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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|Information theory; Shannon’s formula

We have seen that Maxwell’s demon needed information to sort gas particles and thus decrease entropy; and we have seen that when fast and slow particles are mixed so that entropy increases, information is lost. The relationship between entropy and lost or missing information was made quantitative by the Hungarian-American physicist Leo Szilard (1898-1964) and by the American mathematician Claude Shannon (1916-2001). In 1929, Szilard published an important article in Zeitschrift für Physik in which he analyzed Maxwell’s demon. In this famous article, Szilard emphasized the connection between entropy and missing information. He was able to show that the entropy associated with a unit of information is $k \ln 2$, where $k$ is Boltzmann’s constant. We will discuss this relationship in more detail below.

Claude Shannon is usually considered to be the “father of information theory”. Shannon graduated from the University of Michigan in 1936 , and he later obtained a Ph.D. in mathematics from the Massachusetts Institute of Technology. He worked at the Bell Telephone Laboratories, and later became a professor at MIT. In 1949 , motivated by the need of AT\&T to quantify the amount of information that could be transmitted over a given line, Shannon published a pioneering study of information as applied to communication and computers. Shannon first examined the question of how many binary digits are needed to express a given integer $\Omega$. In the decimal system we express an integer by telling how many 1 s it contains, how many $10 \mathrm{~s}$, how many $100 \mathrm{~s}$, how many $1000 \mathrm{~s}$, and so on. Thus, for example, in the decimal system,
$$105=1 \times 10^{2}+0 \times 10^{1}+1 \times 10^{0}$$
Any integer greater than or equal to 100 but less than 1,000 can be expressed with 3 decimal digits; any number greater than or equal to 1,000 but less than 10,000 requires 4 , and so on.

## 数学代写|信息论作业代写information theory代考|Entropy expressed as missing information

From the standpoint of information theory, the thermodynamic entropy $S_{N}$ of an ensemble of $N$ identical weakly-interacting systems in a given macrostate can be interpreted as the missing information which we would need in order to specify the state of each system, i.e. the microstate of the ensemble. Thus, thermodynamic information is defined to be the negative of thermodynamic entropy, i.e. the information that would be needed to specify the microstate of an ensemble in a given macrostate. Shannon’s formula allows this missing information to be measured quantitatively. Applying Shannon’s formula, equation (4.13), to the missing information in Boltzmann’s problem we can identify $W$ with $\Omega, S_{N}$ with $I_{N}$, and $k$ with $K$ :
$$W \rightarrow \Omega \quad S_{N} \rightarrow I_{N} \quad k \rightarrow K=\frac{1}{\ln 2} \text { bits }$$
so that
$$k \ln 2=1 \text { bit }=0.95697 \times 10^{-23} \frac{\text { joule }}{\text { kelvin }}$$
and
$$k=1.442695 \text { bits }$$

This implies that temperature has the dimension energy/bit:
$$1 \text { degree Kelvin }=0.95697 \times 10^{-23} \frac{\text { joule }}{\text { bit }}$$
From this it follows that
$$1 \frac{\text { joule }}{\text { kelvin }}=1.04496 \times 10^{23} \text { bits }$$
If we divide equation (4.28) by Avogadro’s number we have
$$1 \frac{\text { joule }}{\text { kelvin mol }}=\frac{1.04496 \times 10^{23} \text { bits } / \text { molecule }}{6.02217 \times 10^{23} \text { molecules } / \mathrm{mol}}=0.17352 \frac{\text { bits }}{\text { molecule }}$$
Figure $4.1$ shows the experimentally-determined entropy of ammonia, $\mathrm{NH}_{3}$, as a function of the temperature, measured in kelvins. It is usual to express entropy in joule/kelvin-mol; but it follows from equation (4.29) that entropy can also be expressed in bits/molecule, as is shown in the figure. Since
$$1 \text { electron volt }=1.6023 \times 10^{-19} \text { joule }$$

## 数学代写|信息论作业代写information theory代考|Information theory; Shannon’s formula

$$105=1 \times 10^{2}+0 \times 10^{1}+1 \times 10^{0}$$

## 数学代写|信息论作业代写information theory代考|Entropy expressed as missing information

$$W \rightarrow \Omega \quad S_{N} \rightarrow I_{N} \quad k \rightarrow K=\frac{1}{\ln 2} \text { bits }$$
L便
$$k \ln 2=1 \text { bit }=0.95697 \times 10^{-23} \frac{\text { joule }}{\text { kelvin }}$$

$$k=1.442695 \text { bits }$$

$$1 \text { degree Kelvin }=0.95697 \times 10^{-23} \frac{\text { joule }}{\text { bit }}$$

$$1 \frac{\text { joule }}{\text { kelvin }}=1.04496 \times 10^{23} \text { bits }$$

$$1 \frac{\text { joule }}{\text { kelvin mol }}=\frac{1.04496 \times 10^{23} \text { bits } / \text { molecule }}{6.02217 \times 10^{23} \text { molecules } / \mathrm{mol}}=0.17352 \frac{\text { bits }}{\text { molecule }}$$


1 \text { electron volt }=1.6023 \times 10^{-19} \text { joule }

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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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师