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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

## 数学代写|信息论作业代写information theory代考|Statistical Entropy

The probabilistic quantity used to describe the extent of the random fluctuations of the signal’s coefficients is the statistical entropy. This arises in thermodynamics, statistical mechanics, and information theory with slightly different definitions that are all related to each other.

The Shannon (1948) entropy of a discrete random variable $\mathrm{X}$ taking values in a set $\mathscr{X}$ with probability mass function $p_{\mathrm{X}}(x)$ is
\begin{aligned} H &=-\sum_{x \in \mathscr{X}} p(x) \log p(x) \ &=-\mathbb{E}{\mathrm{X}} \log p(\mathrm{X}) \end{aligned} where we define, by continuity, $0 \log 0=0$. The logarithm is taken base two, and this only amounts to choosing the unit of measure to be bits. A change of base simply corresponds to multiplication by a constant factor and thus to a change of units, as $\log {b} p=\log {b} a \log {a} p$

The Shannon entropy represents a measure of the uncertainty of the random variable. Viewing this uncertainty as the “surprise” of observing a given outcome, it also measures the amount of information transferred in the observation process. When the random variable takes only one value with probability one, there is no surprise in the realization and the entropy is zero. When the outcome is unknown, the probability mass has a wider distribution and the entropy becomes positive. When the distribution becomes uniform among all possible outcomes, the entropy is maximized and it corresponds to the logarithm of the number of possible outcomes. A plot of the entropy for the simplest case of a Bernoulli random variable that takes values with probability $p$ and $1-p$ is depicted in Figure $1.15$.

## 数学代写|信息论作业代写information theory代考|Differential Entropy

To appreciate the relationship between statistical entropy and the number of degrees of freedom of an electromagnetic waveform, we introduce the differential entropy of a continuous random variable $\mathrm{X}$ of probability density function $g_{\mathrm{X}}(x)$. This is also due to Shannon (1948), and is given by
\begin{aligned} h_{\mathrm{X}}(g) &=-\int_{\mathscr{X}} g(x) \log g(x) d x \ &=-\mathbb{E}{\mathbf{X}} \log g(\mathrm{X}) \end{aligned} where $\mathscr{X}$ is the support set of the random variable. The joint differential entropy of two random variables is \begin{aligned} h{\mathrm{X}, \mathrm{Y}}(g) &=-\int_{\mathscr{X}} \int_{\mathscr{y}} g(x, y) \log g(x, y) d x d y \ &=-\mathbb{E}{\mathrm{X}, \mathrm{Y}} \log g(\mathrm{X}, \mathrm{Y}) \end{aligned} and when the two variables are independent, we have $$h{\mathrm{X}, \mathrm{Y}}=h_{\mathrm{X}}+h_{\mathrm{Y}} \text {. }$$
The definition naturally extends to random vectors. If $\mathrm{X}{1}, \mathrm{X}{2}, \ldots, \mathrm{X}{N{0}}$ are mutually independent, we have
$$h_{\left(\mathrm{X}{1}, \mathrm{X}{2}, \ldots, \mathrm{X}{N{0}}\right)}=\sum_{n=1}^{N_{0}} h_{\mathrm{X}{n}},$$ and if they also have the same distribution, we have $$h{\left(\mathrm{X}{1}, \mathrm{X}{2}, \ldots, \mathrm{X}{N{0}}\right)}=N_{0} h,$$
where $h=h \mathrm{x}_{1}(f)$ is the differential entropy of a single random variable.

# 信息论代写

## 数学代写|信息论作业代写information theory代考|Statistical Entropy

$$H=-\sum_{x \in \mathscr{X}} p(x) \log p(x) \quad=-\mathbb{E} \mathrm{X} \log p(\mathrm{X})$$

## 数学代写|信息论作业代写information theory代考|Differential Entropy

$$h_{\mathrm{X}}(g)=-\int_{\mathscr{X}} g(x) \log g(x) d x \quad=-\mathbb{E} \mathbf{X} \log g(\mathrm{X})$$

$$h \mathrm{X}, \mathrm{Y}(g)=-\int_{\mathscr{X}} \int_{\mathcal{Y}} g(x, y) \log g(x, y) d x d y \quad=-\mathbb{E} \mathrm{X}, \mathrm{Y} \log g(\mathrm{X}, \mathrm{Y})$$

$$h \mathrm{X}, \mathrm{Y}=h_{\mathrm{X}}+h_{\mathrm{Y}}$$

$$h_{(\mathrm{X} 1, \mathrm{X} 2, \ldots, \mathrm{X} N 0)}=\sum_{n=1}^{N_{0}} h_{\mathrm{X} n}$$

$$h(\mathrm{X} 1, \mathrm{X} 2, \ldots, \mathrm{X} N 0)=N_{0} h$$

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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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