### 统计代写|随机过程作业代写stochastic process代考|Methods of Evaluation of the n-Step Transition Probability

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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

## 统计代写|随机过程作业代写stochastic process代考|Method of Spectral Decomposition

Let $P$ be a NXN matrix with latent roots $\lambda_{1}, \ldots, \lambda_{N}$ all distinct and simple. Then $\left(P-\lambda_{i} I\right) U_{i}=0$ for the column latent vector $U_{i}$ and $V_{i}^{\prime}\left(P-\lambda_{i} I\right)=0$ for the row latent vector $V_{i}$.
$A_{i}=U_{i} V_{i}^{\prime}$ are called latent or spectral matrix associated with $\lambda_{i}, i=1, \ldots, N$. The following properties of $A_{i}$ ‘s are well known:

(i) $A_{i}$ ‘s are idempotent, i.e. $A_{i}^{2}=A_{i}$,
(ii) they are orthogonal, i.e. $A_{i} A_{j}=0(i \neq j)$,
(iii) they give spectral decomposition $P=\sum_{i=1}^{N} \lambda_{i} A_{i}$. It follows from (i) to (iii), that
$$P^{k}=\left(\sum_{i=1}^{N} \lambda_{i} A_{i}\right)^{k}=\sum_{i=1}^{N} \lambda_{i}^{k} A_{i}=\sum_{i=1}^{N} \lambda_{i}^{k} U_{i} V_{i}^{\prime}$$
Also we know that $P^{k}=U D^{k} U^{-1}$ (by Diagonalisation Theorem) where
\begin{aligned} &U=\left(U_{1}, U_{2}, \ldots, U_{N}\right) \ &D=\left[\begin{array}{ccc} \lambda_{1} & 0 . . & 0 \ 0 & \lambda_{2} & \vdots \ 0 & \ldots & \lambda_{N} \end{array}\right] \end{aligned}
Since the latent vectors are determined uniquely only upto a multiplicative constant, we have chosen them such that $U_{i}^{\prime} V_{i}=1$. From $(2.21)$ one can get any power of $P$ knowing $\lambda_{i}$ ‘s and $A_{i}{ }^{\prime}$ s.

## 统计代写|随机过程作业代写stochastic process代考|Method of Caley-Hamilton

Caley-Hamilton Theorem Every square matrix satisfies its own characteristic equation.

The characteristic equation of $P$ is given by $|P-\lambda I|=0$. In the last example
$$\left|\begin{array}{cc} 0.9-\lambda & 0.10 \ 0.01 & 0.99 \end{array}\right|=0 \Rightarrow \lambda^{2}-1.89 \lambda+0.89=0 \text {. }$$
By Caley-Hamilton Theorem,
\begin{aligned} P^{2}-1.89 P+0.89 I &=0 \Rightarrow P^{2}=1.89 P-0.89 I \ \Rightarrow \quad P^{3} &=1.89 P^{2}-0.89 P=1.89(1.89 P-0.89 I)-0.89 P \ &=2.6821 P-1.6821 I . \end{aligned}
Similarly, any power of $P$ can be calculated in this manner.

## 统计代写|随机过程作业代写stochastic process代考|Exercises and Complements

Exercise 2.2 A service agency assigns its jobs to a particular worker in the following way. The maximum number of jobs assigned to him, in addition to one he is working on at any time is $N(\geq 1)$. If he can not finish the assigned jobs on a given day, he starts with the remaining ones the following day. However, if any time of the day he finishes all the jobs assigned to him, he returns to his own work and becomes unavailable for any more agency jobs for that day. Let $p_{j}$ be the probability that $j(\geq 0)$ new jobs arrive during a service period. Let $X_{n}$ be the number of jobs assigned to him at the end of the nth service. Under what conditions is $\left{X_{n}\right}$ a Markov chain? Determine its transition probability martix and classify its states.

Exercise 2.3 (Bartky’s sampling inspection scheme). In a sampling inspection procedure successive sampling of size $N$ is taken. If in the initial sample the number of defective is zero. the lot is accepted. If the number of defective exceeds a predetermined number a, the lot is rejected. From the second sample onward one defective per sample is allowed. Thus after $n$ such samples, the lot will be accepted if the total number of defectives is $\leq n$ and rejected if the number of defectives is $>a+n$. Let $X_{k}=$ number of defectives out of $N$-One at the $k$ th sample. Let $S_{n}=$ Total excess number of defectives in the lot. Find the distribution of $X_{k}$. Show that $\left{S_{n}\right}$ is a Markov chain. Find the transition matrix for $\left{S_{n}\right}$ in terms of distribution of $X_{k}$. Also classify the states of $\left{S_{n}\right}$.
44 Introduction to Stochastic Process
Exercise 2.4 A Simple Waiting Model (Queueing). In a simple queueing model a server serves one customer (if any) at time instant $0,1,2, \ldots$ Let $\xi_{n}$ be the number of customers arrive in the time interval $(n, n+1)$ and we assume $\left{\xi_{n}, n \geq 0\right}$ is a sequence of i.i.d. nonnegative integer valued r.v.’s with $P\left(\xi_{0}=k\right)=p_{k}, \sum_{k=0}^{\infty} p_{k}=1$ and there is a waiting room for at most $m$ customers (including the customer being served). Let $X_{n}$ be the number of customers present at time $n$, including the one being served. Show that $X_{n}$ is a Markov chain with states $0,1, \ldots, m$. Find its transition matrix in terms of $\left{p_{k}\right}_{0}^{-}$.

## 统计代写|随机过程作业代写stochastic process代考|Method of Spectral Decomposition

（一世）一种一世是幂等的，即一种一世2=一种一世,
(ii) 它们是正交的，即一种一世一种j=0(一世≠j),
(iii) 他们给出谱分解磷=∑一世=1ñλ一世一种一世. 从 (i) 到 (iii) 得出，

## 统计代写|随机过程作业代写stochastic process代考|Method of Caley-Hamilton

Caley-Hamilton 定理 每个方阵都满足自己的特征方程。

|0.9−λ0.10 0.010.99|=0⇒λ2−1.89λ+0.89=0.

44 随机过程简介

## 广义线性模型代考

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## MATLAB代写

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