### 统计代写|统计推断代写Statistical inference代考|STAT3923

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference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。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference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|统计推断代写Statistical inference代考|Inequalities involving expectation

In proving convergence results it is often useful to be able to provide bounds for probabilities and expectations. The propositions below provide bounds that are often rather loose. The appeal of these results is their generality.
Proposition 3.4.12 (Markov inequality)
If $Y$ is a positive random variable with $\mathbb{E}(Y)<\infty$, then $\mathrm{P}(Y \geq a) \leq \mathbb{E}(Y)$ / a for any constant $a>0$.
Proof.
We prove this for the continuous case. A similar argument holds in the discrete case.
\begin{aligned} \mathrm{P}(Y \geq a) &=\int_{a}^{\infty} f_{Y}(y) d y & & \text { by definition } \ & \leq \int_{a}^{\infty} \frac{y}{a} f_{Y}(y) d y & & \text { since } 1 \leq \frac{y}{a} \text { for } y \in[a, \infty) \ & \leq \frac{1}{a} \int_{0}^{\infty} y f_{Y}(y) d y & & \text { for positive } g, \int_{a}^{\infty} g(y) d y \leq \int_{0}^{\infty} g(y) d y \ & \leq \frac{1}{a} \mathbb{E}(Y) & & \text { by definition of } \mathbb{E}(Y) . \end{aligned}
The Markov inequality provides us with a bound on the amount of probability in the upper tail of the distribution of a positive random variable. As advertised, this bound is fairly loose. Consider the following illustration. Let us suppose that $Y$ is the length of life of a British man. Life expectancy in Britain is not great, in fact, male life expectancy is around 79 years (it’s all the lager and pies). If we take $\mathrm{B}(Y)=79$, then we can calculate a bound on the probability that a British man lives to be over $158 .$ Using the Markov inequality,
$$\mathrm{P}(Y \geq 158) \leq \mathrm{E}(Y) / 158=79 / 158=1 / 2 \text {. }$$
Clearly, this is a loose bound. We would expect this probability to be pretty close to zero. The beauty of the Markov inequality lies not in tightness of the bounds but generality of application; no distributional assumptions are required.

The Markov inequality can be extended to random variables that are not necessarily positive.

## 统计代写|统计推断代写Statistical inference代考|Moments

We have discussed measures of central tendency and measures of spread. As the names suggest, central tendency gives an indication of the location of the centre of a distribution, and spread measures how widely probability is dispersed. Other characteristics of a distribution that might be of interest include symmetry and the extent to which we find probability in the tails (fatness of tails). We can express commonly used measures of central tendency, spread, symmetry, and tail fatness in terms of moments and central moments.
Definition 3.4.18 (Moments)
For a random variable $X$ and positive integer $r$, the $r^{\text {th }}$ moment of $X$ is denoted $\mu_{r}^{\prime}$, where
$$\mu_{r}^{\prime}=\mathbb{B}\left(X^{r}\right),$$
whenever this is well defined.
Moments depend on the horizontal location of the distribution. When we are measuring a characteristic like spread, we would like to use a quantity that remains unchanged when the distribution is moved left or right along the horizontal axis. This motivates the definition of central moments, in which we perform a translation to account for the value of the mean.
Definition 3.4.19 (Central moments)
For a random variable $X$ and positive integer $r$, the $r^{\text {th }}$ central moment of $X$ is denoted $\mu_{r}$, where
$$\mu_{r}=\mathbb{E}\left[(X-\mathbb{B}(X))^{r}\right]$$
whenever this is well defined.

## 统计代写|统计推断代写Statistical inference代考|Moment-generating functions

For many distributions, all the moments $\mathrm{B}(X), \mathrm{B}\left(X^{2}\right), \ldots$ can be encapsulated in a single function. This function is referred to as the moment-generating function, and it exists for many commonly used distributions. It often provides the most efficient method for calculating moments. Moment-generating functions are also useful in establishing distributional results, such as the properties of sums of random variables, and in proving asymptotic results.

Definition 3.5.1 (Moment-generating function)
The moment-generating function of a random variable $X$ is a function $M_{X}: \mathbb{R} \longrightarrow$ $[0, \infty)$ given by
$$M_{X}(t)=\mathbb{E}\left(e^{t X}\right)= \begin{cases}\sum_{x} e^{l x} f_{X}(x) & \text { if } X \text { discrete } \ \int_{-\infty}^{\infty} e^{t x} f_{X}(x) d x & \text { if } X \text { continuous. }\end{cases}$$
where, for the function to be well defined, we require that $M_{X}(t)<\infty$ for all $t \in[-h, h]$ for some $h>0$.
A few things to note about moment-generating functions.

1. Problems involving moment-generating functions almost always use the definition in terms of expectation as a starting point.
2. The moment-generating function $M_{X}(t)=\mathrm{B}\left(e^{t X}\right)$ is a function of $t$. The $t$ is just a label, so $M_{X}(s)=\mathbb{E}\left(e^{s X}\right), M_{X}(\theta)=\mathbb{E}\left(e^{\theta X}\right), M_{Y}(p)=\mathbb{B}\left(e^{p Y}\right)$, and so on.
3. We need the moment-generating function to be defined in an interval around the origin. Later on we will be taking derivatives of the moment-generating function at zero, $M_{X}^{\prime}(0), M_{X}^{\prime \prime}(0)$, and so on.

The moment-generating function of $X$ is the expected value of an exponential function of $X$. Useful properties of moment-generating functions are inherited from the exponential function, $e^{x}$. The Taylor series expansion around zero, provides an expression for $e^{x}$ as a polynomial in $x$,
$$e^{x}=1+x+\frac{1}{2 !} x^{2}+\ldots+\frac{1}{r !} x^{r}+\ldots=\sum_{j=0}^{\infty} \frac{1}{j !} x^{j}$$

μr′=乙(Xr),

μr=和[(X−乙(X))r]

## 统计代写|统计推断代写Statistical inference代考|Moment-generating functions

1. 涉及矩生成函数的问题几乎总是使用期望方面的定义作为起点。
2. 力矩生成函数米X(吨)=乙(和吨X)是一个函数吨. 这吨只是一个标签，所以米X(s)=和(和sX),米X(θ)=和(和θX),米是(p)=乙(和p是)， 等等。
3. 我们需要在原点周围的区间内定义矩生成函数。稍后我们将在零处取矩生成函数的导数，米X′(0),米X′′(0)， 等等。

## 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。