[Math Review] Statistics Basic: Estimation
Two Types of Estimation
- Point Estimate: the value of sample statistics
Point estimates of average height with multiple samples (Source: Zhihu)
- Confidence Intervals: intervals constructed using a method that contains the population parameter a specified proportion of the time.
95% confidence interval of average height with multiple samples (Source: Zhihu)
Confidence Interval for the Mean
Population Variance is known
Suppose that M is the mean of N samples X1, X2, ......, Xn, i.e.
According to Central Limit Theorem, the the sampling distribution of the mean M is
where μ and σ2 are the mean and variance of the population respectively. If repeated samples were taken and the 95% confidence interval computed for each sample, 95% of the intervals would contain the population mean. So the 95% confidence interval for M is the inverval that is symetric about the point estimate μ so that the area under normal distribution is 0.95.
That is,
Since we don't know the mean of population, we could use the sample mean instead.
Population Variance is Unknown
Dregree of Freedom
The degrees of freedom (df) of an estimate is the number of independent pieces of information on which the estimate is based. In general, the degrees of freedom for an estimate is equal to the number of values minus the number of parameters estimated en route to the estimate in question.
If the variance in a sample is used to estimate the variance in a population, we couldn't calculate the sample variace as
That's because we have two parameters to estimate (i.e., sample mean and sample variance). The degree of freedom should be N-1, so the previous formula underestimates the variance. Instead, we should use the following formula
where s2 is the estimate of the variance and M is the sample mean. The denominator of this formula is the degree of freedom.
Student's t-Distribution
Suppose that X is a random variable of normal distribution, i.e., X ~ N(μ, σ2)
is sample mean and
is sample deviation.
is a random variable of normal distribution.
is a random variable of student's t distribution.
The probability density function of T is
where is the degree of freedom,
is a gamma function.
The t distribution is very similar to the normal distribution when the estimate of variance is based on many degrees of freedom, but has relatively more scores in its tails when there are fewer degrees of freedom. Here are t distributions with 2, 4, and 10 degrees of freedom and the standard normal distribution. Notice that the normal distribution has relatively more scores in the center of the distribution and the t distribution has relatively more in the tails.
The t distribution is therefore leptokurtic. The t distribution approaches the normal distribution as the degrees of freedom increase.
Confidence Interval of t Distribution
Now consider the case in which you have a normal distribution but you do not know the standard deviation. You sample N values and compute the sample mean (M) and estimate the standard error of the mean (σM) with sM. What is the probability that M will be within 1.96 sM of the population mean (μ)? This is a difficult problem because there are two ways in which M could be more than 1.96 sM from μ: (1) M could, by chance, be either very high or very low and (2) sM could, by chance, be very low. Intuitively, it makes sense that the probability of being within 1.96 standard errors of the mean should be smaller than in the case when the standard deviation is known (and cannot be underestimated).
Luckily, however, we can prove that random variable T will be student's t distribution. So we can use t distribution to estimate the mean of a normal distribution population in situations where the sample size is small and population standard deviation is unknown. For 90% confidence interval, it can be calculated as
where A is value of T that contains 90% of the area of the t distribution for n-1 degree of freedom. We can calculate A through the t table.
[Math Review] Statistics Basic: Estimation的更多相关文章
- [Math Review] Statistics Basic: Sampling Distribution
Inferential Statistics Generalizing from a sample to a population that involves determining how far ...
- [Math Review] Statistics Basics: Main Concepts in Hypothesis Testing
Case Study The case study Physicians' Reactions sought to determine whether physicians spend less ti ...
- [Math Review] Linear Algebra for Singular Value Decomposition (SVD)
Matrix and Determinant Let C be an M × N matrix with real-valued entries, i.e. C={cij}mxn Determinan ...
- 统计处理包Statsmodels: statistics in python
http://blog.csdn.net/pipisorry/article/details/52227580 Statsmodels Statsmodels is a Python package ...
- FAQ: Automatic Statistics Collection (文档 ID 1233203.1)
In this Document Purpose Questions and Answers What kind of statistics do the Automated tasks ...
- Machine and Deep Learning with Python
Machine and Deep Learning with Python Education Tutorials and courses Supervised learning superstiti ...
- How do I learn machine learning?
https://www.quora.com/How-do-I-learn-machine-learning-1?redirected_qid=6578644 How Can I Learn X? ...
- 本人AI知识体系导航 - AI menu
Relevant Readable Links Name Interesting topic Comment Edwin Chen 非参贝叶斯 徐亦达老板 Dirichlet Process 学习 ...
- [book]awesome-machine-learning books
https://github.com/josephmisiti/awesome-machine-learning/blob/master/books.md Machine-Learning / Dat ...
随机推荐
- Nuget 异常引用记录
事件描述 Nuget未能将packages.config中的dll成功引入项目中 解决办法 从Nuget中删除对NewtonSoft.Json的引用并重新对NewtonSoft.Json 4.5.0. ...
- Synology DS213J 群晖NAS git server架设方法!
最近单位购入一台Synology DS213J用作数据存储. 本人打算将一些项目的源代码也放在上面,他本身的套件中心提供了SVN SERVER和GIT SERVER. 设置SVN SERVER非常简 ...
- [常识]Windows系统里休眠和睡眠的区别?
睡眠和休眠都是笔记本电脑的节能方式,但有细微的差别: 睡眠还保持着开机状态的,休眠是关机了,但是再次开机之后和关闭时的系统状态是一样的. 睡眠还是保持着系统运行数据在内存中,而休眠则将内存中的数据保存 ...
- 关于jdk与jre的区别
JDK:Java Development Kit JRE顾名思义是java运行时环境,包含了java虚拟机,java基础类库.是使用java语言编写的程序运行所需要的软件环境,是提供给想运行java程 ...
- VMware 密匙
11.0版本 1F04Z-6D111-7Z029-AV0Q4-3AEH8 亲测可用
- php记日志
就是把log追加到文件中 用到了一个方法 file_put_contents <?php file_put_contents('a',date('Y m d h:i:s').' some tex ...
- 2017 多校5 Rikka with String
2017 多校5 Rikka with String(ac自动机+dp) 题意: Yuta has \(n\) \(01\) strings \(s_i\), and he wants to know ...
- poj 1764 Dice Contest
题目戳这里. 首先我要吐槽这个题目描述不清.\(2\)对着选手,那选手朝那边?看完别人写的程序后我才知道选手对着目标所在的方向(或左或右). 然后这道题还是不错的,因为他交给我矩阵乘法不只有常规意义下 ...
- Eclipse EE导入maven工程
Eclipse EE下载地址:https://eclipse.org/downloads/ 启动Eclipse后,点击File->Import,选择Existing Maven Projects ...
- xdebug使用教程
http://www.cnblogs.com/xujian2016/p/5548921.html 配置信息 zend_extension="D:\phpStudy\php53n\ext\ph ...