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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|Birth-and-Death Process

A birth-and-death process is a Markov jump process with a transition rate graph of the form given in Figure 1.7. Imagine that $X_t$ represents the total number of individuals in a population at time $t$. Jumps to the right correspond to births, and jumps to the left to deaths. The birth rates $\left{b_i\right}$ and the death mates $\left{d_i\right}$ may differ from state to state. Many applications of Markov chains involve processes of this kind. Note that the process jumps from one state to the next according to a Markov chain with transition probabilities $K_{0,1}=1$, $K_{i, i+1}=b_i /\left(b_i+d_i\right)$, and $K_{i, i-1}=d_i /\left(b_i+d_i\right), i=1,2, \ldots$. Moreover, it spends an $\operatorname{Exp}\left(b_0\right)$ amount of time in state 0 and $\operatorname{Exp}\left(b_i+d_i\right)$ in the other states.
Limiting Behavior We now formulate the continuous-time analogues of (1.34) and Theorem 1.13.2. Irreducibility and recurrence for Markov jump processes are defined in the same way as for Markov chains. For simplicity, we assume that $\mathscr{E}={1,2, \ldots}$. If $X$ is a recurrent and irreducible Markov jump process, then regardless of $i$,
$$\lim {t \rightarrow \infty} \mathbb{P}\left(X_t=j \mid X_0=i\right)=\pi_j$$ for some number $\pi_j \geqslant 0$. Moreover, $\pi=\left(\pi_1, \pi_2, \ldots\right)$ is the solution to $$\sum{j \neq i} \pi_i q_{i j}=\sum_{j \neq i} \pi_j q_{j i}, \quad \text { for all } i=1, \ldots, m$$
with $\sum_j \pi_j=1$, if such a solution exists, in which case all states are positive recurrent. If such a solution does not exist, all $\pi_j$ are 0 .

As in the Markov chain case, $\left{\pi_j\right}$ is called the limiting distribution of $X$ and is usually identified with the row vector $\pi$. Any solution $\pi$ of (1.42) with $\sum_j \pi_j=1$ is called a stationary distribution, since taking it as the initial distribution of the Markov jump process renders the process stationary.

数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|GAUSSIAN PROCESSES

The normal distribution is also called the Gaussian distribution. Gaussian processes are generalizations of multivariate normal random vectors (discussed in Section 1.10). Specifically, a stochastic process $\left{X_t, t \in \mathscr{T}\right}$ is said to be Gaussian if all its finite-dimensional distributions are Gaussian. That is, if for any choice of $n$ and $t_1, \ldots, t_n \in \mathscr{T}$, it holds that
$$\left(X_{t_1}, \ldots, X_{t_n}\right)^{\top} \sim \mathrm{N}(\boldsymbol{\mu}, \Sigma)$$
for some expectation vector $\mu$ and covariance matrix $\Sigma$ (both of which depend on the choice of $\left.t_1, \ldots, t_n\right)$. Equivalently, $\left{X_t, t \in \mathscr{T}\right}$ is Gaussian if any linear combination $\sum_{i=1}^n b_i X_{t_i}$ has a normal distribution. Note that a Gaussian process is determined completely by its expectation function $\mu_t=\mathbb{E}\left[X_t\right], \quad t \in \mathscr{T}$, and covariance function $\Sigma_{s, t}=\operatorname{Cov}\left(X_s, X_t\right), s, t \in \mathscr{T}$.The quintessential Gaussian process is the Wiener process or (standard) Brownian motion. It can be viewed as a continuous version of a random walk process. Figure $1.8$ gives a typical sample path. The Wiener process plays a central role in probability and forms the basis of many other stochastic processes.

The Wiener process can be defined as a Gaussian process $\left{X_t, t \geqslant 0\right}$ with expectation function $\mu_t=0$ for all $t$ and covariance function $\Sigma_{s, t}=s$ for $0 \leqslant s \leqslant t$. The Wiener process has many fascinating properties (e.g., [11]). For example, it is a Markov process (i.e., it satisfies the Markov property (1.30)) with continuous sample paths that are nowhere differentiable. Moreover, the increments $X_t-X_s$ over intervals $[s, t]$ are independent and normally distributed. Specifically, for any $t_1<t_2 \leqslant t_3<t_4$,
$$X_{t_4}-X_{t_3} \text { and } X_{t_2}-X_{t_1}$$
are independent random variables, and for all $t \geqslant s \geqslant 0$,
$$X_t-X_s \sim \mathrm{N}(0, t-s) .$$
This leads to a simple simulation procedure for Wiener processes, which is discussed in Section 2.8.

模拟和蒙特卡洛方法代写

数学代写|模拟和蒙特卡洛方法作业代写simulation and monte-carlo method代考|Birth-and-Death Process

$$\lim {t \rightarrow \infty} \mathbb{P}\left(X_t=j \mid X_0=i\right)=\pi_j$$。此外，$\pi=\left(\pi_1, \pi_2, \ldots\right)$是$$\sum{j \neq i} \pi_i q_{i j}=\sum_{j \neq i} \pi_j q_{j i}, \quad \text { for all } i=1, \ldots, m$$

数学代写|模拟和蒙特卡洛方法作业代写simulation and monte-carlo method代考|高斯过程

$$\left(X_{t_1}, \ldots, X_{t_n}\right)^{\top} \sim \mathrm{N}(\boldsymbol{\mu}, \Sigma)$$
。同样，如果任何线性组合$\sum_{i=1}^n b_i X_{t_i}$具有正态分布，则$\left{X_t, t \in \mathscr{T}\right}$就是高斯分布。注意，高斯过程完全由它的期望函数$\mu_t=\mathbb{E}\left[X_t\right], \quad t \in \mathscr{T}$和协方差函数$\Sigma_{s, t}=\operatorname{Cov}\left(X_s, X_t\right), s, t \in \mathscr{T}$决定。典型的高斯过程是维纳过程或(标准)布朗运动。它可以被看作是随机游走过程的一个连续版本。图$1.8$给出了一个典型的示例路径。维纳过程在概率论中起着核心作用，并且构成了许多其他随机过程的基础

$$X_{t_4}-X_{t_3} \text { and } X_{t_2}-X_{t_1}$$

$$X_t-X_s \sim \mathrm{N}(0, t-s) .$$

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。

assignmentutor™您的专属作业导师
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