Statistics
The main way is to extract information on the original statistical information in certain operations, draw some representative figures to reflect the characteristics of certain aspects of the data, this number is called statistics. Using statistical language statements, statistics is a function of the sample, it does not depend on any unknown function. Inferential statistics is the important role of proper statistics by taking samples from the population structure.
Common Statistics
Sampling distribution
When the overall distribution of X type is known, if for any natural number n can export statistics mathematical expression distribution, this distribution is called an accurate sampling distribution. It's a small sample size of n statistical inference problem very useful. Most accurate sampling distribution is obtained in the overall normality. Under normal conditions in general, there are distribution, t distribution, F distribution, known as the three major statistical distribution.
Chi-square distribution (Chi-square distribution)
definition:
Random variable independently of each other, and subject to the standard normal distribution N (0,1), and the square of their compliance n degrees of freedom distribution, chi-square distribution for reading.
DOF is a concept commonly used in statistics, it can be interpreted as the number of independent variables, it may also be interpreted as a quadratic Rank [2]. For example, a 1 degree of freedom of the distribution ; is n degrees of freedom distribution .
Mathematical expectation distributed as follows:
The variance of the distribution is as follows:
Distribution is additive, i.e. if ,, and independently, is
When a sufficiently large degree of freedom, the probability density distribution curve tends to be symmetrical. At that time, limiting distribution distribution is a normal distribution.
t distribution (Distribution t)
definition:
Random variable X and Y are independently ,, and then
This distribution is called the t-distribution, referred to as t (n-), where, n-degree of freedom.
When , the mathematical expectation of the t-distribution . When the time, t distribution variance .
从上图可以看出，t分布的密度函数曲线与标准正态分布的密度函数曲线非常相似，都是单峰偶函数。只是，的密度函数的两侧尾部要比的两侧尾部粗一些。的方差比的方差大一些。
F分布(F distribution)
定义：
设随机变量Y与Z相互独立，且Y和Z分别服从自由度为m和n的卡方分布，随机变量X有如下表达式：
则称X服从第一自由度为m，第二自由度为n的F分布，记为，简记为。F分布的密度函数的图形如下图。
设随机变量X服从分布，则数学期望和方差分别为：
样本均值的分布与中心极限定理
当总体分布为正态分布时，可以得到下面的结果： 的抽样分布(sampling distribution)仍为正态分布，的数学期望为，方差为，则
上面的结果表明，的期望值与总体均值相同，而方差则缩小为总体方差的。这说明当用样本均值去估计总体均值时，平均来说没有偏差（这一点称为无偏性）；当n越来越大时，的散布程度越来越小，即用估计越来越准确。实际问题中，总体的分布并不总是正态分布或近似正态分布，此时的分布将取决于总体分布的情况。不过当抽样个数n比较大时，人们证明了如下的中心极限定理。该定理告诉我们不管总体的分布是什么，样本均值的分布总是近似正态分布，只要总体的方差有限。因为无论是什么总体分布，设总体均值为，总体方差为，总有：
所以当n比较大时，近似服从，等价地有
中心极限定理(central limit theorem) 定义：设从均值为、方差为(有限)的任意一个总体中抽取样本量为n的样本，当n充分大时，样本均值的抽样分布近似服从均值为、方差为的正态分布。
总结
中心极限定理的作用在大样本情况下，可以认为样本均值的抽样分布服从正态分布，从而完成样本均值概率的计算。
正态总体下的几个常用统计量的抽样分布，因为获得了较为完整的分布数据，一旦确认统计量符合这几类抽样分布，可以通过查表的方式对概率值进行计算。
转自：https://mp.weixin.qq.com/s/vxBYqAFxt0MTBcux1SZlxg