network with ten stations. Assume that, at a given moment, each node can be active with probability $p=0.1$. What is the probability that (a) one station is active, (b) five stations are active, (c) all ten stations are active?
Solution:
Assuming that the stations are independent, the number of active stations can be modeled by a binomial distribution with parameters ( $10,0.1)$. From the formula for $p(i)$, we get
a. $p(1)=\left(\begin{array}{c}10 \ 1\end{array}\right) 0.1^{1} 0.9^{9}=0.38$
b. $p(5)=\left(\begin{array}{c}10 \ 5\end{array}\right) 0.1^{5} 0.9^{5}=1.49 \times 10^{-3}$
c. $p(10)=\left(\begin{array}{l}10 \ 10\end{array}\right) 0.1^{10} 0.9^{0}=1 \times 10^{-10}$
This is shown in Figure 1.3. Note how the probability of one station being active is $0.38$, which is greater than the probability of any single station being active. Note also how rapidly the probability of multiple active stations drops. This is what drives spatial statistical multiplexing: the provisioning of a link with a capacity smaller than the sum of the demands of the stations.

电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Geometric Distribution

Consider a sequence of independent Bernoulli experiments, each of which succeeds with probability $a$. In section $1.5 .2$, we wanted to count the number of successes; now, we want to compute the probability mass function of a random variable $X$ that represents the number of trials before the first success. Such a variable is called a geometric random variable and has a probability mass function
$$p(i)=(1-a)^{i-1} a$$
The expected value of a geometrically distributed variable with parameter $a$ is $1 / a$.
EXAMPLE 1.30: GEOMETRIC RANDOM VARIABLE
Assume that a link has a loss probability of $10 \%$ and that packet losses are independent, although this is rarely true in practice. Suppose that when a packet gets lost, this is detected and the packet is retransmitted until it is correctly received. What is the probability that it would be transmitted exactly one, two, and three times?
Solution:
Assuming that the packet transmissions are independent events, we note that the probability of success $=p=0.9$. Therefore, $p(1)=0.1^{0} * 0.9=0.9 ; p(2)=$ $0.1^{1} * 0.9=0.09 ; p(3)=0.1^{2} * 0.9=0.009$. Note the rapid decrease in the probability of more than two transmissions, even with a fairly high packet loss rate of $10 \%$. Indeed, the expected number of transmissions is only $1 / 0.9=1 . \overline{1}$.

计算数学基础代考

电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Binomial Distribution

$$p(i)=(n i) a^{i}(1-a)^{n-i}$$

C。 $p(10)=(1010) 0.1^{10} 0.9^{0}=1 \times 10^{-10}$

电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Geometric Distribution

$$p(i)=(1-a)^{i-1} a$$

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。

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