如何理解中心极限定理？

2024-03-02 23:41

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公众号统计 R 中心极限定理正态分布概率论

正态分布（normal distribution）在统计学中的江湖地位，应该是由中心极限定理（CLT，Central Limit Theorem）确立的。该定理的表述是：对于独立同分布的随机变量，无论其分布如何（即使并非正态分布），其样本均值的抽样分布也会趋向于正态分布。

所谓“纸上得来终觉浅，绝知此事要躬行”，下面就用代码来进行模拟抽样：

    r
    
      
      
    
  
 1# Demonstration of the Central Limit Theorem (CLT)
 2
 3# Set common parameters
 4n <- 1000        # Number of samples
 5sample_size <- 30   # Size of each sample
 6
 7# Prepare a 2x2 plot layout
 8par(mfrow=c(2,2))
 9
10# 1. Uniform Distribution
11sample_means_uniform <- replicate(n, mean(runif(sample_size, min=0, max=1)))
12hist(sample_means_uniform, breaks=40, probability=TRUE, main="Uniform Distribution", xlab="Sample Mean")
13curve(dnorm(x, mean=mean(sample_means_uniform), sd=sd(sample_means_uniform)), add=TRUE, col="red", lwd=2)
14
15# 2. Exponential Distribution
16sample_means_exponential <- replicate(n, mean(rexp(sample_size, rate=1)))
17hist(sample_means_exponential, breaks=40, probability=TRUE, main="Exponential Distribution", xlab="Sample Mean")
18curve(dnorm(x, mean=mean(sample_means_exponential), sd=sd(sample_means_exponential)), add=TRUE, col="red", lwd=2)
19
20# 3. Poisson Distribution
21sample_means_poisson <- replicate(n, mean(rpois(sample_size, lambda=2)))
22hist(sample_means_poisson, breaks=40, probability=TRUE, main="Poisson Distribution", xlab="Sample Mean")
23curve(dnorm(x, mean=mean(sample_means_poisson), sd=sd(sample_means_poisson)), add=TRUE, col="red", lwd=2)
24
25# 4. Binomial Distribution
26sample_means_binomial <- replicate(n, mean(rbinom(sample_size, size=10, prob=0.5)))
27hist(sample_means_binomial, breaks=40, probability=TRUE, main="Binomial Distribution", xlab="Sample Mean")
28curve(dnorm(x, mean=mean(sample_means_binomial), sd=sd(sample_means_binomial)), add=TRUE, col="red", lwd=2)
29
30# Reset plot layout
31par(mfrow=c(1,1))

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这里选用了四种分布：均匀分布（Uniform Distribution）、指数分布（Exponential Distribution）、泊松分布（Poisson Distribution）和二项分布（Binomial Distribution）。分别抽样1000次，每次抽取30个符合该分布的实数，计算这30个数的均值，最后对这1000个均值绘制直方图，以展示它们的分布，同时在图上叠加理想的正态分布曲线。

可以看到直方图形状的确“趋向于正态分布”，如下图结果所示：