### 统计代写|Generalized linear model代考广义线性模型代写|Probability and the Central Limit Theorem

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Generalized linear model代考广义线性模型代写|Basic Probability

Statistics is based entirely on a branch of mathematics called probability, which is concerned with the likelihood of outcomes for an event. In probability theory, mathematicians use information about

all possible outcomes for an event in order to determine the likelihood of any given outcome. The first section of this chapter will explain the basics of probability before advancing to the foundational issues of the rest of the textbook.

Imagine an event repeated 100 times. Each of these repetitions – called a trial – would be recorded. Mathematically, the formula for probability (abbreviated $p$ ) is:
$$p=\frac{\text { Number of trials with the same outcome }}{\text { Total number of trials }}$$
(Formula 6.1)
To calculate the probability of any given outcome, it is necessary to count the number of trials that resulted in that particular outcome and divide it by the total number of trials. Thus, if the 100 trials consisted of coin flips – which have two outcomes, heads and tails – and 50 trials resulted in “heads” and 50 trials resulted in “tails,” the probability of the coin landing heads side up would then be $\frac{50}{100}$ ” To simplify this notation, it is standard to reduce the fraction to its simplest form or to convert it to a decimal: $1 / 2$ or $.50$ in this example.

This is called the empirical probability because it is based on empirical data collected from actual trials. Some readers will notice that this is the same method and formula for calculating the relative frequency in a frequency table (see Chapter 3). Therefore, empirical probabilities have the same mathematical properties as relative frequencies. That is, probability values always range from 0 to 1 . A probability of zero indicates that the outcome never occurred, and a probability value of 1 indicates that the outcome occurred for every trial (and that no other outcome occurred). Also, all probabilities for a trial must add up to 1 – just as all relative frequencies in a dataset must add up to 1 .

Interpreting probability statistics only requires an understanding of percentages, fractions, and decimals. In the coin-flipping example, a probability of $.50$ indicates that we can expect that half of those trials would result in an outcome of “heads.” Similarly, the chances that any particular trial will result in an outcome of “heads” is $50 \%$. Mathematically, it really doesn’t matter whether probability values are expressed as percentages (e.g., $50 \%$ ), fractions (e.g., 1/2), or decimals (e.g., .50). Statisticians, though, prefer to express probabilities as decimals, and this textbook will stick to that convention.

## 统计代写|Generalized linear model代考广义线性模型代写|The Logic of Inferential Statistics and the CLT

The beginning of this book, especially Chapter 4, discussed descriptive statistics, which is the branch of statistics concerned with describing data that have been collected. Descriptive statistics are indispensable for understanding data, but they are often of limited usefulness because most social scientists wish to make conclusions about the entire population of interest – not just the subjects in the sample that provided data to the researcher. For example, in a study of bipolar mood disorders, Kupka et al. (2007) collected data from 507 patients to examine how frequently they were in manic, hypomanic, and depressed mood states. The descriptive statistics that Kupka et al. provided are interesting – but they are of limited use if they only apply to the 507 people in the study. The authors of this study – and the vast majority of researchers – want to apply their conclusions to the entire population of people they are studying, even people who are not in the sample. The process of drawing conclusions about the entire population based on data from a sample is called generalization, and it requires inferential statistics in order to be possible. Inferential statistics is the branch of statistics that builds on the foundation of probability in order to make generalizations.
The logic of inferential statistics is diagrammed in Figure 6.6. It starts with a population, which, for a continuous, interval- or ratio-level variable, has an unknown mean and unknown standard deviation (represented as the circle in the top left of Figure 6.6). A researcher then draws a random sample from the population and uses the techniques discussed in previous chapters to calculate a sample mean and create a sample histogram. The problem with using a single sample to learn about the population is that there is no way of knowing whether that sample is typical – or representative – of the population from which it is drawn. It is possible (although not likely if the

sampling method was truly random) that the sample consists of several outliers that distort the sample mean and standard deviation.

The solution to this conundrum is to take multiple random samples with the same $n$ from the population. Because of natural random variation in which population members are selected for a sample, we expect some slight variation in mean values from sample to sample. This natural variation across samples is called sampling error. With several samples that all have the same sample size, it becomes easier to judge whether any particular sample is typical or unusual because we can see which samples differ from the others (and therefore probably have more sampling error). However, this still does not tell the researcher anything about the population. Further steps are needed to make inferences about the population from sample data.

After finding a mean for each sample, it is necessary to create a separate histogram (in the middle of Figure 6.6) that consists solely of sample means. The histogram of means from a series of samples is called a sampling distribution of means. Sampling distributions can be created from

other sample statistics (e.g., standard deviations, medians, ranges), but for the purposes of this chapter it is only necessary to talk about a sampling distribution of means.

Because a sampling distribution of means is produced by a purely random process, its properties are governed by the same principles of probability that govern other random outcomes, such as dice throws and coin tosses. Therefore, there are some regular predictions that can be made about the properties of the sampling distribution as the number of contributing samples increases. First, with a large sample size within each sample, the shape of the sampling distribution of means is a normal distribution. Given the convergence to a normal distribution, as shown in Figures 6.4a-6.4d, this should not be surprising. What is surprising to some people is that this convergence towards a normal distribution occurs regardless of the shape of the population, as long as the $\mathrm{n}$ for each sample is at least 25 and all samples have the same sample size. This is the main principle of the CLT.

## 统计代写|Generalized linear model代考广义线性模型代写|Summary

The basics of probability form the foundation of inferential statistics. Probability is the branch of mathematics concerned with estimating the likelihood of outcomes of trials. There are two types of probabilities that can be estimated. The first is the empirical probability, which is calculated by conducting a large number of trials and finding the proportion of trials that resulted in each outcome, using Formula 6.1. The second type of probability is the theoretical probability, which is calculated by dividing the number of methods of obtaining an outcome by the total number of possible outcomes. Adding together the probabilities of two different events will produce the probability that either one will occur. Multiplying the probabilities of two events together will produce the joint probability, which is the likelihood that the two events will occur at the same time or in succession.
With a small or moderate number of trials, there may be discrepancies between the empirical and theoretical probabilities. However, as the number of trials increases, the empirical probability converges to the value of the theoretical probability. Additionally, it is possible to build a histogram of outcomes of trials from multiple events; dividing the number of trials that resulted in each outcome by the total number of trials produces an empirical probability distribution. As the number of trials increases, this empirical probability distribution gradually converges to the theoretical probability distribution, which is a histogram of the theoretical probabilities.

If an outcome is produced by adding together the results of multiple independent events, the theoretical probability distribution will be normally distributed. Additionally, with a large number of trials, the empirical probability distribution will also be normally distributed. This is a result of the CLT.

The CLT states that a sampling distribution of means will be normally distributed if the size of each sample is at least 25. As a result of the CLT, it is possible to make inferences about the population based on sample data – a process called generalization. Additionally, the mean of the sample means converges to the population mean as the number of samples in a sampling distribution increases. Likewise, the standard deviation of means in the sampling distribution (called the standard error) converges on the value of $\frac{\sigma}{\sqrt{n}}$.

## 统计代写|Generalized linear model代考广义线性模型代写|Basic Probability

p= 相同结果的试验次数  试验总数
（公式 6.1）

## 统计代写|Generalized linear model代考广义线性模型代写|Summary

CLT 指出，如果每个样本的大小至少为 25，则均值的抽样分布将呈正态分布。由于 CLT，可以根据样本数据对总体进行推断——这一过程称为泛化。此外，随着抽样分布中样本数量的增加，样本均值的均值会收敛到总体均值。同样，抽样分布中均值的标准差（称为标准误差）收敛于σn.

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## MATLAB代写

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