Understanding of law of large numbers and central limit theorem

Catalog

1、 Law of large Numbers

2、 Central limit theorem

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1、 Law of large Numbers

When the number of samples is large enough , The statistical characteristics of the sample can approximately represent the statistical characteristics of the population .“ Large number ”  It means that the number of samples is large enough or the number of tests is large enough .

2、 Central limit theorem

Set the overall as   As a whole N Sample sets , Each sample set contains a certain number of samples from the population A Sampled samples , And the mean values of these sample sets are  . The central limit theorem tells us , When N Sufficiently large （ Law of large Numbers ）, Regardless of the overall A What distribution does it follow ,A Of the sample set mean value They all follow a normal distribution .“ center ”  It can be understood as near the mean ,“ limit ”  Is the law of large numbers .

for instance ：

For example, we should count the salary level in Wuhan . All Wuhan people are a whole A, Draw multiple groups of samples randomly from the whole  , There are K people , The salaries are （  ）, Calculate the average salary of each group of survey samples  , Get a set of mean values  , When the sample set N Enough time ,   The distribution of is approximately subject to normal distribution , As shown in the figure below .

The official definition of mathematics is ：

Let the sequence of random variables   Mutually independent and identically distributed , And both mathematical expectation and variance exist ,  ,  , For any real number x There is always ：

among ,   Is the standard normal distribution function .