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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Poisson Distribution

The Poisson distribution is widely encountered in networking situations, usually to model the arrival of packets or new end-to-end connections to a switch or a router. A discrete random variable $X$ with the domain ${0,1,2,3, \ldots}$ is said to be a Poisson random variable with parameter $\lambda$ if, for some $\lambda>0$ :
$$\left.P(X=i)=e^{-\lambda\left(\lambda^{i}\right.}\right)$$
Poisson variables are often used to model the number of events that happen in a fixed time interval. If the events are reasonably rare, the probability that multiple events occur in a fixed time interval drops off rapidly, due to the $i$ ! term in the denominator. The first use of Poisson variables, indeed, was to investigate the number of soldier deaths due to being kicked by a horse in Napoleon’s army!

The Poisson distribution, which has only a single parameter $\lambda$, can be used to model a binomial distribution with two parameters ( $n$ and $a$ ) when $n$ is “large” and $a$ is “small.” In this case, the Poisson variable’s parameter $\lambda$ corresponds to the product of the two binomial parameters (i.e., $\lambda=n_{\text {Binomial }}{ }^{*} a_{\text {Binomial }}$ ). Recall that a binomial distribution arises naturally when we conduct independent trials. The Poisson distribution, therefore, arises when the number of such independent trials is large, and the probability of success of each trial is small. The expected value of a Poisson distributed random variable with parameter $\lambda$ is also $\lambda$.

Consider an endpoint sending a packet on a link. We can model the transmission of a packet by the endpoint in a given time interval as a trial as follows: If the source sends a packet in a particular interval, we will call the trial a success; if the source does not send a packet, we will call the trial a failure. When the load generated by each source is light, the probability of success of a trial defined in this manner, which is just the packet transmission probability, is small. Therefore, as the number of endpoints grows, and if we can assume the endpoints to be independent, the sum of their loads will be well modeled by a Poisson random variable. This is heartening because systems subjected to a Poisson load are mathematically tractable, as we will see in our discussion of queueing theory. Unfortunately, over the last two decades, numerous measurements have shown that actual traffic can be far from Poisson. Therefore, this modeling assumption should be used with care and only as a rough approximation to reality.

电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Gaussian, or Normal, Distribution

A random variable is Gaussian, or normally distributed, with parameters $\mu$ and $\sigma^{2}$ if its density is given by
$$f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}$$
We denote a Gaussian random variable $X$ with parameters $\mu$ and $\sigma^{2}$ as $X \sim$ $N\left(\mu, \sigma^{2}\right)$, where we read the ” $”$ as “is distributed as.”

The Gaussian distribution can be obtained as the limiting case of the binomial distribution as $n$ tends to infinity and $p$ is kept constant. That is, if we have a very large number of independent trials, such that the random variable measures the number of trials that succeed, the random variable is Gaussian. Thus, Gaussian random variables naturally occur when we want to study the statistical properties of aggregates.

The Gaussian distribution is called normal because many quantities, such as the heights of people, the slight variations in the size of a manufactured item, and the time taken to complete an activity approximately follow the well-known bell-shaped curve. ${ }^{4}$When performing experiments or simulations, it is often the case that the same quantity assumes different values during different trials. For instance, if five students were each measuring the $\mathrm{pH}$ of a reagent, it is likely that they would get five slightly different values. In such situations, it is common to assume that these quantities, which are supposed to be the same, are in fact normally distributed about some mean. Generally speaking, if you know that a quantity is supposed to have a certain standard value but you also know that there can be small variations in this value due to many small and independent random effects, it is reasonable to assume that the quantity is a Gaussian random variable with its mean centered on the expected value.

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|Poisson Distribution

$$\left.P(X=i)=e^{-\lambda\left(\lambda^{i}\right.}\right)$$

电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Gaussian, or Normal, Distribution

$$f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x \mu}{\sigma}\right)^{2}}$$
䖸们表示一个高斯随机变量 $X$ 带参数 $\mu$ 和 $\sigma^{2}$ 作为 $X \sim N\left(\mu, \sigma^{2}\right)$ ，我们在哪里读到”“作为”分发为”。

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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