Posted on Categories:Statistical inference, 统计代写, 统计代考, 统计推断

# 统计代写|统计推断代考Statistical Inference代写|STAT360 Expectation, variance, and higher moments

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## 统计代写|统计推断代考Statistical Inference代写|Mean of a random variable

Central tendency is among the first concepts taught on any course in descriptive statistics. The hope is that calculating central tendency will provide us with some sense of the usual or average values taken by an observed variable. Among sample statistics commonly considered are the mode (most commonly occurring value), the median (middle value when observations are ordered) and the arithmetic mean. If we have a massless ruler with points of equal mass placed at locations corresponding to the observed values, the arithmetic mean is the point where we should place a fulcrum in order for the ruler to balance. We will follow the usual convention and refer to the arithmetic mean as just the mean.

These ideas transfer neatly to describing features of distributions. The measures of central tendency that are applied to describe data can also be applied to our models. For example, suppose that $X$ is a continuous random variable with density $f_X$ and cumulative distribution function $F_X$. We define $\operatorname{mode}(X)=\arg \max _x f_X(x)$ and median $(X)=m$, where $m$ is the value satisfying $F_X(m)=0.5$. We will now focus our attention on the mean.

Definition 3.4.1 (Mean)
The mean of a random variable $X$, denoted $\mathbb{E}(X)$, is given by
$$\mathbb{E}(X)= \begin{cases}\sum_x x f_X(x) & \text { if } X \text { discrete } \ \int_{-\infty}^{\infty} x f_X(x) d x & \text { if } X \text { continuous, }\end{cases}$$
where, to guarantee that $\mathbb{E}(X)$ is well defined, we usually insist that $\sum_x|x| f_X(x)<\infty$ in the discrete case and $\int_{-\infty}^{\infty}|x| f_X(x) d x<\infty$ in the continuous case.

## 统计代写|统计推断代考Statistical Inference代写|Expectation operator

The process of finding the mean is so commonly used that we adopt a special notation for it. The expectation operator, $\mathbb{E}$, when applied to a random variable yields its mean; if $\mu$ is the mean of $X$ then $\mathbb{E}(X)=\mu$. We will often want to evaluate the mean of a function of a random variable, $g(X)$. We could work out the mass or density of $g(X)$ then use Definition 3.4.1 to evaluate $\mathbb{E}(g(X))$. However, this is not always a straightforward process. The following theorem provides a simple mechanism for calculating the expectation of a random variable without requiring us to derive its mass/density. We will discuss methods to find the mass/density of a function of a random variable in section 3.6.
Theorem 3.4.4 (Expectation of a function of a random variable) For any well-behaved function $g: \mathbb{R} \longrightarrow \mathbb{R}$, the expectation of $g(X)$ is defined as
$$\mathbb{E}[g(X)]= \begin{cases}\sum_x g(x) f_X(x) & \text { if } X \text { discrete, } \ \int_{-\infty}^{\infty} g(x) f_X(x) d x & \text { if } X \text { continuous, }\end{cases}$$
where, to guarantee that expectation is well defined, we usually insist that $\sum_x|g(x)| f_X(x)<\infty$ in the discrete case and $\int_{-\infty}^{\infty}|g(x)| f_X(x) d x<\infty$ in the continuous case.

In the discrete case this is sometimes referred to (pejoratively by mathematicians) as the law of the unconscious statistician. We will prove the continuous case in section 3.6.
One of the key properties of expectation is its linearity.

# 统计推断代写

## 统计代写|统计推断代考统计推断代写|随机变量的平均值

3.4.1(均值)

$$\mathbb{E}(X)= \begin{cases}\sum_x x f_X(x) & \text { if } X \text { discrete } \ \int_{-\infty}^{\infty} x f_X(x) d x & \text { if } X \text { continuous, }\end{cases}$$

## 统计代写|统计推断代考统计推断代写|期望运算符

$$\mathbb{E}[g(X)]= \begin{cases}\sum_x g(x) f_X(x) & \text { if } X \text { discrete, } \ \int_{-\infty}^{\infty} g(x) f_X(x) d x & \text { if } X \text { continuous, }\end{cases}$$

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## 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。