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# 统计代写|统计推断代考Statistical Inference代写|Joint sample moments

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## 统计代写|统计推断代考Statistical Inference代写|Joint sample moments

Joint sample moments provide information about the dependence structure between two variables. We begin with some definitions.
Definition 7.2.7 (Joint sample moments and joint central sample moments)
Suppose we have a random sample of pairs $\left{\left(X_1, Y_1\right), \ldots,\left(X_n, Y_n\right)\right}$. The $(r, s)^{\text {th }}$ joint sample moment is
$$m_{r, s}^{\prime}=\frac{1}{n} \sum_{i=1}^n X_i^r Y_i^s$$
and the $(r, s)^{\text {th }}$ joint central sample moment is
$$m_{r, s}=\frac{1}{n} \sum_{i=1}^n\left(X_i-\bar{X}\right)^r\left(Y_i-\bar{Y}\right)^s \text {. }$$
Definition 7.2.8 (Sample covariance)
Given a random sample of pairs, $\left{\left(X_1, Y_1\right), \ldots,\left(X_n, Y_n\right)\right}$, the sample covariance is defined as
$$c_{X, Y}=\frac{1}{n-1} \sum_{i=1}^n\left(X_i-\bar{X}\right)\left(Y_i-\bar{Y}\right)=\frac{n}{n-1} m_{1,1},$$
or, alternatively,
$$c_{X, Y}=\frac{1}{n-1}\left(\sum_{i=1}^n X_i Y_i-n \bar{X} \bar{Y}\right)$$

## 统计代写|统计推断代考Statistical Inference代写|Sample mean and variance for a normal population

Suppose that the population distribution is normal. We can find the distribution of both the sample mean and sample variance in terms of well-known parametric forms. Perhaps more surprisingly, we can show that the sample mean and sample variance are independent.

Lemma 7.3.1 (Independence of sample mean and variance for normal population) Suppose that $Y_1, \ldots, Y_n$ is a random sample from a normal population, that is $Y \sim$ $\mathrm{N}\left(\mu, \sigma^2\right)$. Then
i. the sample mean, $\bar{Y}$, and terms of the form $\left(Y_j-\bar{Y}\right)$ are independent, $\bar{Y} \Perp\left(Y_j-\bar{Y}\right)$ for $j=1, \ldots, n$,
ii. the sample mean and the sample variance are independent, $\bar{Y} \Perp S^2$.
Proof.
For a normal distribution, $\mu_3=0$. Jointly normal random variables that are uncorrelated are independent. Using these facts, the result is an immediate corollary of Proposition 7.2.6.

In constructing the sample variance, we consider sums of the squares of a sequence of random variables. If the variables are standard normal, the sum of the squares has a chi-squared distribution chi is pronounced like “eye” preceded by a ” $\mathrm{k}$ “.

Proof.
For a normal distribution, $\mu_3=0$. Jointly normal random variables that are uncorrelated are independent. Using these facts, the result is an immediate corollary of Proposition 7.2.6.

In constructing the sample variance, we consider sums of the squares of a sequence of random variables. If the variables are standard normal, the sum of the squares has a chi-squared distribution (chi is pronounced like “eye” preceded by a ” $k$ “).

# 统计推断代写

## 统计代写|统计推断代考Statistical Inference代写|Joint sample moments

$$m_{r, s}^{\prime}=\frac{1}{n} \sum_{i=1}^n X_i^r Y_i^s$$

$$m_{r, s}=\frac{1}{n} \sum_{i=1}^n\left(X_i-\bar{X}\right)^r\left(Y_i-\bar{Y}\right)^s \text {. }$$

$$c_{X, Y}=\frac{1}{n-1} \sum_{i=1}^n\left(X_i-\bar{X}\right)\left(Y_i-\bar{Y}\right)=\frac{n}{n-1} m_{1,1},$$

$$c_{X, Y}=\frac{1}{n-1}\left(\sum_{i=1}^n X_i Y_i-n \bar{X} \bar{Y}\right)$$

## 统计代写|统计推断代考Statistical Inference代写|Sample mean and variance for a normal population

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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。