## 统计代写|随机分析作业代写stochastic analysis代写|MA53200

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl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。我们的专家在代写随机分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl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

## 统计代写|随机分析作业代写stochastic analysis代写|Continuous Distributions

Consider now the general case when $\Omega$ is not necessarily enumerable. Let us begin with the definition of a random variable. Denote by $\mathcal{R}$ the Borel $\sigma$-algebra on $\mathbb{R}$, the smallest $\sigma$-algebra containing all open sets.

Definition 1.10. A random variable $X$ is an $\mathcal{F}$-measurable real-valued function $X: \Omega \rightarrow \mathbb{R}$; i.e., for any $B \in \mathcal{R}, X^{-1}(B) \in \mathcal{F}$.

Definition 1.11. The distribution of the random variable $X$ is a probability measure $\mu$ on $\mathbb{R}$, defined for any set $B \in \mathcal{R}$ by
$$\mu(B)=\mathbb{P}(X \in B)=\mathbb{P} \circ X^{-1}(B) .$$
In particular, we define the distribution function $F(x)=\mathbb{P}(X \leq x)$ when $B=(-\infty, x]$

If there exists an integrable function $\rho(x)$ such that
$$\mu(B)=\int_B \rho(x) d x$$
for any $B \in \mathcal{R}$, then $\rho$ is called the probability density function ( $\mathrm{PDF}$ ) of $X$. Here $\rho(x)=d \mu / d m$ is the Radon-Nikodym derivative of $\mu(d x)$ with respect to the Lebesgue measure $m(d x)$ if $\mu(d x)$ is absolutely continuous with respect to $m(d x)$; i.e., for any set $B \in \mathcal{R}$, if $m(B)=0$, then $\mu(B)=0$ (see also Section C of the appendix) [Bil79]. In this case, we write $\mu \ll m$.
Definition 1.12. The expectation of a random variable $X$ is defined as
$$\mathbb{E} X=\int_{\Omega} X(\omega) \mathbb{P}(d \omega)=\int_{\mathbb{R}} x \mu(d x)$$
if the integrals are well-defined.
The variance of $X$ is defined as
$$\operatorname{Var}(X)=\mathbb{E}(X-\mathbb{E} X)^2 .$$
For two random variables $X$ and $Y$, we can define their covariance as (1.15) $\operatorname{Cov}(X, Y)=\mathbb{E}(X-\mathbb{E} X)(Y-\mathbb{E} Y)$.
$X$ and $Y$ are called uncorrelated if $\operatorname{Cov}(X, Y)=0$.

## 统计代写|随机分析作业代写stochastic analysis代写|Independence

We now come to one of the most distinctive notions in probability theory, the notion of independence. Let us start by defining the independence of events. Two events $A, B \in \mathcal{F}$ are independent if
$$\mathbb{P}(A \cap B)=\mathbb{P}(A) \mathbb{P}(B) .$$
Definition 1.21. Two random variables $X$ and $Y$ are said to be independent if for any two Borel sets $A$ and $B, X^{-1}(A)$ and $Y^{-1}(B)$ are independent; i.e.,
(1.30) $\quad \mathbb{P}\left(X^{-1}(A) \cap Y^{-1}(B)\right)=\mathbb{P}\left(X^{-1}(A)\right) \mathbb{P}\left(Y^{-1}(B)\right)$.

The joint distribution of the two random variables $X$ and $Y$ is defined to be the distribution of the random vector $(X, Y)$. Let $\mu_1$ and $\mu_2$ be the probability distribution of $X$ and $Y$, respectively, and let $\mu$ be their joint distribution. If $X$ and $Y$ are independent, then for any two Borel sets $A$ and $B$, we have
$$\mu(A \times B)=\mu_1(A) \mu_2(B) .$$
Consequently, we have
$$\mu=\mu_1 \mu_2 ;$$
i.e., the joint distribution of two independent random variables is the product distribution. If both $\mu_1$ and $\mu_2$ are absolutely continuous, with densities $p_1$ and $p_2$, respectively, then $\mu$ is also absolutely continuous, with density given by
$$p(x, y)=p_1(x) p_2(y) .$$
One can also understand independence from the viewpoint of expectations. Let $f_1$ and $f_2$ be two continuous functions. If $X$ and $Y$ are two independent random variables, then
$$\mathbb{E} f_1(X) f_2(Y)=\mathbb{E} f_1(X) \mathbb{E} f_2(Y) .$$
In fact, this can also be used as the definition of independence.

# 随机分析代考

## 统计代写|随机分析作业代写stochastic analysis代写|Continuous Distributions

$$\mu(B)=\mathbb{P}(X \in B)=\mathbb{P} \circ X^{-1}(B) .$$

$$\mu(B)=\int_B \rho(x) d x$$

$$\mathbb{E} X=\int_{\Omega} X(\omega) \mathbb{P}(d \omega)=\int_{\mathbb{R}} x \mu(d x)$$

$$\operatorname{Var}(X)=\mathbb{E}(X-\mathbb{E} X)^2$$

$\operatorname{Cov}(X, Y)=\mathbb{E}(X-\mathbb{E} X)(Y-\mathbb{E} Y)$.
$X$ 和 $Y$ 被称为不相关的，如果 $\operatorname{Cov}(X, Y)=0$.

## 统计代写|随机分析作业代写stochastic analysis代写|Independence

$$\mathbb{P}(A \cap B)=\mathbb{P}(A) \mathbb{P}(B) .$$

(1.30) $\mathbb{P}\left(X^{-1}(A) \cap Y^{-1}(B)\right)=\mathbb{P}\left(X^{-1}(A)\right) \mathbb{P}\left(Y^{-1}(B)\right)$.

$$\mu(A \times B)=\mu_1(A) \mu_2(B) .$$

$$\mu=\mu_1 \mu_2 ;$$

$$p(x, y)=p_1(x) p_2(y)$$

$$\mathbb{E} f_1(X) f_2(Y)=\mathbb{E} f_1(X) \mathbb{E} f_2(Y)$$

## 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。

## 统计代写|随机分析作业代写stochastic analysis代写|MATH477

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl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。我们的专家在代写随机分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl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

## 统计代写|随机分析作业代写stochastic analysis代写|Conditional Probability

Let $A, B \in \mathcal{F}$ and assume that $\mathbb{P}(B) \neq 0$. Then the conditional probability of $A$ given $B$ is defined as
$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} .$$
This is the proportion of events that both $A$ and $B$ occur given that $B$ occurs. For instance, the probability to obtain two tails in two tosses of a fair coin is $1 / 4$, but the conditional probability to obtain two tails is $1 / 2$ given that the first toss is a tail, and it is zero given that the first toss is a head.
Since $\mathbb{P}(A \cap B)=\mathbb{P}(A \mid B) \mathbb{P}(B)$ by definition, we also have
$$\mathbb{P}(A \cap B \cap C)=\mathbb{P}(A \mid B \cap C) \mathbb{P}(B \mid C) \mathbb{P}(C),$$
and so on. It is straightforward to obtain
$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A) \mathbb{P}(B \mid A)}{\mathbb{P}(B)}$$
from the definition of conditional probability. This is called Bayes’s rule.

Proposition 1.6 (Bayes’s theorem). If $A_1, A_2, \ldots$ are disjoint sets such that $\bigcup_{j=1}^{\infty} A_j=\Omega$, then we have
$$\mathbb{P}\left(A_j \mid B\right)=\frac{\mathbb{P}\left(A_j\right) \mathbb{P}\left(B \mid A_j\right)}{\sum_{n=1}^{\infty} \mathbb{P}\left(A_n\right) \mathbb{P}\left(B \mid A_n\right)} \quad \text { for any } j \in \mathbb{N} \text {. }$$
This is useful in Bayesian statistics where $A_j$ corresponds to the hypothesis and $\mathbb{P}\left(A_j\right)$ is the prior probability of the hypothesis $A_j$. The conditional probability $\mathbb{P}\left(A_j \mid B\right)$ is the posterior probability of $A_j$ given that the event $B$ occurs.

## 统计代写|随机分析作业代写stochastic analysis代写|Discrete Distributions

If the elements in $\Omega$ are finite or enumerable, say, $\Omega=\left{\omega_1, \omega_2, \ldots\right}$, we have a situation of discrete probability space and discrete distribution. In this case, let $X\left(\omega_j\right)=x_j$ and
$$p_j=\mathbb{P}\left(X=x_j\right), \quad j=0,1, \ldots$$
Of course, we have to have
$$0 \leq p_j \leq 1, \quad \sum_j p_j=1 .$$
Given a function $f$ of $X$, its expectation is given by
$$\mathbb{E} f(X)=\sum_j f\left(x_j\right) p_j$$
if the sum is well-defined. In particular, the $p$ th moment of the distribution is defined as
$$m_p=\sum_j x_j^p p_j .$$
When $p=1$, it is called the mean of the random variable and is also denoted by mean $(X)$. Another important quantity is its variance, defined as
$$\operatorname{Var}(X)=m_2-m_1^2=\sum_j\left(x_j-m_1\right)^2 p_j .$$
Example 1.7 (Bernoulli distribution). The Bernoulli distribution has the form
$$\mathbb{P}(X=j)= \begin{cases}p, & j=1, \ q, & j=0,\end{cases}$$
$p+q=1$ and $p, q \geq 0$. When $p=q=1 / 2$, it corresponds to the toss of a fair coin. The mean and variance can be calculated directly:
$$\mathbb{E} X=p, \quad \operatorname{Var}(X)=p q$$

# 随机分析代考

## 统计代写|随机分析作业代写stochastic analysis代写|Conditional Probability

$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} .$$

$$\mathbb{P}(A \cap B \cap C)=\mathbb{P}(A \mid B \cap C) \mathbb{P}(B \mid C) \mathbb{P}(C),$$

$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A) \mathbb{P}(B \mid A)}{\mathbb{P}(B)}$$

$$\mathbb{P}\left(A_j \mid B\right)=\frac{\mathbb{P}\left(A_j\right) \mathbb{P}\left(B \mid A_j\right)}{\sum_{n=1}^{\infty} \mathbb{P}\left(A_n\right) \mathbb{P}\left(B \mid A_n\right)} \quad \text { for any } j \in \mathbb{N} \text {. }$$

## 统计代写|随机分析作业代写stochastic analysis代写|Discrete Distributions

$$p_j=\mathbb{P}\left(X=x_j\right), \quad j=0,1, \ldots$$

$$0 \leq p_j \leq 1, \quad \sum_j p_j=1 .$$

$$\mathbb{E} f(X)=\sum_j f\left(x_j\right) p_j$$

$$m_p=\sum_j x_j^p p_j .$$

$$\operatorname{Var}(X)=m_2-m_1^2=\sum_j\left(x_j-m_1\right)^2 p_j .$$

$$\mathbb{P}(X=j)={p, \quad j=1, q, \quad j=0,$$
$p+q=1$ 和 $p, q \geq 0$. 什么时候 $p=q=1 / 2$ ，它对应于公平硬币的抛郑。可以直接计算均值和方差:
$$\mathbb{E} X=p, \quad \operatorname{Var}(X)=p q$$

## 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。

## 统计代写|随机分析作业代写stochastic analysis代写|STAT342

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl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。我们的专家在代写随机分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl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

## 统计代写|随机分析作业代写stochastic analysis代写|Elementary Examples

We will start with some elementary examples of probability. The most wellknown example is that of a fair coin: if flipped, the probability of getting a head or tail both equal to $1 / 2$. If we perform $n$ independent tosses, then the probability of obtaining $n$ heads is equal to $1 / 2^n$ : among the $2^n$ equally possible outcomes only one gives the result that we look for. More generally, let $S_n=X_1+X_2+\cdots+X_n$, where
$$X_j= \begin{cases}1, & \text { if the result of the } n \text {th trial is a head, } \ 0, & \text { if the result of the } n \text {th trial is a tail. }\end{cases}$$
Then the probability that we get $k$ heads out of $n$ tosses is equal to
$$\operatorname{Prob}\left(S_n=k\right)=\frac{1}{2^n}\left(\begin{array}{l} n \ k \end{array}\right) .$$
Applying Stirling’s formula
$$n ! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n, \quad n \rightarrow \infty,$$
we can calculate, for example, the asymptotic probability of obtaining heads exactly half of the time:
$$\operatorname{Prob}\left(S_{2 n}=n\right)=\frac{1}{2^{2 n}}\left(\begin{array}{c} 2 n \ n \end{array}\right)=\frac{1}{2^{2 n}} \frac{(2 n) !}{(n !)^2} \sim \frac{1}{\sqrt{\pi n}} \rightarrow 0,$$
as $n \rightarrow \infty$.

On the other hand, since we have a fair coin, we do expect to obtain heads roughly half of the time; i.e.,
$$\frac{S_{2 n}}{2 n} \approx \frac{1}{2},$$
for large $n$. Such a statement is indeed true and is embodied in the law of large numbers that we will discuss in the next chapter. For the moment let us simply observe that while the probability that $S_{2 n}$ equals $n$ goes to zero as $n \rightarrow \infty$, the probability that $S_{2 n}$ is close to $n$ goes to 1 as $n \rightarrow \infty$. More precisely, for any $\epsilon>0$,
$$\operatorname{Prob}\left(\left|\frac{S_{2 n}}{2 n}-\frac{1}{2}\right|>\epsilon\right) \rightarrow 0,$$
as $n \rightarrow \infty$. This can be seen as follows.

## 统计代写|随机分析作业代写stochastic analysis代写|Probability Space

It is useful to put these intuitive notions of probability on a firm mathematical basis, as was done by Kolmogorov. For this purpose, we need the notion of probability space, often written as a triplet $(\Omega, \mathcal{F}, \mathbb{P})$, defined as follows.
Definition $1.1$ (Sample space). The sample space $\Omega$ is the set of all possible outcomes. Each element $\omega \in \Omega$ is called a sample point.

Definition $1.2$ ( $\sigma$-algebra). A $\sigma$-algebra (or $\sigma$-field) $\mathcal{F}$ is a collection of subsets of $\Omega$ that satisfies the following conditions:
(i) $\Omega \in \mathcal{F}$;
(ii) if $A \in \mathcal{F}$, then $A^c \in \mathcal{F}$, where $A^c=\Omega \backslash A$ is the complement of $A$ in $\Omega$;
(iii) if $A_1, A_2, \ldots \in \mathcal{F}$, then $\bigcup_{n=1}^{\infty} A_n \in \mathcal{F}$.
Each set $A$ in $\mathcal{F}$ is called an event. Let $\mathcal{B}$ be a collection of subsets of $\Omega$. We denote by $\sigma(\mathcal{B})$ the smallest $\sigma$-algebra generated by the sets in $\mathcal{B}$, i.e., the smallest $\sigma$-algebra that contains $\mathcal{B}$. The pair $(\Omega, \mathcal{F})$ with the above properties is called a measurable space.

Definition $1.3$ (Probability measure). The probability measure $\mathbb{P}: \mathcal{F} \rightarrow$ $[0,1]$ is a set function defined on $\mathcal{F}$ which satisfies
(a) $\mathbb{P}(\emptyset)=0, \mathbb{P}(\Omega)=1$;
(b) if $A_1, A_2, \ldots \in \mathcal{F}$ are pairwise disjoint, i.e., $A_i \cap A_j=\emptyset$ if $i \neq j$, then
$$\mathbb{P}\left(\bigcup_{n=1}^{\infty} A_n\right)=\sum_{n=1}^{\infty} \mathbb{P}\left(A_n\right) .$$
(1.1) is called countable additivity or $\sigma$-additivity.

# 随机分析代考

## 统计代写|随机分析作业代写stochastic analysis代写|Elementary Examples

$X_j={1, \quad$ if the result of the $n$th trial is a head, $0, \quad$ if the result of the $n$th trial is a tail.

$$\operatorname{Prob}\left(S_n=k\right)=\frac{1}{2^n}(n k) .$$

$$n ! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n, \quad n \rightarrow \infty,$$

$$\operatorname{Prob}\left(S_{2 n}=n\right)=\frac{1}{2^{2 n}}(2 n n)=\frac{1}{2^{2 n}} \frac{(2 n) !}{(n !)^2} \sim \frac{1}{\sqrt{\pi n}} \rightarrow 0,$$

$$\frac{S_{2 n}}{2 n} \approx \frac{1}{2},$$

$$\operatorname{Prob}\left(\left|\frac{S_{2 n}}{2 n}-\frac{1}{2}\right|>\epsilon\right) \rightarrow 0,$$

## 统计代写|随机分析作业代写stochastic analysis代写|Probability Space

(ii) 如果 $A \in \mathcal{F}$ ，然后 $A^c \in \mathcal{F}$ ，在哪里 $A^c=\Omega \backslash A$ 是的补充 $A$ 在 $\Omega$;
(iii) 如果 $A_1, A_2, \ldots \in \mathcal{F}$ ，然后 $\bigcup_{n=1}^{\infty} A_n \in \mathcal{F}$.

(a) $\mathbb{P}(\emptyset)=0, \mathbb{P}(\Omega)=1$;
(b) 如果 $A_1, A_2, \ldots \in \mathcal{F}$ 成对不相交，即 $A_i \cap A_j=\emptyset$ 如果 $i \neq j$ ，然后
$$\mathbb{P}\left(\bigcup_{n=1}^{\infty} A_n\right)=\sum_{n=1}^{\infty} \mathbb{P}\left(A_n\right) .$$
(1.1) 称为可数加性或 $\sigma$-可加性。

## 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。

## 统计代写|随机分析作业代写stochastic analysis代写|MA53200

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl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。我们的专家在代写随机分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl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

## 统计代写|随机分析作业代写stochastic analysis代写|Loynes’s scheme

Here we will consider the case where the state space $E$ is equipped with a partial ordering $\preceq$ (see section A.3), and admits a minimal point $\mathbf{0}$ such that $\mathbf{0} \preceq x$ for all $x \in E$. We will assume that on $E$ there exists a metric $d_{E}$ such that all $\preceq$-increasing sequences converge in $\bar{E}$, the adherence of $E$.
DEFINITION 2.5.- A function $\varphi: E \times F^{\mathbf{Z}} \rightarrow E$ is said $\preceq$-increasing when
$$\eta \preceq \eta^{\prime} \Longrightarrow \varphi(\eta, \omega) \preceq \varphi\left(\eta^{\prime}, \omega\right), \mathbf{P}{X}-a . s . .$$ It is said continuous with respect to its first variable when for $\mathbf{P}{X}$-almost all $\omega$, the function $(\eta \mapsto \varphi(\eta, \omega))$ is continuous for the metric $d_{E}$.

THEOREM $2.4$ (LOYNES’s THEOREM).- If $\varphi$ is $\preceq$-increasing and continuous, the equation [2.7] admits a solution $M_{\infty}$ with values in the adherence $\bar{E}$ of $E$.

Proof. Let us recall that we have assumed that we know the stimulus through the quadruple $\mathfrak{O}$, whose generic element is denoted $\omega$. We look for a random variable $Y$ valued in $E$ and satisfying [2.7]. We will get $Y$ as the limit of a sequence converging almost surely. To do this, we consider Loynes’s sequence $\left(M_{n}, n \in \mathbf{N}\right)$, defined by
$$M_{0}(\omega)=\mathbf{0} \text { and } M_{n+1}(\omega)=\varphi\left(M_{n} \circ \theta^{-1}(\omega), \theta^{-1} \omega\right), \forall n \geq 1 .$$
By the definition of $\mathbf{0}$, we have $M_{0}=\mathbf{0} \preceq M_{1}$, and assuming that for some $n>1$, $M_{n-1} \preceq M_{n}$ a.s., since $\varphi$ is increasing we have
$$M_{n}(\omega)=\varphi\left(M_{n-1}\left(\theta^{-1} \omega\right), \theta^{-1} \omega\right) \preceq \varphi\left(M_{n}\left(\theta^{-1} \omega\right), \theta^{-1} \omega\right)=M_{n+1}(\omega) \mathbf{P}_{X} \text {-a.s.. }$$

## 统计代写|随机分析作业代写stochastic analysis代写|Coupling

The idea of coupling plays a central role in the asymptotic study of SRS. It is in fact possible to state the conditions under which the trajectories of two SRS (or possibly those of the corresponding backward schemes) coincide at a certain point. These properties imply naturally, in particular, more traditional properties of convergence for random sequences such as convergence in distribution.

Hereafter we only state the results that will be useful to us in the applications to queueing, in their simplest form.

Secondly, we develop the theory of renovating events of Borovkov, which gives sufficient conditions for coupling, and even strong backward coupling. In addition, the results of Borovkov and Foss also allow in many cases to solve the equation [2.7], even when the conditions of continuity and monotonicity of Theorem $2.4$ are not satisfied. Particularly, in this framework we can also deal with the intricate question of the transient behavior depending on the initial conditions. In what follows, $\mathfrak{O}=$ $(\Omega, \mathcal{F}, \mathbf{P}, \theta)$ is a stationary ergodic quadruple.

## 统计代写|随机分析作业代写stochastic analysis代写|Loynes’s scheme

$$\eta \preceq \eta^{\prime} \Longrightarrow \varphi(\eta, \omega) \preceq \varphi\left(\eta^{\prime}, \omega\right), \mathbf{P} X-a . s . .$$

$$M_{0}(\omega)=\mathbf{0} \text { and } M_{n+1}(\omega)=\varphi\left(M_{n} \circ \theta^{-1}(\omega), \theta^{-1} \omega\right), \forall n \geq 1$$

$$M_{n}(\omega)=\varphi\left(M_{n-1}\left(\theta^{-1} \omega\right), \theta^{-1} \omega\right) \preceq \varphi\left(M_{n}\left(\theta^{-1} \omega\right), \theta^{-1} \omega\right)=M_{n+1}(\omega) \mathbf{P}_{X} \text {-a.s.. }$$

## 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。

## 统计代写|随机分析作业代写stochastic analysis代写|MATH477

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl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。我们的专家在代写随机分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl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

## 统计代写|随机分析作业代写stochastic analysis代写|Fluid model

A fluid model consists of replacing a queue which is a discrete-time event system by a reservoir of infinite capacity which empties itself at unit speed and is fed by some continuous data flow. We can then obtain qualitative results on models whose study supports no other approaches. On the one hand, the method does not require precise knowledge about the rate of the input process, and on the other hand, it is particularly well adapted to the study of extreme cases: low and high loads, superposition of heterogeneous traffic.

We work in continuous time and we assume that all the processes are rightcontinuous with left limits. We denote:
1) $S(t)$ : the total service time for the requests arrived up to time $t$;
2) $W(t)$ : the virtual waiting time of a customer arriving at time $t$, that is the time that the customer must wait before starting to be served;
3) $X(t)=S(t)-t$.
As the system has no losses, we have
$$W(t)=X(t)-\left(t-\int_{0}^{t} \mathbf{1}_{{0}}(W(s)) \mathrm{d} s\right) .$$
We will focus on showing an equivalent formulation of this equation.

## 统计代写|随机分析作业代写stochastic analysis代写|Canonical space

The concept of stationarity implies invariance in time, that is : a shift in time does not change the global picture. If the idea is easily understood, its formalization quickly clouds the basic concept.

Let us consider the set $F^{\mathbf{N}}$ of sequences of elements of a set $F$. The shift operator $\theta$ on $F^{\mathbf{N}}$ is then defined by
$$\theta: \begin{cases}F^{\mathbf{N}} & \longrightarrow F^{\mathbf{N}} \ \left(\omega_{n}, n \geq 0\right) & \longmapsto\left(\omega_{n+1}, n \geq 0\right)=\left(\omega_{n}, n \geq 1\right)\end{cases}$$
Defined in this way, this operator has the drawback of not being bijective: if we consider a sequence $\beta=\left(\beta_{n}, n \geq 0\right)$, all the sequences obtained by concatenation of any element of $F$ and $\beta$ are mapped onto $\beta$ by $\theta$. To overcome this problem, it is customary to work with sequences indexed by $\mathbf{Z}$ and not by $\mathbf{N}$. This change has no crucial mathematical consequence, as the indexation space remains countable. Philosophically, however, it implies that there is no more origin of time…
The shift operator is thus defined on $F^{\mathbf{Z}}$ by
$$\theta\left(\omega_{n}, n \in \mathbf{Z}\right)=\left(\omega_{n+1}, n \in \mathbf{Z}\right)$$
and thus becomes bijective!

## 统计代写|随机分析作业代写stochastic analysis代写|Fluid model

1) $S(t)$ ：请求的总服务时间到达时间 $t$;
2) $W(t)$ ：客户到达时间的虚拟等待时间 $t$ ，即客户在开始服务之前必须等待的时间；
3) $X(t)=S(t)-t$.

$$W(t)=X(t)-\left(t-\int_{0}^{t} \mathbf{1}_{0}(W(s)) \mathrm{d} s\right) .$$

## 统计代写|随机分析作业代写stochastic analysis代写|Canonical space

$$\theta:\left{F^{\mathbf{N}} \longrightarrow F^{\mathbf{N}}\left(\omega_{n}, n \geq 0\right) \longmapsto\left(\omega_{n+1}, n \geq 0\right)=\left(\omega_{n}, n \geq 1\right)\right.$$

$$\theta\left(\omega_{n}, n \in \mathbf{Z}\right)=\left(\omega_{n+1}, n \in \mathbf{Z}\right)$$

## 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。

## 统计代写|随机分析作业代写stochastic analysis代写|STAT342

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl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。我们的专家在代写随机分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl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

## 统计代写|随机分析作业代写stochastic analysis代写|Traffic, load, Erlang, etc.

In electricity, we count the amps or volts; in meteorology, we measure the pressure; in telecommunications, we count the Erlangs.

The telephone came into existence in 1870. Most of the concepts and notations were derived during this period. Looking at a telephone connection over a time period of length $T$, we define its observed traffic flow as the percentage of time during which the connection is busy
$$\rho=\frac{\sum_{i} t_{i}}{T}$$
A priori, traffic is a dimensionless quantity since it is the ratio of the occupation time to the total time. However, it still has a unit, Erlang, in remembrance of Erlang who, along with Palm, was one of the pioneers of the performance assessment of telephone networks. Therefore, a load of 1 Erlang corresponds to an always busy connection.

Looking at several connections, the traffic carried by this trunk is the sum of the traffic of each connection
$$\rho_{\text {trunk }}=\sum_{\text {connections }} \rho_{\text {connection }}$$
This is no longer a percentage, but we can give a physical interpretation to this quantity according to the ergodic hypothesis. In fact, assume that the number of junctions is large, then we can calculate the average occupation rate in two different ways: either by calculating the percentage of the occupation time of a particular connection over a large period of time; or by computing the percentage of busy connections at a given time.

## 统计代写|随机分析作业代写stochastic analysis代写|Lindley and Beneˇs

We often consider the number of customers present in the system but the quantity that contains the most information is the system load, defined at each moment as the time required for the system to empty itself in the absence of new arrivals. The server works at unit speed: it serves a unit of work per unit time. Consequently, the load decreases with speed 1 between two arrivals. Figure $1.8$ which represents the load over time depending on the arrivals and required service times is easily constructed.

DEFINITION 1.2.- A busy period of a queue is a period that begins with the arrival of a customer in an empty system (server plus buffer) and ends with the end of a service after which the system is empty again.

A cycle is a time period that begins with the arrival of a customer in an empty system and ends on the next arrival of a customer in an empty system. This is the concatenation of a busy period and an idle period, that is the time elapsed between the departure of the last customer of the busy period and the arrival of the next customer.

NOTE.- In Figure 1.8, a busy period begins at $T_{1}$ and ends at $D_{4}$. The corresponding cycle begins at $T_{1}$ and ends at $T_{5}$.

Note that as long as a service policy is conservative, the size of a busy period is independent of it: for waiting rooms of infinite size, the busy periods have, for example, the same length for the FIFO policy as that for the non-preemptive or preemptive resume LIFO policy.

## 统计代写|随机分析作业代写stochastic analysis代写|Traffic, load, Erlang, etc.

$$\rho=\frac{\sum_{i} t_{i}}{T}$$

$$\rho_{\text {trunk }}=\sum_{\text {connections }} \rho_{\text {connection }}$$

## 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。

## 统计代写|随机分析作业代写stochastic analysis代写|MA53200

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl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。我们的专家在代写随机分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl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

## 统计代写|随机分析作业代写stochastic analysis代写|Discrete Distributions

If the elements in $\Omega$ are finite or enumerable, say, $\Omega=\left{\omega_{1}, \omega_{2}, \ldots\right}$, we have a situation of discrete probability space and discrete distribution. In this case, let $X\left(\omega_{j}\right)=x_{j}$ and
$$p_{j}=\mathbb{P}\left(X=x_{j}\right), \quad j=0,1, \ldots$$
Of course, we have to have
$$0 \leq p_{j} \leq 1, \quad \sum_{j} p_{j}=1 .$$
Given a function $f$ of $X$, its expectation is given by
$$\mathbb{E} f(X)=\sum_{j} f\left(x_{j}\right) p_{j}$$
if the sum is well-defined. In particular, the $p$ th moment of the distribution is defined as
$$m_{p}=\sum_{j} x_{j}^{p} p_{j} .$$
When $p=1$, it is called the mean of the random variable and is also denoted by mean $(X)$. Another important quantity is its variance, defined as
$$\operatorname{Var}(X)=m_{2}-m_{1}^{2}=\sum_{j}\left(x_{j}-m_{1}\right)^{2} p_{j}$$
Example 1.7 (Bernoulli distribution). The Bernoulli distribution has the form
$$\mathbb{P}(X=j)= \begin{cases}p, & j=1 \ q, & j=0\end{cases}$$
$p+q=1$ and $p, q \geq 0$. When $p=q=1 / 2$, it corresponds to the toss of a fair coin. The mean and variance can be calculated directly:
$$\mathbb{E} X=p, \quad \operatorname{Var}(X)=p q .$$

## 统计代写|随机分析作业代写stochastic analysis代写|Continuous Distributions

Consider now the general case when $\Omega$ is not necessarily enumerable. Let us begin with the definition of a random variable. Denote by $\mathcal{R}$ the Borel $\sigma$-algebra on $\mathbb{R}$, the smallest $\sigma$-algebra containing all open sets.

Definition 1.10. A random variable $X$ is an $\mathcal{F}$-measurable real-valued function $X: \Omega \rightarrow \mathbb{R}$; i.e., for any $B \in \mathcal{R}, X^{-1}(B) \in \mathcal{F}$.

Definition 1.11. The distribution of the random variable $X$ is a probability measure $\mu$ on $\mathbb{R}$, defined for any set $B \in \mathcal{R}$ by
$$\mu(B)=\mathbb{P}(X \in B)=\mathbb{P} \circ X^{-1}(B) .$$
In particular, we define the distribution function $F(x)=\mathbb{P}(X \leq x)$ when $B=(-\infty, x]$

If there exists an integrable function $\rho(x)$ such that
$$\mu(B)=\int_{B} \rho(x) d x$$
for any $B \in \mathcal{R}$, then $\rho$ is called the probability density function (PDF) of $X$. Here $\rho(x)=d \mu / d m$ is the Radon-Nikodym derivative of $\mu(d x)$ with respect to the Lebesgue measure $m(d x)$ if $\mu(d x)$ is absolutely continuous with respect to $m(d x)$; i.e., for any set $B \in \mathcal{R}$, if $m(B)=0$, then $\mu(B)=0$ (see also Section C of the appendix) [Bil79]. In this case, we write $\mu \ll m$.

## 统计代写|随机分析作业代写stochastic analysis代写|Probability Space

(i) $\Omega \in \mathcal{F}$
(ii) 如果 $A \in \mathcal{F}$ ，然后 $A^{c} \in \mathcal{F}$ ，在哪里 $A^{c}=\Omega \backslash A$ 是的补码 $A$ 在 $\Omega$;
(iii) 如果 $A_{1}, A_{2}, \ldots \in \mathcal{F}$ ，然后 $\bigcup_{n=1}^{\infty} A_{n} \in \mathcal{F}$.

$\Omega$. 我们表示 $\sigma(\mathcal{B})$ 最小的 $\sigma$ – 由集合生成的代数 $\mathcal{B}$ ，即最小的 $\sigma$-代数包含 $\mathcal{B}$. 这对 $(\Omega, \mathcal{F})$ 具有上述性质的空间称为可 测空间。

(a) $\mathbb{P}(\emptyset)=0, \mathbb{P}(\Omega)=1$;
(b) 如果 $A_{1}, A_{2}, \ldots \in \mathcal{F}$ 是成对不相交的，即 $A_{i} \cap A_{j}=\emptyset$ 如果 $i \neq j$ ，然后
$$\mathbb{P}\left(\bigcup_{n=1}^{\infty} A_{n}\right)=\sum_{n=1}^{\infty} \mathbb{P}\left(A_{n}\right)$$
(1.1) 称为可数可加性或 $\sigma$-可加性。

## 统计代写|随机分析作业代写stochastic analysis代写|Conditional Probability

$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}$$

$$\mathbb{P}(A \cap B \cap C)=\mathbb{P}(A \mid B \cap C) \mathbb{P}(B \mid C) \mathbb{P}(C)$$

$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A) \mathbb{P}(B \mid A)}{\mathbb{P}(B)}$$

## 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。

## 统计代写|随机分析作业代写stochastic analysis代写|MATH477

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl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。我们的专家在代写随机分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl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

## 统计代写|随机分析作业代写stochastic analysis代写|Probability Space

It is useful to put these intuitive notions of probability on a firm mathematical basis, as was done by Kolmogorov. For this purpose, we need the notion of probability space, often written as a triplet $(\Omega, \mathcal{F}, \mathbb{P})$, defined as follows.
Definition 1.1 (Sample space). The sample space $\Omega$ is the set of all possible outcomes. Each element $\omega \in \Omega$ is called a sample point.

Definition $1.2$ ( $\sigma$-algebra). A $\sigma$-algebra (or $\sigma$-field) $\mathcal{F}$ is a collection of subsets of $\Omega$ that satisfies the following conditions:
(i) $\Omega \in \mathcal{F}$
(ii) if $A \in \mathcal{F}$, then $A^{c} \in \mathcal{F}$, where $A^{c}=\Omega \backslash A$ is the complement of $A$ in $\Omega$;
(iii) if $A_{1}, A_{2}, \ldots \in \mathcal{F}$, then $\bigcup_{n=1}^{\infty} A_{n} \in \mathcal{F}$.
Each set $A$ in $\mathcal{F}$ is called an event. Let $\mathcal{B}$ be a collection of subsets of
$\Omega$. We denote by $\sigma(\mathcal{B})$ the smallest $\sigma$-algebra generated by the sets in $\mathcal{B}$, i.e., the smallest $\sigma$-algebra that contains $\mathcal{B}$. The pair $(\Omega, \mathcal{F})$ with the above properties is called a measurable space.

Definition $1.3$ (Probability measure). The probability measure $\mathbb{P}: \mathcal{F} \rightarrow$ $[0,1]$ is a set function defined on $\mathcal{F}$ which satisfies
(a) $\mathbb{P}(\emptyset)=0, \mathbb{P}(\Omega)=1$;
(b) if $A_{1}, A_{2}, \ldots \in \mathcal{F}$ are pairwise disjoint, i.e., $A_{i} \cap A_{j}=\emptyset$ if $i \neq j$, then
$$\mathbb{P}\left(\bigcup_{n=1}^{\infty} A_{n}\right)=\sum_{n=1}^{\infty} \mathbb{P}\left(A_{n}\right)$$
(1.1) is called countable additivity or $\sigma$-additivity.

## 统计代写|随机分析作业代写stochastic analysis代写|Conditional Probability

Let $A, B \in \mathcal{F}$ and assume that $\mathbb{P}(B) \neq 0$. Then the conditional probability of $A$ given $B$ is defined as
$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}$$
This is the proportion of events that both $A$ and $B$ occur given that $B$ occurs. For instance, the probability to obtain two tails in two tosses of a fair coin is $1 / 4$, but the conditional probability to obtain two tails is $1 / 2$ given that the first toss is a tail, and it is zero given that the first toss is a head.
Since $\mathbb{P}(A \cap B)=\mathbb{P}(A \mid B) \mathbb{P}(B)$ by definition, we also have
$$\mathbb{P}(A \cap B \cap C)=\mathbb{P}(A \mid B \cap C) \mathbb{P}(B \mid C) \mathbb{P}(C),$$
and so on. It is straightforward to obtain
$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A) \mathbb{P}(B \mid A)}{\mathbb{P}(B)}$$
from the definition of conditional probability. This is called Bayes’s rule.

## 统计代写|随机分析作业代写stochastic analysis代写|Probability Space

(i) $\Omega \in \mathcal{F}$
(ii) 如果 $A \in \mathcal{F}$ ，然后 $A^{c} \in \mathcal{F}$ ，在哪里 $A^{c}=\Omega \backslash A$ 是的补码 $A$ 在 $\Omega$;
(iii) 如果 $A_{1}, A_{2}, \ldots \in \mathcal{F}$ ，然后 $\bigcup_{n=1}^{\infty} A_{n} \in \mathcal{F}$.

$\Omega$. 我们表示 $\sigma(\mathcal{B})$ 最小的 $\sigma$ – 由集合生成的代数 $\mathcal{B}$ ，即最小的 $\sigma$-代数包含 $\mathcal{B}$. 这对 $(\Omega, \mathcal{F})$ 具有上述性质的空间称为可 测空间。

(a) $\mathbb{P}(\emptyset)=0, \mathbb{P}(\Omega)=1$;
(b) 如果 $A_{1}, A_{2}, \ldots \in \mathcal{F}$ 是成对不相交的，即 $A_{i} \cap A_{j}=\emptyset$ 如果 $i \neq j$ ，然后
$$\mathbb{P}\left(\bigcup_{n=1}^{\infty} A_{n}\right)=\sum_{n=1}^{\infty} \mathbb{P}\left(A_{n}\right)$$
(1.1) 称为可数可加性或 $\sigma$-可加性。

## 统计代写|随机分析作业代写stochastic analysis代写|Conditional Probability

$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}$$

$$\mathbb{P}(A \cap B \cap C)=\mathbb{P}(A \mid B \cap C) \mathbb{P}(B \mid C) \mathbb{P}(C)$$

$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A) \mathbb{P}(B \mid A)}{\mathbb{P}(B)}$$

## 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。

## 统计代写|随机分析作业代写stochastic analysis代写|STAT342

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl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。我们的专家在代写随机分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl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

## 统计代写|随机分析作业代写stochastic analysis代写|Elementary Examples

We will start with some elementary examples of probability. The most wellknown example is that of a fair coin: if flipped, the probability of getting a head or tail both equal to $1 / 2$. If we perform $n$ independent tosses, then the probability of obtaining $n$ heads is equal to $1 / 2^{n}$ : among the $2^{n}$ equally possible outcomes only one gives the result that we look for. More generally, let $S_{n}=X_{1}+X_{2}+\cdots+X_{n}$, where
$$X_{j}= \begin{cases}1, & \text { if the result of the } n \text {th trial is a head, } \ 0, & \text { if the result of the } n \text {th trial is a tail. }\end{cases}$$
Then the probability that we get $k$ heads out of $n$ tosses is equal to
$$\operatorname{Prob}\left(S_{n}=k\right)=\frac{1}{2^{n}}\left(\begin{array}{l} n \ k \end{array}\right)$$
Applying Stirling’s formula
$$n ! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^{n}, \quad n \rightarrow \infty$$
we can calculate, for example, the asymptotic probability of obtaining heads exactly half of the time:
$$\operatorname{Prob}\left(S_{2 n}=n\right)=\frac{1}{2^{2 n}}\left(\begin{array}{c} 2 n \ n \end{array}\right)=\frac{1}{2^{2 n}} \frac{(2 n) !}{(n !)^{2}} \sim \frac{1}{\sqrt{\pi n}} \rightarrow 0$$
as $n \rightarrow \infty$

## 统计代写|随机分析作业代写stochastic analysis代写| Random Variables

On the other hand, since we have a fair coin, we do expect to obtain heads roughly half of the time; i.e.,
$$\frac{S_{2 n}}{2 n} \approx \frac{1}{2},$$
for large $n$. Such a statement is indeed true and is embodied in the law of large numbers that we will discuss in the next chapter. For the moment let us simply observe that while the probability that $S_{2 n}$ equals $n$ goes to zero as $n \rightarrow \infty$, the probability that $S_{2 n}$ is close to $n$ goes to 1 as $n \rightarrow \infty$. More precisely, for any $\epsilon>0$,
$$\operatorname{Prob}\left(\left|\frac{S_{2 n}}{2 n}-\frac{1}{2}\right|>\epsilon\right) \rightarrow 0,$$
as $n \rightarrow \infty$. This can be seen as follows. Noting that the distribution $\operatorname{Prob}\left{S_{2 n}=k\right}$ is unimodal and symmetric around the state $k=n$, we have
$\operatorname{Prob}\left(\left|\frac{S_{2 n}}{2 n}-\frac{1}{2}\right|>\epsilon\right) \leq 2 \cdot \frac{1}{2^{2 n}} \sum_{k>n+2 n \epsilon} \frac{(2 n) !}{k !(2 n-k) !}$
$\leq 2(n-2 n \epsilon) \cdot \frac{1}{2^{2 n}} \frac{(2 n) !}{\lceil n+2 n \epsilon\rceil !\lfloor n-2 n \epsilon\rfloor !}$
$\sim \frac{2 \sqrt{1-2 \epsilon}}{\sqrt{\pi(1+2 \epsilon)}} \cdot \frac{\sqrt{n}}{(1-2 \epsilon)^{n(1-2 \epsilon)}(1+2 \epsilon)^{n(1+2 \epsilon)}} \rightarrow 0$
for sufficiently small $\epsilon$ and $n \gg 1$, where $\lceil\cdot\rceil$ and $\lfloor\cdot\rfloor$ are the ceil and floor functions, respectively, defined by $\lceil x\rceil=m+1$ and $\lfloor x\rfloor=m$ if $x \in[m, m+1)$ for $m \in \mathbb{Z}$. This is the weak law of large numbers for this particular example.
In the example of a fair coin, the number of outcomes in an experiment is finite. In contrast, the second class of examples involves a continuous set of possible outcomes. Consider the orientation of a unit vector $\boldsymbol{\tau}$. Denote by $\mathbb{S}^{2}$ the unit sphere in $\mathbb{R}^{3}$. Define $\rho(\boldsymbol{n}), \boldsymbol{n} \in \mathbb{S}^{2}$, as the orientation distribution density; i.e., for $A \subset \mathbb{S}^{2}$,
$$\operatorname{Prob}(\boldsymbol{\tau} \in A)=\int_{A} \rho(\boldsymbol{n}) d S,$$
where $d S$ is the surface area element on $\mathbb{S}^{2}$. If $\boldsymbol{\tau}$ does not have a preferred orientation, i.e., it has equal probability of pointing at any direction, then
$$\rho(\boldsymbol{n})=\frac{1}{4 \pi} .$$
In this case, we say that $\tau$ is isotropic. On the other hand, if $\boldsymbol{\tau}$ does have a preferred orientation, say $\boldsymbol{n}{0}$, then we expect $\rho(\boldsymbol{n})$ to be peaked at $\boldsymbol{n}{0}$.

## 统计代写|随机分析作业代写stochastic analysis代写|Elementary Examples

$X_{j}={1, \quad$ if the result of the $n$th trial is a head, $0, \quad$ if the result of the $n$th trial is a tail.

$$\operatorname{Prob}\left(S_{n}=k\right)=\frac{1}{2^{n}}(n k)$$

$$n ! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^{n}, \quad n \rightarrow \infty$$

$$\operatorname{Prob}\left(S_{2 n}=n\right)=\frac{1}{2^{2 n}}(2 n n)=\frac{1}{2^{2 n}} \frac{(2 n) !}{(n !)^{2}} \sim \frac{1}{\sqrt{\pi n}} \rightarrow 0$$

## 统计代写|随机分析作业代写stochastic analysis代写| Random Variables

$$\frac{S_{2 n}}{2 n} \approx \frac{1}{2},$$

$$\operatorname{Prob}\left(\left|\frac{S_{2 n}}{2 n}-\frac{1}{2}\right|>\epsilon\right) \rightarrow 0,$$

\begin{aligned} &\operatorname{Prob}\left(\left|\frac{S_{2 n}}{2 n}-\frac{1}{2}\right|>\epsilon\right) \leq 2 \cdot \frac{1}{2^{2 n}} \sum_{k>n+2 n \epsilon} \frac{(2 n) !}{k !(2 n-k) !} \ &\leq 2(n-2 n \epsilon) \cdot \frac{1}{2^{2 n}} \frac{(2 n) !}{[n+2 n \epsilon] ![n-2 n \epsilon] !} \ &\sim \frac{2 \sqrt{1-2 \epsilon}}{\sqrt{\pi(1+2 \epsilon)}} \cdot \frac{\sqrt{n}}{(1-2 \epsilon)^{n(1-2 \epsilon)}(1+2 \epsilon \epsilon)^{n(1+2 \epsilon)}} \rightarrow 0 \ &\text { 对于足够小的 } 6 \text { 和 } n \gg 1 \text { ，在哪里 }\lceil\cdot\rceil \text { 和 }[\cdot \text { 分别是 ceil 和 floor 函数，由下式定义 }\lceil x\rceil=m+1 \text { 和 }[x\rfloor=m \text { 如果 } \end{aligned} $x \in[m, m+1)$ 为了 $m \in \mathbb{Z}$. 这是这个特定示例的弱大数定律。

$$\operatorname{Prob}(\boldsymbol{\tau} \in A)=\int_{A} \rho(\boldsymbol{n}) d S,$$

$$\rho(\boldsymbol{n})=\frac{1}{4 \pi} .$$

## 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。

## 统计代写|应用随机过程代写Stochastic process代考|MA53200

statistics-lab™ 为您的留学生涯保驾护航 在代写应用随机过程Stochastic process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。我们的专家在代写应用随机过程Stochastic process代写方面经验极为丰富，各种代写应用随机过程Stochastic process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

## 统计代写|应用随机过程代写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|应用随机过程代写Stochastic process代考|A Few More Examples

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) Súccéss 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
$$P_{i j}=P\left(X_{n}=j \mid X_{n-1}=i\right)=\left{\begin{array}{l} p \text { if } j=i+1 \ q \text { if } j=0 \ 0 \text { otherwise. } \end{array}\right.$$

## 统计代写|应用随机过程代写Stochastic process代考|Definition and Transition Probabilities

$P\left[X_{n+1}=j \mid X_{0}=i_{0}, X_{1}=i_{1}, \ldots, X_{n}=i_{n}\right]=P\left[X_{n+1}=j \mid X_{n}=i_{n}\right]$ ，马尔可夫性质。然后是随 机过程 lleft{X_{{n}, n Igeq OIright $}$ 称为马尔可夫链 $(\mathrm{MC})$ 。我们将假设 $\mathrm{MC}$ 是静止的，即 $P\left[X_{n+1}=j \mid X_{n}=\right.$ $i]=p_{i j}$ 独立于 $n$ 对所有人 $i, j \in, S$. 让 $P=\left(P_{i j}\right) ; i, j \in S$ 是具有元素的有限或可数无限维矩阵 $p_{i j}$.

$$p_{j k}=0 \text { if } k \neq j-1 \text { or } j+1$$

$$S=\ldots,-3,-2,-1,0,1,2,3, \ldots .$$

$$P\left[X_{0}=i_{0}, X_{1}=i_{i}, \ldots, X_{n}=i_{n}\right] \quad=P\left[X_{n}=i_{n} \mid X_{n-1}=i_{n-1}, X_{n-2}=i_{n-2}, \ldots, X_{1}=i_{1} \ldots\right.$$

## 统计代写|应用随机过程代写Stochastic process代考|A Few More Examples

(a) 独立试验
$P^{n}=P$ 对所有人 $n \geq 1$ ， 在哪里 $p_{i j}=p_{j}$ 即所有行都是相同的。
(b) 成功运行

$\$ \$$p_{-}{i j} \wedge{(n)}=\backslash left { q p^{j} for j=0,1,2, \ldots, i+n-1 p^{j} for j=j+n 0 otherwise \正确的。 This followseitherdirectlyor fromChapman - Kolmogorov’ sequation. Itcanbeshownthat \ P^{n} \ c P_{-}{i j}=P \backslash l e f t\left(X_{-}{n}=j \backslash m i d X_{-}{n-1}=i \backslash\right. right )=V left { p if j=i+1 q if j=00 otherwise. 【正确的。 \ \$$

## 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。