R验证中心极限定理

中心极限定理: 对于任意分布(均值为\(\mu\), 方差为\(\sigma^2\)), 每次从总体中抽取n个样本, n足够大时(一般n>=30) 样本均值\(\overline{X}\)的分布近似服从\(N(\mu, \sigma^2/n)\), 即 \[ \begin{align*} \overline{X} \sim N(\mu,\sigma^2/n), \frac{\overline{X} - \mu}{\sigma / \sqrt{n}} \sim N(0,1) \end{align*} \] 下面是验证:

library(tidyverse)
set.seed(3124)

# 原始分布指数分布, 均值为1
data = tibble(x = rexp(10000, rate=1))
ggplot(data, aes(x)) +
  geom_histogram(fill = "#4E84C4", color = "black") +
  labs(title="原始总体分布（指数分布）", x="X", y="count")

# 每次抽样样本数
n = 100
# 不同抽样次数
sample_sizes = c(5, 10, 40, 100, 500, 1000)

results <- map_dfr(sample_sizes, ~ {
  tibble(
    sample_size = .x,
    mean = map_dbl(1:.x, ~mean(sample_n(data, n)$x))
  )
})

ggplot(results, aes(x = mean)) +
  geom_histogram(fill = "#4E84C4", color = "black") +
  facet_wrap(~ sample_size, scales = "free") +
  labs(
    title = "不同抽样次数下样本均值的分布",
    x = "样本均值",
    y = "频数"
  ) +
  scale_color_brewer(palette = "Paired")