摘要：分享兴趣，传播快乐，增长见闻，留下美好。亲爱的您，这里是LearingYard学苑！今天小编为您带来"概率论与数理统计的学习"欢迎您的访问！Share interest, spread happiness,increase knowledge, and lea

分享兴趣，传播快乐，

增长见闻，留下美好。

亲爱的您，这里是LearingYard学苑！

今天小编为您带来"概率论与数理统计的学习"

欢迎您的访问！

Share interest, spread happiness,

increase knowledge, and leave beautiful.

Dear, this is the LearingYard Academy!

Today, the editor will bring you" Learning of probability theory and mathematical statistics"

Welcome to visit!

思维导图

Mind mapping

概率论是数学的一个分支，主要研究随机事件及其概率。在概率论中，样本和抽样分布是两个非常重要的概念。

Probability theory is a branch of mathematics that primarily studies random events and their probabilities. In probability theory, sample and sampling distribution are two very important concepts.

样本是从总体中抽取的一部分个体或者观察值的集合。在进行统计分析时，通常不可能对总体中的每一个个体都进行研究，因此我们会抽取样本来进行研究，并通过对样本的分析来推断总体的特性。

A sample is a collection of individuals or observations taken from a population. When conducting statistical analysis, it is usually not possible to study every individual in the population, so we extract samples for study and infer the characteristics of the population through the analysis of the sample.

Types of samples:

简单随机样本：从总体中随机抽取n个个体，每个个体被抽中的概率相等。系统抽样：按照一定的规则从总体中抽取样本，比如每隔一定数量的单位抽取一个。分层抽样：将总体分成不同的层，然后从每一层中随机抽取样本。聚类抽样：将总体分成若干群体（簇），随机抽取一些群体，然后对这些群体中的所有个体进行研究。

Simple random sample: Randomly select n individuals from the population, with each individual having an equal probability of being chosen.Systematic sampling: Extract samples from the population according to certain rules, such as selecting one every certain number of units.Stratified sampling: Divide the population into different layers and then randomly select samples from each layer.Cluster sampling: Divide the population into several groups (clusters) and randomly select some groups to study all individuals within these groups.

抽样分布是指从同一个总体中抽取所有可能样本的统计量（如样本均值、样本比例、样本方差等）的概率分布。抽样分布对于理解和进行统计推断至关重要。

A sampling distribution refers to the probability distribution of all possible statistics (such as sample mean, sample proportion, sample variance, etc.) from samples taken from the same population. The sampling distribution is crucial for understanding and conducting statistical inference.

抽样分布的特性：

Characteristics of sampling distribution:

期望值：抽样分布的均值通常等于总体的参数。方差：表示抽样分布的离散程度，通常与样本大小有关。标准误差：是抽样分布的标准差，用于估计参数的精确度。

Mean: The mean of the sampling distribution usually equals the parameter of the population.Variance: It represents the degree of dispersion of the sampling distribution, usually related to the sample size.Standard Error: It is the standard deviation of the sampling distribution, used to estimate the precision of the parameter.

常见的抽样分布：

Common sampling distributions:

二项分布：当样本量为固定n，每次试验只有两种可能结果，且每次试验成功的概率相同，这时样本中成功的次数服从二项分布。正态分布：根据中心极限定理，当样本量足够大时，样本均值的分布近似正态分布。t分布：当总体标准差未知且样本量较小时，样本均值服从t分布。卡方分布：当样本量为n的多个独立的服从标准正态分布的随机变量平方和时，该随机变量服从自由度为n的卡方分布。

Binomial distribution: When the sample size is fixed at n, each trial has only two possible outcomes, and the probability of success is the same for each trial, the number of successes in the sample follows a binomial distribution.Normal distribution: According to the Central Limit Theorem, when the sample size is sufficiently large, the distribution of the sample mean approximates a normal distribution.t-distribution: When the population standard deviation is unknown and the sample size is small, the sample mean follows a t-distribution.Chi-square distribution: When the sum of the squares of n independent random variables that follow a standard normal distribution, the random variable follows a chi-square distribution with n degrees of freedom.

理解抽样分布是进行假设检验和构建置信区间的基础，也是统计学中推断统计部分的重要内容。通过对抽样分布的研究，我们可以了解样本统计量围绕总体参数的波动情况，从而对总体参数进行有效的估计。

Understanding the sampling distribution is the basis for hypothesis testing and constructing confidence intervals, and it is an important part of inferential statistics. By studying the sampling distribution, we can understand how the sample statistics fluctuate around the population parameters, thus effectively estimating the population parameters.

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来源：关爱中小学教育