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# 数学代写|概率论代考Probability Theory代写|Pólya’s urn scheme, spread of contagion

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## 数学代写|概率论代考Probability Theory代写|Pólya’s urn scheme, spread of contagion

A town has a population of $r+s$ individuals belonging to two rival sects: the royalists are initially $r$ in number with the seditionists numbering $s$. At each epoch a group of a new arrivals makes its way to the town and casts its lot with one sect or the other depending on the allegiance of the first inhabitant it meets. What can be said about the ebb and flow of the proportions of royalists and seditionists in the town?

The setting, shorn of political colour, is that of Pólya’s urn scheme. An urn initially contains $r$ red balls and s black balls. A ball is selected at random but not removed and a balls of the same colour as the selection are added to the urn. The process is then repeated with a balls of one colour or the other added to the urn at each epoch. With each addition the population of the urn increases by $a$ and it is helpful to imagine that the urn has unbounded capacity. For the model it is not necessary, however, that $a$ is a positive integer; we may take $a$ as having any integral value. If $a=0$ then the status quo in the urn is preserved from epoch to epoch. If a is negative, however, the population of the urn decreases with each step and, assuming both $r$ and $s$ are divisible by $|a|$, terminates with first one subpopulation being eliminated, then the other.

To obviate trivialities, suppose now that $a \neq 0$. The probability of selecting a red ball initially is $r /(r+s)$ and in this case the red population increases to $r+a$ after one epoch; contrariwise, a black ball is selected with probability $s /(r+s)$ in which case the black population increases to $s+a$ after one epoch. (It should be clear that in this context words like “add” and “increase” should be interpreted depending on the sign of $a$; if $a$ is negative they correspond to “subtract” and “decrease”, respectively.)

The probability of selecting two reds in succession is obtained by a simple conditional argument with an obvious notation as
$$\mathbf{P}\left(R_1 \cap R_2\right)=\mathbf{P}\left(R_2 \mid R_1\right) \mathbf{P}\left(R_1\right)=\frac{r+a}{r+s+a} \cdot \frac{r}{r+s},$$
so that the probability that the urn contains $r+2 a$ red balls after two epochs is $r(r+a) /((r+s)(r+s+a))$. Arguing likewise, the probability of selecting two blacks in succession is given by
$$P\left(B_1 \cap B_2\right)=\frac{s+a}{r+s+a} \cdot \frac{s}{r+s}$$
so that, after two epochs, the probability that there are $s+2 a$ black balls in the urn is $s(s+a) /((r+s)(r+s+a))$. As one red ball and one black ball may be selected in the first two epochs with red first and black next or vice versa, by total probability the chance of observing $r+a$ red balls and $s+a$ black balls in the urn after two epochs is given by entirely similar conditioning considerations to be
$$\frac{s}{r+s+a} \cdot \frac{r}{r+s}+\frac{r}{r+s+a} \cdot \frac{s}{r+s}=\frac{2 r s}{(r+s)(r+s+a)} .$$

## 数学代写|概率论代考Probability Theory代写|The Ehrenfest model of diffusion

Two chambers labelled, say, left and right, and separated by a permeable membrane contain $\mathrm{N}$ indistinguishable particles distributed between them. At each epoch a randomly selected particle exchanges chambers and passes through the membrane to the other chamber, the more populous chamber being more likely to proffer the exchange particle. The setting that has been described provides a primitive model of a diffusion or osmotic process and was proposed by $\mathrm{P}$. and $T$. Ehrenfest. ${ }^5$ The model has been expanded widely and continues to be important in the theory of diffusion processes.

The state of the system at any epoch may be described by the number of particles in, say, the left chamber. The state then is an integer between 0 and $\mathrm{N}$. If the current state is $k$ then, at the next transition epoch, the state becomes either $k-1$ or $k+1$ depending on whether the left chamber proffers the exchange particle or receives it. It is natural in this setting to assume that the chance of the exchange particle being drawn from a given chamber is proportional to its current population. Thus, a particle is more likely to be lost by the more populous chamber so that there is a stochastic restoring force towards population equilibrium in the two chambers. This picture is consonant with the kind of stochastic diffusion or osmosis from regions of higher concentration to regions of lesser concentration that one sees in a variety of natural situations.

To keep with later terminology and notation, let $p_{j k}$ denote the conditional probability that at a given transition epoch the system state changes from $j$ to $k$. The quantities $p_{j k}$ are naturally called transition probabilities for the system. In the Ehrenfest model transitions are only possible to neighbouring states with transition probabilities proportional to the population of the chamber that offers the exchange particle; accordingly, we must have $p_{k, k-1}=k / N$ and $p_{k, k+1}=(\mathrm{N}-\mathrm{k}) / \mathrm{N}$ for $0 \leq k \leq N$. Here, as in the previous examples, it is most natural to specify the underlying probability measure in terms of conditional probabilities.

# 概率论代写

## 数学代写|概率论代咢Probability Theory代写|Pólya’s urn scheme, spread of contagion

$$\mathbf{P}\left(R_1 \cap R_2\right)=\mathbf{P}\left(R_2 \mid R_1\right) \mathbf{P}\left(R_1\right)=\frac{r+a}{r+s+a} \cdot \frac{r}{r+s}$$

$$P\left(B_1 \cap B_2\right)=\frac{s+a}{r+s+a} \cdot \frac{s}{r+s}$$

$$\frac{s}{r+s+a} \cdot \frac{r}{r+s}+\frac{r}{r+s+a} \cdot \frac{s}{r+s}=\frac{2 r s}{(r+s)(r+s+a)} .$$

## 数学代写概率论代考Probability Theory代写|The Ehrenfest model of diffusion

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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。