### 数学代写|概率模型和随机过程代写Probability Models and Stochastic Processes代考|MATH 355

statistics-lab™ 为您的留学生涯保驾护航 在代写概率模型和随机过程方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写概率模型和随机过程代写方面经验极为丰富，各种代写概率模型和随机过程相关的作业也就用不着说。

• 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|概率模型和随机过程代写Probability Models and Stochastic Processes代考|Solutions to Assignment 1

1. Let $X$ and $Y$ be random variables defined on a common probability space. Assuming $\operatorname{Var}(X)<\infty$, show that $\operatorname{Var}(X)=\mathbb{E} \operatorname{Var}(X \mid Y)+\operatorname{Var}(\mathbb{E}[X \mid Y])$. [Hint: Use defn. of $\operatorname{Var}(X)$ and conditional expectation tricks.]

Solution: Recall that provided $\mathbb{E}|Z|<\infty$, we have that $\mathbb{E} Z=\mathbb{E E}[Z \mid Y]$. Applying this to $X$ and $X^{2}$, we know that $\mathbb{E} X=\mathbb{E}[X \mid Y]$ and $\mathbb{E} X^{2}=\mathbb{E E}\left[X^{2} \mid Y\right]$, both of which exist and are finite since $\operatorname{Var}(X)<\infty$ by assumption. Now, $\operatorname{Var}(X)=$ $\mathbb{E} X^{2}-(\mathbb{E} X)^{2}$, and $\operatorname{Var}(X \mid Y)=\mathbb{E}\left[X^{2} \mid Y\right]-(\mathbb{E}[X \mid Y])^{2}$. Hence
\begin{aligned} \operatorname{Var}(X) &=\mathbb{E} E\left[X^{2} \mid Y\right]-(\mathbb{E} E[X \mid Y])^{2} \ &=\mathbb{E E}\left[X^{2} \mid Y\right]-\mathbb{E}\left[(\mathbb{E}[X \mid Y])^{2}\right]+\mathbb{E}\left[(\mathbb{E}[X \mid Y])^{2}\right]-(\mathbb{E} E[X \mid Y])^{2} \ &=\mathbb{E} \operatorname{Var}(X \mid Y)+\operatorname{Var}(\mathbb{E}[X \mid Y]) \end{aligned}
$[1]$

1. Let $X$ be a non-negative random variable with probability density function (pdf) $f$
(a) Show that $\mathbb{E} X=\int_{0}^{\infty} \mathbb{P}(X \geqslant x) \mathrm{d} x$.
Solution: Note that $\mathbb{P}(X \geqslant x)=\int_{x}^{\infty} f(u)$ du. Hence
\begin{aligned} \int_{0}^{\infty} \mathbb{P}(X \geqslant x) \mathrm{d} x &=\int_{0}^{\infty} \int_{x}^{\infty} f(u) \mathrm{d} u \mathrm{~d} x \ &=\int_{0}^{\infty} f(u) \int_{0}^{u} 1 \mathrm{~d} x \mathrm{~d} u \ &=\int_{0}^{\infty} u f(u) \mathrm{d} u=\mathbb{E} X \end{aligned}
where the second line follows from swapping integration order.
$[1]$
(b) Using (a), show that $\mathbb{E}\left[X^{\alpha}\right]=\int_{0}^{\infty} \alpha x^{\alpha-1} \mathbb{P}(X \geqslant x) \mathrm{d} x$ for any $\alpha>0$.
Solution: Write $Y=X^{\alpha}$ which is still a non-negative random variable with some pdf. Then, from (a), we know that $\mathbb{E Y = \int _ { 0 } ^ { \infty } \mathbb { P } ( Y \geqslant y ) \mathrm { d } y \text { . Change }}$ variables via $y=x^{\alpha}$ so $\mathrm{d} y=\alpha x^{\alpha-1} \mathrm{~d} x$; note that $x=y^{1 / \alpha}$ has the same limits as $y$ for any $\alpha>0$. Hence $\mathbb{E}\left[X^{\alpha}\right]=\mathbb{E} Y=\int_{0}^{\infty} \mathbb{P}(X \geqslant x) \alpha x^{\alpha-1} \mathrm{~d} x$. A minor rearrangement yields the result.
2. Suppose $X_{1}, X_{2}, \ldots, X_{n}$ are independent random variables, with cdfs $F_{1}, F_{2}, \ldots$, $F_{n}$, respectively. Express the cdf of $M=\min \left(X_{1}, \ldots, X_{n}\right)$ in terms of the $\left{F_{i}\right}$.

## 数学代写|概率模型和随机过程代写Probability Models and Stochastic Processes代考|Assignment 2

1. Suppose a stochastic wallaby is hopping along-side the length of an infinitely long road in search of a tasty snack in the form of Livistona Rand. palm seeds. Each hop that our stochastic wallaby takes is of size 1. Our stochastic wallaby hops up the road with probability $p \in(0,1)$, and down the road with probability $q=1-p$.
(a) Suppose there is a single Livistona Rand. palm which has dropped its seeds $u$ hops up the road from where our stochastic wallaby currently is, and no such palm anywhere down the road. As a function of $p, q$, and $u$, what is the probability that the wallaby ever gets the opportunity to have its tasty snack of palm seeds?
(b) Continuing, suppose $p=q=\frac{1}{2}$. As a function of $u$, what is the expected number of hops required?
(c) * Continuing, as a function of $u$, what is the variance of the number of hops required?
(d) * Now for general $p \in(0,1)$, determine the expected number of hops and the variance of the number of hops required as a function of $p, q$, and $u$.
(e) Suppose now that is an additional Livistona Rand. palm which has dropped its seeds $d$ hops down the road from where our stochastic wallaby currently is. As a function of $p, q, u$, and $d$, what is the probability that the wallaby ever gets the opportunity to have its tasty snack of palm seeds?
$[1]$
(f) Continuing, as a function of $p, q, u$, and $d$, what is the expected number of hops required?
2. Let $X=\left(X_{n}, n=0,1, \ldots\right)$ be a Markov chain with state-space $E={0,1,2}$, initial distribution $\pi^{(0)}=(0,1,0)$, and one-step transition matrix
$$\mathbf{P}=\frac{1}{15}\left(\begin{array}{lll} 8 & 1 & 6 \ 3 & 5 & 7 \ 4 & 9 & 2 \end{array}\right)$$
(a) Draw the transition diagram for this Markov chain.
$[1]$
(b) Calculate the probability that $X_{3}=1$.
$[1]$
(c) Find the unique stationary (and limiting) distribution of the chain.
$[1]$

## 数学代写|概率模型和随机过程代写Probability Models and Stochastic Processes代考|Solutions to Assignment 2

1. Suppose a stochastic wallaby is hopping along-side the length of an infinitely long road in search of a tasty snack in the form of Livistona Rand. palm seeds. Each hop that our stochastic wallaby takes is of size 1. Our stochastic wallaby hops up the road with probability $p \in(0,1)$, and down the road with probability $q=1-p$.
(a) Suppose there is a single Livistona Rand. palm which has dropped its seeds $u$ hops up the road from where our stochastic wallaby currently is, and no such palm anywhere down the road. As a function of $p, q$, and $u$, what is the probability that the wallaby ever gets the opportunity to have its tasty snack of palm seeds?

Solution: Let $\left(S_{n}, n=0,1, \ldots\right)$ be the position of the stochastic wallaby, and denote its initial location as 0 ; that is, $S_{0}=0$. Then its position on the $(n+1)$-st hop can be written in terms of its position at the $n$-th step as $S_{n+1}=S_{n}+2 B_{n+1}-1$, for $n=0,1,2, \ldots$, where $B_{1}, B_{2}, \ldots$ iid $\operatorname{Ber}(p)$. Denote the (random) time at which the stochastic wallaby first visits position $x$ as $\tau_{x}=\inf \left{n \in \mathbb{N}: S_{n}=x\right}$. Then we seek $\mathbb{P}{0}\left(\tau{u}<\infty\right)$.

Let us introduce a new Livistona Rand. palm at position $-d$, and consider $r_{x}^{u,-d}=\mathbb{P}{x}\left(\tau{u}<\tau_{-d}\right)$. Then in particular we know that $\lim {d \rightarrow \infty} \tau{-d} \rightarrow \infty$, and so the original quantity of interest can be obtained as $\mathbb{P}{0}\left(\tau{u}<\infty\right)=$ $\lim {d \rightarrow \infty} r{0}^{u,-d}$.
Now, we know $r_{u}^{u,-d}=\mathbb{P}{u}\left(\tau{u}<\tau_{-d}\right)=1$ and $r_{-d}^{u,-d}=\mathbb{P}{-d}\left(\tau{u}<\tau_{-d}\right)=0$.
By one step analysis we also have that, for $-d<x<u$,
$$r_{x}^{u,-d}=p r_{x+1}^{u,-d}+q r_{x-1}^{u,-d} .$$
Recalling $1=p+q$, we may rearrange this to read
$$(p+q) r_{x}^{u,-d}=p r_{x+1}^{u,-d}+q r_{x-1}^{u,-d} \Longleftrightarrow \underbrace{\left(r_{x+1}^{u,-d}-r_{x}^{u,-d}\right)}{v{x+1}}=\underbrace{\frac{q}{p}}{e} \cdot \underbrace{\left(r{x}^{u,-d}-r_{x-1}^{u,-d}\right)}{v{x}} .$$
That is, $v_{x+1}=\varrho v_{x}$. Repeated application of this recursion yields $v_{x}=$ $\varrho^{x+d-1} v_{-(d-1)}$

Now, we also have (dropping the superscripts for notational convenience) that
$$r_{x}=r_{x}-r_{-d}=\sum_{y=-(d-1)}^{x} v_{y},$$
and so
\begin{aligned} r_{x} &=v_{-(d-1)} \sum_{y=-(d-1)}^{x} \varrho^{y+d-1}=\left(r_{-(d-1)}-r_{-d}\right) \sum_{z=0}^{x+d-1} \varrho^{z} \ &=r_{-(d-1)} \times \begin{cases}\frac{1-g^{z+d}}{1-\varrho}, & \varrho \neq 1, \ (x+d), & \varrho=1 .\end{cases} \end{aligned}

## 数学代写|概率模型和随机过程代写Probability Models and Stochastic Processes代考|Solutions to Assignment 1

1. 让X和是是在公共概率空间上定义的随机变量。假设曾是⁡(X)<∞， 显示曾是⁡(X)=和曾是⁡(X∣是)+曾是⁡(和[X∣是]). [提示：使用定义。的曾是⁡(X)和条件期望技巧。]

[1]

1. 让X是具有概率密度函数的非负随机变量 (pdf)F
(a) 证明和X=∫0∞磷(X⩾X)dX.
解决方案：注意磷(X⩾X)=∫X∞F(在)你。因此
∫0∞磷(X⩾X)dX=∫0∞∫X∞F(在)d在 dX =∫0∞F(在)∫0在1 dX d在 =∫0∞在F(在)d在=和X
其中第二行来自交换集成顺序。
[1]
(b) 使用 (a)，证明和[X一个]=∫0∞一个X一个−1磷(X⩾X)dX对于任何一个>0.
解决方案：写是=X一个这仍然是一个带有一些 pdf 的非负随机变量。那么，从（a），我们知道和是=∫0∞磷(是⩾是)d是 . 改变 变量通过是=X一个所以d是=一个X一个−1 dX; 注意X=是1/一个具有相同的限制是对于任何一个>0. 因此和[X一个]=和是=∫0∞磷(X⩾X)一个X一个−1 dX. 一个小的重排产生结果。
2. 认为X1,X2,…,Xn是独立的随机变量，用 cdfsF1,F2,…, Fn， 分别。表达 cdf 的米=分钟(X1,…,Xn)方面\left{F_{i}\right}\left{F_{i}\right}.

## 数学代写|概率模型和随机过程代写Probability Models and Stochastic Processes代考|Assignment 2

1. 假设一只随机的小袋鼠沿着一条无限长的道路跳跃，以寻找 Livistona Rand 形式的美味小吃。棕榈种子。我们的随机小袋鼠跳的每一跳的大小都是 1。我们的随机小袋鼠有概率在路上跳跃p∈(0,1), 并且有概率地走下去q=1−p.
(a) 假设有一个 Livistona Rand。落下种子的棕榈在从我们的随机小袋鼠目前所在的地方跳上马路，而且在路上的任何地方都没有这样的手掌。作为一个函数p,q， 和在，小袋鼠有机会吃到美味的棕榈籽零食的概率是多少？
(b) 继续，假设p=q=12. 作为一个函数在，所需的预期跳数是多少？
(c) * 继续，作为以下函数的函数在，所需跳数的方差是多少？
(d) * 现在一般p∈(0,1), 确定预期的跳数和所需跳数的方差p,q， 和在.
(e) 现在假设这是一个额外的 Livistona Rand。落下种子的棕榈d从我们的随机小袋鼠目前所在的地方跳下来。作为一个函数p,q,在， 和d，小袋鼠有机会吃到美味的棕榈籽零食的概率是多少？
[1]
(f) 继续，作为p,q,在， 和d，所需的预期跳数是多少？
2. 让X=(Xn,n=0,1,…)是具有状态空间的马尔可夫链和=0,1,2, 初始分布圆周率(0)=(0,1,0), 和一步转移矩阵
磷=115(816 357 492)
(a) 画出这条马尔可夫链的转移图。
[1]
(b) 计算概率X3=1.
[1]
(c) 找出链的唯一平稳（和限制）分布。
[1]

## 数学代写|概率模型和随机过程代写Probability Models and Stochastic Processes代考|Solutions to Assignment 2

1. 假设一只随机的小袋鼠沿着一条无限长的道路跳跃，以寻找 Livistona Rand 形式的美味小吃。棕榈种子。我们的随机小袋鼠跳的每一跳的大小都是 1。我们的随机小袋鼠有概率在路上跳跃p∈(0,1), 并且有概率地走下去q=1−p.
(a) 假设有一个 Livistona Rand。落下种子的棕榈在从我们的随机小袋鼠目前所在的地方跳上马路，而且在路上的任何地方都没有这样的手掌。作为一个函数p,q， 和在，小袋鼠有机会吃到美味的棕榈籽零食的概率是多少？

rX在,−d=prX+1在,−d+qrX−1在,−d.

(p+q)rX在,−d=prX+1在,−d+qrX−1在,−d⟺(rX+1在,−d−rX在,−d)⏟在X+1=qp⏟和⋅(rX在,−d−rX−1在,−d)⏟在X.

rX=rX−r−d=∑是=−(d−1)X在是,

rX=在−(d−1)∑是=−(d−1)Xϱ是+d−1=(r−(d−1)−r−d)∑和=0X+d−1ϱ和 =r−(d−1)×{1−G和+d1−ϱ,ϱ≠1, (X+d),ϱ=1.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。