## 数学代写|离散数学作业代写discrete mathematics代考|MATH200

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## 数学代写|离散数学作业代写discrete mathematics代考|Reflection principle

This function defines a bijection between the two sets of paths. Indeed, if we now take a path $\left(\widetilde{S}_m, \ldots, \widetilde{S}_n\right)$ from $(m, a)$ to $(n,-b)$, as it takes steps whose amplitude is 1 , since $a>0$ and $-b<0$, this path must necessarily pass through 0 . I et $p$ be the first instant where the path is at 0 . We then construct a path $\left(S_m, \ldots, S_n\right)$ from $(m, a)$ to $(n, b)$ passing through 0 writing
$$S_k= \begin{cases}\widetilde{S}_k & \text { if } m \leq k \leq p, \ -\widetilde{S}_k & \text { if } p \leq k \leq n .\end{cases}$$
There are, thus, the same number of paths of both types.
An initial application of this property is the following result, called the ballot theorem.

THEOREM 3.1.-During an election with two opposing candidates, $A$ and $B$, candidate $A$ has obtained a votes and candidate $B$ botes, with $a>b$. Thus, the probability that $A$ has obtained the majority (in the broad sense) throughout the counting is
$$p=1-\frac{b}{a+1} .$$
PROOF.- With all the counts being equiprobable, $p$ is obtained as the ratio of the number of counts with $A$ in the lead all the time to that of the total number of counts. A count can be modeled by a random walk $\left(S_n\right)$, where $S_n$ is the number of votes by which $A$ is ahead of $B$ after counting the $n$-th ballot.

It is now assumed that the walk starts from $0: S_0=0$ and we wish to know whether the walk will return to 0 almost surely. Let us start by observing that if $S_n=0$, then $n$ is necessarily even.
Proposition 3.9.- For any $n \in \mathbb{N}$, we have
$$\mathbb{P}\left(S_{2 n}=0\right)=\frac{1}{4^n} C_{2 n}^n .$$
Proof.- This probability corresponds to the number of paths from $(0,0)$ to $(2 n, 0)$ divided by the total number of paths with length $2 n$, since all the paths are equiprobable. We thus directly have $\mathbb{P}\left(S_{2_n}=0\right)=\frac{C_{2 n}^n}{4^n}$.

We now look at the first instant of the return to 0 . It is denoted by $T_0$. It is, therefore, the random variable
$$T_0=\inf \left{n \geq 1 ; S_n=0\right},$$
using the convention that $\inf \emptyset=+\infty$. It is, therefore, a discrete random variable taking values in $\mathbb{N}^* \cup{+\infty}$. The distribution of $T_0$ can be explicitly calculated.
PROPOSITION 3.10.- For any $n \in \mathbb{N}^*$, we have
$$\mathbb{P}\left(T_0=2 n\right)=\frac{(2 n-2) !}{2^{2 n-1} n !(n-1) !} .$$
Proof.- The event $\left(T_0=2 n\right)$ corresponds to $\left(S_2 \neq 0, \ldots, S_{2 n-2} \neq 0, S_{2 n}=0\right)$ because then $2 n$ is the first time the walk returns to 0 . In particular, between the instant 0 and the instant $2 n$, the walk does not change in sign and retains the same sign as $S_1$. Therefore, we have
\begin{aligned} \mathbb{P}\left(T_0=2 n\right)= & \mathbb{P}\left(S_1=1, S_2>0, \ldots, S_{2 n-2}>0, S_{2 n}=0\right) \ & +\mathbb{P}\left(S_1=-1, S_2<0, \ldots, S_{2 n-2}<0, S_{2 n}=0\right) \end{aligned}

# 离散数学代写

## 数学代写|离散数学作业代写discrete mathematics代考|Reflection principle

$$p=1-\frac{b}{a+1} .$$

$$\mathbb{P}\left(S_{2 n}=0\right)=\frac{1}{4^n} C_{2 n}^n .$$

$$\mathbb{P}\left(T_0=2 n\right)=\frac{(2 n-2) !}{2^{2 n-1} n !(n-1) !}$$

$\left(S_2 \neq 0, \ldots, S_{2 n-2} \neq 0, S_{2 n}=0\right)$ 因为那时 $2 n$ 是 第一次步行返回 0 。特别地，在时刻 0 和时刻之间 $2 n$, 行走的符号不变并保持与 $S_1$. 因此，我们有
$$\mathbb{P}\left(T_0=2 n\right)=\mathbb{P}\left(S_1=1, S_2>0, \ldots, S_{2 n-2}>0\right.$$

## 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。