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# 数学代写| 随机过程代考|Introduction

A population starts with one individual at time $n=0: Z_{0}=1$.

After one unit of time (at time $n=1$ ) the sole individual produces $Z_{1}$ identical clones of itself and dies. $Z_{1}$ is an $\mathbb{N}_{0}$-valued random variable.

(a) If $Z_{1}$ happens to be equal to 0 the population is dead and nothing happens at any future time $n \geq 2$.

(b) If $Z_{1}>0$, a unit of time later, each of $Z_{1}$ individuals gives birth to a random number of children and dies. The first one has $Z_{1,1}$ children, the second one $Z_{1,2}$ children, etc. The last, $Z_{1}^{\text {th }}$ one, gives birth to $Z_{1, Z_{1}}$ children. We assume that the distribution of the number of children is the same for each individual in every generation and independent of either the number of individuals in the generation and of the number of children the others have. This distribution, shared by all $Z_{n, i}$ and $Z_{1}$, is called the offspring distribution. The total number of individuals in the second generation is now
$$Z_{2}=\sum_{k=1}^{Z_{1}} Z_{1, k}$$
(c) The third, fourth, etc. generations are produced in the same way. If it ever happens that $Z_{n}=0$, for some $n$, then $Z_{m}=0$ for all $m \geq n$ – the population is extinct. Otherwise,
$$Z_{n+1}=\sum_{k=1}^{Z_{n}} Z_{n, k}$$

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• 粒子滤波 Particle Filter
• 采样理论 sampling theory

## 数学代写| 随机过程代考|mathematical

Other possible choices for $T$ include $\mathbb{R}^{n}$ and $\mathbb{Z}^{n}$, whilst $S$ might be an uncountable set such as $\mathbb{R}$. The mathematical analysis of a random process varies greatly depending on whether $S$ and $T$ are countable or uncountable, just as discrete random variables are distinguishable from continuous variables. The main differences are indicated by those cases in which
(a) $T={0,1,2, \ldots}$ or $T=[0, \infty)$,
(b) $S=\mathbb{Z}$ or $S=\mathbb{R}$.
There are two levels at which we can observe the evolution of a random process $X$.
(a) Each $X_{t}$ is a function which maps $\Omega$ into $S$. For any fixed $\omega \in \Omega$, there is a corresponding collection $\left{X_{t}(\omega): t \in T\right}$ of members of $S$; this is called the realization or sample path of $X$ at $\omega$. We can study properties of sample paths.
(b) The $X_{t}$ are not independent in general. If $S \subseteq \mathbb{R}$ and $\mathbf{t}=\left(t_{1}, t_{2}, \ldots, t_{n}\right)$ is a vector of members of $T$, then the vector $\left(X_{t_{1}}, X_{t_{2}}, \ldots, X_{t_{n}}\right)$ has joint distribution function $F_{\mathbf{t}}: \mathbb{R}^{n} \rightarrow[0,1]$ given by $F_{\mathbf{t}}(\mathbf{x})=\mathbb{P}\left(X_{t_{1}} \leq x_{1}, \ldots, X_{t_{n}} \leq x_{n}\right)$. The collection $\left{F_{\mathbf{t}}\right}$, as $\mathbf{t}$ ranges over all vectors of members of $T$ of any finite length, is called the collection of finite-dimensional distributions (abbreviated to $f d d s$ ) of $X$, and it contains all the information which is available about $X$ from the distributions of its component variables $X_{t}$. We can study the distributional properties of $X$ by using its fdds.
These two approaches do not generally yield the same information about the process in question, since knowledge of the fdds does not yield complete information about the properties of the sample paths. We shall see an example of this in the final section of this chapter.

## 数学代写| 随机过程代考|random processes

We are not concerned here with the general theory of random processes, but prefer to study certain specific collections of processes which are characterized by one or more special properties. This is not a new approach for us. In Chapter 6 we devoted our attention to processes which satisfy the Markov property, whilst large parts of Chapter 7 were devoted to sequences $\left{S_{n}\right}$ which wereeither martingales or the partial sums of independent sequences. In this short chapter we introduce certain other types of process and their characteristic properties. These can be divided broadly under four headings, covering ‘stationary processes’, ‘renewal processes’, ‘queues’, and ‘diffusions’; their detailed analysis is left for Chapters $9,10,11$, and 13 respectively.

We shall only be concerned with the cases when $T$ is one of the sets $\mathbb{Z},{0,1,2, \ldots}, \mathbb{R}$, or $[0, \infty)$. If $T$ is an uncountable subset of $\mathbb{R}$, representing continuous time say, then we shall usually write $X(t)$ rather than $X_{t}$ for ease of notation. Evaluation of $X(t)$ at some $\omega \in \Omega$ yields a point in $S$, which we shall denote by $X(t ; \omega)$.

## 数学代写| 随机过程代考|MATHEMATICAL

b 小号=和要么小号=R.

b这X吨一般不独立。如果小号⊆R和吨=(吨1,吨2,…,吨n)是成员的向量吨, 然后向量(X吨1,X吨2,…,X吨n)具有联合分布函数F吨:Rn→[0,1]由F吨(X)=磷(X吨1≤X1,…,X吨n≤Xn). 该系列\left{F_{\mathbf{t}}\right}\left{F_{\mathbf{t}}\right}， 作为吨范围在成员的所有向量吨任何有限长度的，称为有限维分布的集合一种bbr和v一世一种吨和d吨○$Fdds$的X, 它包含所有可用的信息X从其组成变量的分布X吨. 我们可以研究分布特性X通过使用它的 fdds。

## 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。