### 统计代写|统计推断作业代写statistical inference代考|The Poisson distribution

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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

## 统计代写|统计推断作业代写statistical inference代考|The Poisson distribution

The Poisson distribution is used to model counts. It is perhaps only second to the normal distribution usefulness. In fact, the Bernoulli, binomial and multinomial distributions can all be modeled by clever uses of the Poisson.

The Poisson distribution is especially useful for modeling unbounded counts or counts per unit of time (rates). Like the number of clicks on advertisements, or the number of people who show up at a bus stop. (While these are in principle bounded, it would be hard to actually put an upper limit on it.) There is also a deep connection between the Poisson distribution and popular models for so-called

event-time data. In addition, the Poisson distribution is the default model for socalled contingency table data, which is simply tabulations of discrete characteristics. Finally, when $n$ is large and $p$ is small, the Poisson is an accurate approximation to the binomial distribution.
The Poisson mass function is:
$$P(X=x ; \lambda)=\frac{\lambda^{x} e^{-\lambda}}{x !}$$
for $x=0,1, \ldots$. The mean of this distribution is $\lambda$. The variance of this distribution is also $\lambda$. Notice that $x$ ranges from 0 to $\infty$. Therefore, the Poisson distribution is especially useful for modeling unbounded counts.

## 统计代写|统计推断作业代写statistical inference代考|Limits of random variables

We’ll only talk about the limiting behavior of one statistic, the sample mean.
Fortunately, for the sample mean there’s a set of powerful results. These results allow us to talk about the large sample distribution of sample means of a collection of iid observations.

The first of these results we intuitively already know. It says that the average limits to what its estimating, the population mean. This result is called the Law of Large Numbers. It simply says that if you go to the trouble of collecting an infinite amount of data, you estimate the population mean perfectly. Note there’s sampling assumptions that have to hold for this result to be true. The data have to be iid.

A great example of this comes from coin flipping. Imagine if $\bar{X}_{n}$ is the average of the result of $n$ coin flips (i.e. the sample proportion of heads). The Law of Large Numbers states that as we flip a coin over and over, it eventually converges to the true probability of a head.

## 统计代写|统计推断作业代写statistical inference代考|The Central Limit Theorem

The Central Limit Theorem (CIT) is nne of the mnst important theorems in statistics. For our purposes, the CLT states that the distribution of averages of iid variables becomes that of a standard normal as the sample size increases.
Consider this fact for a second. We already know the mean and standard deviation of the distribution of averages from iid samples. The CLT gives us an approximation to the full distribution! Thus, for iid samples, we have a good sense of distribution of the average event though: (1) we only observed one average and (2) we don’t know what the population distribution is. Because of this, the CLT applies in an endless variety of settings and is one of the most important theorems ever discovered.
The formal result is that $\frac{\bar{X}{n}-\mu}{\sigma / \sqrt{n}}=\frac{\sqrt{n}\left(\bar{X}{n}-\mu\right)}{\sigma}=\frac{\text { Estimate }-\text { Mean of estimate }}{\text { Std. Err. of estimate }}$ has a distribution like that of a standard normal for large $n$. Replacing the standard error by its estimated value doesn’t change the CLT.
The useful way to think about the $\mathrm{CLT}$ is that $\bar{X}_{n}$ is approximately $N\left(\mu, \sigma^{2} / n\right)$.

## 广义线性模型代考

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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。