## 统计代写|概率与统计作业代写Probability and Statistics代考|STA312

2022年9月29日

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## 统计代写|概率与统计作业代写Probability and Statistics代考|Simple Random Sampling

Simple random sampling is a way of collecting samples such that each unit from the population has the exact same probability of becoming part of the sample. Simple random sampling is a conceptually easy method of forming random samples but it can prove hard in practice. Because of its importance in statistical theory we discuss it in more detail. ${ }^1$

To illustrate simple random sampling, suppose that the entire population consists of six units $(N=6)$ only, numbered from 1 to 6 . It is decided to collect a sample of three units $(n=3)$ from this population. For a simple random sample each of 20 combinations of three units could possibly form the sample $S$, i.e., the possible samples are:

$S_1=(1,2,3) \quad S_2=(1,2,4) \quad S_3=(1,2,5) \quad S_4=(1,2,6)$
$S_5=(1,3,4) \quad S_6=(1,3,5) \quad S_7=(1,3,6)$
$S_8=(1,4,5) \quad S_9=(1,4,6)$
$S_{10}=(1,5,6)$
$S_{11}-(2,3,4) S_{12}-(2,3,5) S_{13}-(2,3,6)$
$S_{14}=(2,4,5) S_{15}=(2,4,6)$
$S_{16}=(2,5,6)$
$S_{17}=(3,4,5) S_{18}=(3,4,6)$
$S_{19}=(3,5,6)$
$S_{20}=(4,5,6)$
The simple random sample can now be collected by generating one number $k$
between 1 and 20 (using our $K=20$-sided die) and then selecting $S_k$ when $k$ appears
on top of the die. Note that each sample has the same probability $(1 / 20)$ of being
collected and that each unit has the same probability (1/2) of being collected..$^{13}$
This is a general property of simple random sampling: each unique sample has
the same probability of being selected, and, as a result, each unit has the same
probability of being selected (the numbers depend on the population and sample size).
Hence, simple random sampling guarantees that each unit has the same probability
of becoming part of the sample.
In $\mathrm{R}$, this can be conducted by applying the function sample. ${ }^{14}$ The function
sample has (at least) three arguments: the data on the variable of interest (here we
choose $x$ ), the number of samples drawn from the data (here we choose 1 ), and the
indicator that tells us whether sampling is done with replacement (here we choose
FALSE) $^{15}$ :
$>x<-c(1: 20)$
set. seed $(575757)$
sample $(x, 1$, FALSE $)$
[1] 15

## 统计代写|概率与统计作业代写Probability and Statistics代考|Systematic Sampling

To obtain a sample of size $n$ using systematic sampling, a few steps are required. First the population should be divided into $n$ groups and the order of the units (if some order exists) should he maintained (or otherwise fix the order). Now suppose that each group consists of $m$ units (thus the population size is $N=n m$ ) ordered from 1 to $m$ in each group. From the first group one unit is randomly collected with probability $1 / m$. Say the $p$ th unit was the result. Then from each of the $n$ groups the $p$ th unit is collected too, forming the sample of $n$ units. Note that systematic sampling provides only $m$ possible sets of samples, i.e., $S_1, S_2, \ldots, S_m$.

Consider the population of six units again where we wish to collect a sample of three units. Splitting the population in to three groups for example provides the subgroups $(1,2) ;(3,4)$; and $(5,6)$. From the first group, which consists of only two units, one unit should be randomly collected with probability $1 / m=0.5$. Thus the sample can now only consist of $S_6=(1,3,5)$ or $S_{15}=(2,4,6)$. Note that we have used the notation or index of the set of possible samples from simple random sampling. The possible samples from systematic sampling are quite different from the set of samples that can be obtained with simple random sampling. However, similar to simple random sampling, each unit in the population still has the same probability of being collected. The probability that a unit enters the sample is $p=1 / m$, which is the same as the probability of selecting one of the $m$ possible sample sets.

The most important advantage of systematic sampling over simple random sampling is the ease with which the sample may be collected. Systematic sampling is often used in manufacturing in relation to a time period, for instance taking a unit every half hour. With a constant production speed a systematic sample is created if the first time point within the first half hour is taken randomly. This is clearly much easier than collecting a simple random sample at the end of production. It probably leads to fewer mistakes or to improper “short cuts” in sampling that would lead to a haphazard or convenience sample. Systematic sampling can also lead to more precise descriptive statistics than simple random sampling (see Cochran 2007). A clear disadvantage of systematic sampling is that the “period” for systematic sampling may coincide with particular patterns in the process or population.

# 概率与统计作业代考

## 统计代写|概率与统计作业代写概率统计代考|简单随机抽样

.

$S_1=(1,2,3) \quad S_2=(1,2,4) \quad S_3=(1,2,5) \quad S_4=(1,2,6)$
$S_5=(1,3,4) \quad S_6=(1,3,5) \quad S_7=(1,3,6)$
$S_8=(1,4,5) \quad S_9=(1,4,6)$
$S_{10}=(1,5,6)$
$S_{11}-(2,3,4) S_{12}-(2,3,5) S_{13}-(2,3,6)$
$S_{14}=(2,4,5) S_{15}=(2,4,6)$
$S_{16}=(2,5,6)$
$S_{17}=(3,4,5) S_{18}=(3,4,6)$
$S_{19}=(3,5,6)$
$S_{20}=(4,5,6)$现在可以通过生成一个数字来收集简单的随机样本 $k$

。注意每个样本都有相同的概率 $(1 / 20)$ 收集
，每个单位有相同的概率(1/2)被收集..$^{13}$这是简单随机抽样的一般性质:每个唯一样本有

FALSE) $^{15}$ :
$>x<-c(1: 20)$

[1] 15

.系统抽样

## 有限元方法代写

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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。