### 统计代写|随机过程作业代写stochastic process代考|Discrete Time Markov Chain

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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|Definition and Transition Probabilities

Here $S=$ a countable set, $T={0,1,2, \ldots},\left{X_{n}, n \geq 0\right}$ is a stochastic process satisfying $P\left[X_{n+1}=j \mid X_{0}=i_{0}, X_{1}=i_{1}, \ldots, X_{n}=i_{n}\right]=P\left[X_{n+1}=j \mid X_{n}=i_{n}\right]$, the Markov property. Then the stochastic process $\left{X_{n}, n \geq 0\right}$ is called a Markov chain (M.C.). We shall assume that the M.C. is stationary i.e. $P\left[X_{n+1}=j \mid X_{n}=\right.$ $i]=p_{i j}$ is independent of $n$ for all $i, j \in, S$. Let $P=\left(P_{i j}\right) ; i, j \in S$ be a finite or countably infinite dimensional matrix with elements $p_{i j}$.

The matrix $P$ is called the one step transition matrix of the M.C. or simply the Transition matrix or the Probability matrix of the M.C.
Example (Random Walk) A random walk on the (real) line is a Markov chain such that
$$p_{j k}=0 \text { if } k \neq j-1 \text { or } j+1 .$$
Transition is possible only to neighbouring states (from $j$ to $j-1$ and $j+1$ ). Here state space is
$$S={\ldots,-3,-2,-1,0,1,2,3, \ldots} .$$
Theorem 2.1 The Markov chain $\left{X_{n}, n \geq 0\right}$ is completely determined by the transition matrix $P$ and the initial distribution $\left{p_{k}\right}$, defined as $P\left[X_{0}=k\right]=p_{k} \geq 0$, $\sum_{k \in s} p_{k}=1$.
Proof
\begin{aligned} P\left[X_{0}\right.&\left.=i_{0}, X_{1}=i_{i}, \ldots, X_{n}=i_{n}\right] \ &=P\left[X_{n}=i_{n} \mid X_{n-1}=i_{n-1}, X_{n-2}=i_{n-2}, \ldots, X_{1}=i_{1} \ldots X_{0}=i_{0}\right] \ P\left[X_{n-1}\right.&\left.=i_{n-1}, X_{n-2}=i_{n-2}, \ldots, X_{1}=i_{1}, X_{0}=i_{0}\right] \ &=P\left[X_{n}=i_{n} \mid X_{n-1}=i_{n-1}\right] P\left[X_{n-1}=i_{n-1}, \ldots, X_{0}=i_{0}\right] \ &=p_{i_{n-1} i_{n}} p_{i_{n-2} i_{n-1}} P\left[X_{n-2}=i_{n-2}, \ldots, X_{0}=i_{0}\right] \ &=p_{i_{n-1} i_{n}} p_{i_{n-2} i_{n-1}} \ldots p_{i_{1} i_{2}} p_{i_{0} i_{1}} p_{i_{0}} \text { (by induction). } \end{aligned}

## 统计代写|随机过程作业代写stochastic process代考|Problems

1. Suppose $P$ is a stochastic matrix, then show that $P^{n}$ is also a stochastic matrix for all $n>1$.
2. If $P^{n}$ is stochastic, is $P$ stochastic?
3. Show that 1 is an eigenvalue if $A$ is a stochastic matrix, i.e.
$$|\lambda I-A|=0 \Rightarrow \lambda=1 \text {. }$$
Consider a sequence of trials with possible outcomes $E_{1}, E_{2}, \ldots, E_{k} \ldots$ To the pairs of outcomes $\left(E_{j}, E_{k}\right)$ we can associate some numbers (i.e. conditional probabilities) $P_{j k}$. The $\left{E_{k}\right}$ are referred to as the possible states of the system. Instead of saying that the $n$th trial results in $E_{k}$ one says that the $n$th step leads to $E_{k}$ or that $E_{k}$ is entered at the $n$th step.

We shall denote by $P_{j k}^{(n)}$ the probability of transition from $E_{j}$ to $E_{k}$ in exactly $n$ steps i.e. the conditional probability of entering $E_{k}$ at the $n$th step from $E_{j}$. This is the sum of all the probabilities of all possible paths $E_{j} \rightarrow E_{j_{1}} \rightarrow \ldots E_{j_{n-1}} \rightarrow E_{k}$ of length $n$ starting at $E_{j}$. and ending at $E_{k}$
In particular,
$$p_{j k}^{(1)}=p_{j k}$$

## 统计代写|随机过程作业代写stochastic process代考|A Few More Examples

(a) Independent trials $P^{n}=P$ for all $n \geq 1$, where $p_{i j}=p_{j}$ i.e. all the rows are same.
(b) Success runs
Consider an infinite sequence of Bernoulli trials and at the $n$th trial the system is in the state $E_{j}$ if the last failure occurred at the trial number $n-j, j=0,1$, $2, \ldots$ and zero-th trial counts as failure. In other words, the index $j$ equals the length of uninterrupted run of successes ending at $n$th trial.
Here
$$p_{i j}^{(n)}=\left{\begin{array}{l} q p^{j} \text { for } j=0,1,2, \ldots, i+n-1 \ p^{j} \text { for } j=j+n \ 0 \text { otherwise } \end{array}\right.$$
This follows either directly or from Chapman-Kolmogorov’s equation. It can

be shown that $P^{n}$ converges to a matrix whose all elements in the column $j$ equals $q p^{j}$, where the transition matrix $P$ is given by
$$P_{i j}=P\left(X_{n}=j \mid X_{n-1}=i\right)=\left{\begin{array}{l} p \text { if } j=i+1 \ q \text { if } j=0 \ 0 \text { otherwise } \end{array}\right.$$
(c) Two state M.C.
There are two possible states $E_{1}$ and $E_{2}$ in which the matrix of transition probability is of the form
$$P=\left(\begin{array}{cc} 1-p & p \ a & 1-a \end{array}\right), 0<p<1 \text { and } 0<a<1 .$$
The system is said to be in state $E_{1}$ if a particle moves in the positive direction and in $E_{2}$ if the direction is negative.

## 统计代写|随机过程作业代写stochastic process代考|Definition and Transition Probabilities

pjķ=0 如果 ķ≠j−1 或者 j+1.

## 统计代写|随机过程作业代写stochastic process代考|Problems

1. 认为磷是一个随机矩阵，那么证明磷n也是一个随机矩阵n>1.
2. 如果磷n是随机的，是磷随机？
3. 证明 1 是一个特征值，如果一种是一个随机矩阵，即
|λ一世−一种|=0⇒λ=1.
考虑一系列具有可能结果的试验和1,和2,…,和ķ…到成对的结果(和j,和ķ)我们可以关联一些数字（即条件概率）磷jķ. 这\left{E_{k}\right}\left{E_{k}\right}称为系统的可能状态。而不是说n试验结果和ķ有人说n这一步导致和ķ或者那个和ķ被输入在n第一步。

pjķ(1)=pjķ

## 统计代写|随机过程作业代写stochastic process代考|A Few More Examples

(a) 独立审判磷n=磷对全部n≥1， 在哪里p一世j=pj即所有行都是相同的。
(b) 成功运行

$$p_{ij}^{(n)}=\left{qpj 为了 j=0,1,2,…,一世+n−1 pj 为了 j=j+n 0 除此以外 \对。$$

$$P_{ij}=P\left(X_{n}=j \mid X_{n-1}=i\right)=\left{ 给出p 如果 j=一世+1 q 如果 j=0 0 除此以外 \对。 (C)吨在这s吨一种吨和米.C.吨H和r和一种r和吨在这p这ss一世bl和s吨一种吨和s和1一种nd和2一世n在H一世CH吨H和米一种吨r一世X这F吨r一种ns一世吨一世这npr这b一种b一世l一世吨是一世s这F吨H和F这r米 P=\左(1−pp 一种1−一种\right), 0<p<1 \text { 和 } 0<a<1 。$$

## 广义线性模型代考

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