Posted on Categories:Stochastic Porcesses, 数学代写, 随机过程

# 数学代写|随机过程Stochastic Porcesses代考|STAT6540 Poisson process

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

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

Poisson processes are continuous time, discrete space processes that we shall analyze in detail in Chapter 5. Here, we shall distinguish between homogeneous and nonhomogeneous Poisson processes.

Definition 1.12: Suppose that the stochastic process $\left{X_t\right}_{t \in T}$ describes the number of events of a certain type produced until time t and has the following properties:

1. The number of events in nonoverlapping intervals are independent.
2. There is a constant $\lambda$ such that the probabilities of occurrence of events over ‘small’ intervals of duration $\triangle t$ are:

$P$ (number of events in $(t, t+\Delta t]=1)=\lambda \Delta t+o(\Delta t)$.

$P($ number of events in $(t, t+\Delta t]>1)=o(\Delta t)$, where oo $\Delta t)$ is such that $o(\Delta t) / \Delta t \rightarrow 0$ when $\Delta t \rightarrow 0$

Then, we say that $\left{X_t\right}$ is an homogeneous Poisson process with parameter $\lambda$, characterized by the fact that $X_t \sim P o(\lambda t)$.

For such a process, it can be proved that the times between successive events are IID random variables with distribution $\operatorname{Ex}(\lambda)$.

The Poisson process is a particular case of many important generic types of processes. Among others, it is an example of a renewal process, that is, a process describing the number of events of a phenomenon of interest occurring until a certain time such that the times between events are IID random variables (exponential in the case of the Poisson process). Poisson processes are also a special case of continuous time Markov chains, with transition probabilities $p_{i, i+1}=1, \forall i$ and $\lambda_i=\lambda$.

## 数学代写|随机过程Stochastic Porcesses代考|Gaussian processes

The Gaussian process is continuous in both time and state spaces. Let $\left{X_t\right}$ be a stochastic process such that for any $n$ times $\left{t_1, t_2, \ldots, t_n\right}$ the joint distribution of $X_{t_i}, i=1,2, \ldots, n$, is $n$-variate normal. Then, the process is Gaussian. Moreover, if for any finite set of time instants $\left{t_i\right}, i=1,2, \ldots$ the random variables are mutually independent and $X_t$ is normally distributed for every $t$, we call it a purely random Gaussian process.

Because of the specific properties of the normal distribution, we may easily specify many properties of a Gaussian process. For example, if a Gaussian process is weakly stationary, then it is strictly stationary.

This continuous time and state-space process has the following properties:

The process $\left{X_t, t \geq 0\right}$ has independent, stationary increments: for $t_1, t_2 \in T$ and $t_10$

and, for nonoverlapping intervals $\left(t_1, t_2\right)$ and $\left(t_3, t_4\right)$, with $t_1<t_2<t_3<t_4$, the random variables $X_{t_2}-X_{t_1}$ and $X_{t_4}-X_{t_3}$ are independent.

For any time interval $\left(t_1, t_2\right)$, the random variable $X_{t_2}-X_{t_1}$ has distribution $\mathrm{N}\left(0, \sigma^2\left(t_2-t_1\right)\right)$

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

1. 非重聖间隔中的事件数是独立的。
2. 有一个常数 $\lambda$ 这样事件在“小”持䋨时间间隔内发生的概率 $\triangle t$ 是:
$P($ 事件的数量 $(t, t+\Delta t]=1)=\lambda \Delta t+o(\Delta t)$.
$P($ 中的事件数 $(t, t+\Delta t]>1)=o(\Delta t)$, 哪里哦 $\Delta t)$ 是这样的 $o(\Delta t) / \Delta t \rightarrow 0$ 什么时候 $\Delta t \rightarrow 0$
那么，我们说 left 缺少或无法识别的分隔符
是具有参数的齐次泊松过程 $\lambda$, 其特点是 $X_t \sim P o(\lambda t)$.
对于这样一个过程，可以证明连续事件之间的时间是具有分布的IID随机变量 $\operatorname{Ex}(\lambda)$.
泊松过程是许多重要的一般过程类型的特例。其中，它是更新过程的一个例子，即描述直到某个时间发生的感兴趣现象的事件数的 过程，使得事件之间的时间是 转移概率 $p_{i, i+1}=1, \forall i$ 和 $\lambda_i=\lambda$.

## 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。