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## 数学代写|矩阵方法代写Applied Matrix Theory代考|Discrete Phase-Type Distributions

Let $\left{X_n\right}_{n \in \mathbb{N}}$ be a Markov chain with state space ${1,2, \ldots, p, p+1}$, where the states $1,2, \ldots, p$ are transient, and consequently, state $p+1$ is absorbing. Then $\left{X_n\right}_{n \in \mathbb{N}}$ has a transition matrix $\boldsymbol{P}$ of the form
$$\boldsymbol{P}=\left(\begin{array}{ll} \boldsymbol{T} & \boldsymbol{t} \ \mathbf{0} & 1 \end{array}\right)$$

where $T$ is a $p \times p$ subtransition matrix (i.e., a matrix of nonnegative numbers in which the rows sum to numbers less than or equal to one, written as $\boldsymbol{T e} \leq \boldsymbol{e}$ ), and $\boldsymbol{t}$ is a p-dimensional column vector. Since $t_i$ is the probability of jumping to an absorbing state directly from state $i$, we shall refer to these probabilities as exit probabilities. Since the rows sum to 1 , we must have that
$$\boldsymbol{t}=\boldsymbol{e}-\boldsymbol{T e}=(\boldsymbol{I}-\boldsymbol{T}) \boldsymbol{e},$$
where $\boldsymbol{e}^{\prime}=(1,1, \ldots, 1)$ is the column vector of ones. Thus $\boldsymbol{t}$ can be obtained from $\boldsymbol{T}$ and hence discarded when the necessary parameters are specified. Let $\pi_i=$ $\mathbb{P}\left(X_0=i\right), \boldsymbol{\pi}=\left(\pi_1, \ldots, \pi_p\right)$ and assume that $\boldsymbol{\pi} \boldsymbol{e}=\pi_1+\cdots+\pi_p=1$.

Definition 1.2.54. Let $\tau=\inf \left{n \geq 1 \mid X_n=p+1\right}$ be the time until absorption. Then we say that $\tau$ has a (discrete) phase-type distribution with initial distribution $\pi$ and subtransition matrix $\boldsymbol{T}$, and we write
$$\tau \sim \mathrm{DPH}_p(\boldsymbol{\pi}, \boldsymbol{T})$$

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Markov Jump Processes

In this section we consider Markov processes in continuous time that take values in a discrete (finite or at most countable) state space. By nature, such processes are piecewise constant, and transitions occur via jumps. They are often referred to as Markov jump processes or continuous-time Markov chains. Which value the process takes at the time of a jump can be assigned arbitrarily, however, we will always assume that the process takes the value of the state to which it jumps. This assumption makes Markov jump processes continuous from the right (and with limits from the left), i.e., they are so-called càdlàg processes.

Definition 1.3.1. A continuous-time stochastic process $\left{X_t\right}_t \geq 0$ taking values in a countable set $E$ is called a Markov jump process with state space $E$ if for all $t_n>$ $t_{n-1}>\cdots>t_1>0$ and $i_n, i_{n-1}, \ldots, i_0 \in E$, we have that
$$\mathbb{P}\left(X_{t_n}=i_n \mid X_{t_{n-1}}=i_{n-1}, \ldots, X_{t_1}=i_1, X_0=i_0\right)=\mathbb{P}\left(X_{t_n}=i_n \mid X_{t_{n-1}}=i_{n-1}\right) .$$
The process is called time-homogeneous if the transition probabilities $\mathbb{P}\left(X_{t+h}=\right.$ $\left.j \mid X_t=i\right)$ depend only on $h$, in which case it is denoted by $p_{i j}(h)$ and referred to as an $h$-step transition probability. Throughout, we assume that all Markov jump processes are time-homogeneous.
The transition probabilities are then arranged in transition matrices
$$\boldsymbol{P}(h)=\left{p_{i j}(h)\right}_{i, j \in E}, \quad h \geq 0 .$$

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Discrete Phase-Type Distributions

$$\boldsymbol{P}=\left(\begin{array}{lll} \boldsymbol{T} & \boldsymbol{t} 0 & 1 \end{array}\right)$$

$$t=e-T e=(I-T) e,$$
$\mathbb{P}\left(X_0=i\right), \pi=\left(\pi_1, \ldots, \pi_p\right)$ 并假设 $\pi e=\pi_1+\cdots+\pi_p=1$.

$$\tau \sim \operatorname{DPH}p(\pi, \boldsymbol{T})$$

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Markov Jump Processes

$$\mathbb{P}\left(X_{t_n}=i_n \mid X_{t_{n-1}}=i_{n-1}, \ldots, X_{t_1}=i_1, X_0=i_0\right)=\mathbb{P}\left(X_{t_n}=i_n \mid X_{t_{n-1}}=i_{n-1}\right) .$$

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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。