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# 数学代写|概率论代考Probability Theory代写|Math461 Building a Model

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## 数学代写|概率论代考Probability Theory代写|Building a Model

When we try to model the occurrences of the various processes described above, then there are a number of characteristics that we may want to build in. To name a few:

There is a certain amount of regularity in the processes described above. Although individual earthquakes are impossible to predict and certainly do not occur in a strictly regular pattern, there is perhaps some statistical regularity in the sense that when we observe the earthquakes during 10 years, say, without knowing the absolute time frame, then we have no way to decide whether we observe the time period 1910-1920 or 1990-2000. In probabilistic terms, the process is stationary in time. In other words, the course of time should not change the probabilistic properties of the process. (Of course, in the case of a shop this can only be realistic as long as the shop is open, and even then one can ask whether there will typically be more customers around closing time than around 3 p.m., say. More about this in Exercise 7.5.8)

The fact that there is an occurrence at a particular time, says nothing about the probability of an occurrence at, or around, a later or earlier time. In other words, there seems to be some kind of independence with respect to various occurrences.

The next occurrence can not be predicted form current and past information. In other words, the process of occurrences seems to have no memory. The fact that something happened in the past has no effect on the probabilities for future occurrences.

There is no accumulation of occurrences at any time. In other words, in each finite time interval, there are only finitely many occurrences.

## 数学代写|概率论代考Probability Theory代写|Basic Properties

In this section we prove a number of basic facts which should increase our understanding of the Poisson process, and which also justify the definition, given our objectives as mentioned at the beginning of the previous section. First of all, we should note that we have computed the distribution of $S_n$ in Example 5.7.1. Indeed, the distribution of $S_n$, being the sum of $n$ independent exponentially distributed random variables with parameter $\lambda$, is a gamma distribution and its density is given by
$$f_{S_n}(x)=\frac{\lambda^n}{(n-1) !} x^{n-1} e^{-\lambda x},$$
for $x \geq 0$, and $f_{S_n}(x)=0$ for $x<0$. We can use this to prove the following fact, which provides a link between Approach 1 and Approach 2 above.

# 概率论代写

## 数学代写|概率论代考Probability Theory代写|Basic Properties

$$f_{S_n}(x)=\frac{\lambda^n}{(n-1) !} x^{n-1} e^{-\lambda x}$$

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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。