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数学代写|随机过程Stochastic Porcesses代考|STATS217 The Poisson process

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数学代写|随机过程Stochastic Porcesses代考|The Poisson process

We already mentioned the Poisson process in Chapters 2 and 3 . It is a particular continuous-time Markov chain. In Chapter 4, we asserted that the Wiener process and the Poisson process are the two most important stochastic processes for applications. The Poisson process is notably used in the basic queueing models.

The Poisson process, which will be denoted by ${N(t), t \geq 0}$, is also a pure birth process (see Subsection 3.3.4). That is, $N(t)$ designates the number of births (or of events, in general) that occurred from 0 up to time $t$. A process of this type is called a counting process.

Definition 5.1.1. Let $N(t)$ be the number of events that occurred in the interval $[0, t]$. The stochastic process ${N(t), t \geq 0}$ is called a counting process.
Counting processes have the following properties, which are deduced directly from their definition.

Properties. i) $N(t)$ is a random variable whose possible values are $0,1, \ldots$.
ii) The function $N(t)$ is nondecreasing: $N\left(t_2\right)-N\left(t_1\right) \geq 0$ if $t_2>t_1 \geq 0$. Moreover, $N\left(t_2\right)-N\left(t_1\right)$ is the number of events that occurred in the interval $\left(t_1, t_2\right]$

数学代写|随机过程Stochastic Porcesses代考|The telegraph signal

As we did with the Brownian motion and in Examples 5.1.4 and 5.1.5, we can define stochastic processes from a Poisson process. An interesting particular transformation of the Poisson process is the telegraph signal ${X(t), t \geq 0}$, defined as follows:
X(t)=(-1)^{N(t)}=\left{\begin{aligned} 1 & \text { if } N(t)=0,2,4, \ldots \ -1 & \text { if } N(t)=1,3,5, \ldots \end{aligned}\right.
An example of a trajectory of a telegraph signal is shown in Fig. 5.2.
Remark . Note that $X(0)=1$, because $N(0)=0$. Thus, the initial value of the process is deterministic. To make the starting point of the process random, we can simply multiply $X(t)$ by a random variable $Z$ that is independent of $X(t)$, for all $t$, and that takes on the value 1 or $-1$ with probability $1 / 2$. It is as if we tossed a fair coin at time $t \geq 0$ to determine whether $X(t)=1$ or $-1$.
The process ${Y(t), t \geq 0}$, where $Y(t):=Z \cdot X(t)$, for all $t \geq 0$, is called a random telegraph signal. We may write that $Z=Y(0)$. Moreover, to be precise, we then use the expression semirandom telegraph signal to designate the process ${X(t), t \geq 0}$. We already encountered the random telegraph signal in Example 2.3.2.

To obtain the distribution of the random variable $X(t)$, it suffices to calculate
\begin{aligned} P[X(t)=1] &=\sum_{k=0}^{\infty} P[N(t)=2 k]=\sum_{k=0}^{\infty} e^{-\lambda t} \frac{(\lambda t)^{2 k}}{(2 k) !} \ &=e^{-\lambda t} \frac{e^{\lambda t}+e^{-\lambda t}}{2}=\frac{1+e^{-2 \lambda t}}{2} \quad \forall t \geq 0 \end{aligned}
because
$$\cosh \lambda t:=\frac{e^{\lambda t}+e^{-\lambda t}}{2}=\sum_{k=0}^{\infty} \frac{(\lambda t)^{2 k}}{(2 k) !}$$

数学代与写随机过程STOCHASTIC PORCESSES代考|THE POISSON PROCESS

ii) 功能 $N(t)$ 是非递减的: $N\left(t_2\right)-N\left(t_1\right) \geq 0$ 如果 $t_2>t_1 \geq 0$. 而且， $N\left(t_2\right)-N\left(t_1\right)$ 是区间内发生的事件数 $\left(t_1, t_2\right]$

数学代寻|随机过程STOCHASTIC PORCESSES代考|THE TELEGRAPH SIGNAL

$$P[X(t)=1]=\sum_{k=0}^{\infty} P[N(t)=2 k]=\sum_{k=0}^{\infty} e^{-\lambda t} \frac{(\lambda t)^{2 k}}{(2 k) !} \quad=e^{-\lambda t} \frac{e^{\lambda t}+e^{-\lambda t}}{2}=\frac{1+e^{-2 \lambda t}}{2} \quad \forall t \geq 0$$

$$\cosh \lambda t:=\frac{e^{\lambda t}+e^{-\lambda t}}{2}=\sum_{k=0}^{\infty} \frac{(\lambda t)^{2 k}}{(2 k) !}$$

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。