用R语言证明极限中心定理
时间: 2024-01-25 16:10:15 浏览: 46
以下是使用R语言证明中心极限定理的步骤:
1.生成服从不同分布的随机样本数据,例如正态分布、均匀分布等。
2.对于每个分布,分别计算样本均值,并将这些均值存储在一个向量中。
3.绘制这些均值的直方图,并将其与相应分布的概率密度函数进行比较。
4.根据中心极限定理,当样本量足够大时,这些均值应该近似于正态分布。
5.为了验证这一点,可以计算这些均值的平均值和标准差,并将其与正态分布的理论平均值和标准差进行比较。
以下是一个使用R语言证明中心极限定理的示例代码:
```R
# 生成服从不同分布的随机样本数据
set.seed(123)
normal_sample <- rnorm(10000, mean = 10, sd = 2)
uniform_sample <- runif(10000, min = 0, max = 20)
poisson_sample <- rpois(10000, lambda = 5)
# 计算样本均值
normal_mean <- numeric(1000)
uniform_mean <- numeric(1000)
poisson_mean <- numeric(1000)
for (i in 1:1000) {
normal_mean[i] <- mean(sample(normal_sample, size = 30, replace = TRUE))
uniform_mean[i] <- mean(sample(uniform_sample, size = 30, replace = TRUE))
poisson_mean[i] <- mean(sample(poisson_sample, size = 30, replace = TRUE))
}
# 绘制直方图
par(mfrow = c(3, 1))
hist(normal_mean, breaks = 30, freq = FALSE, main = "Normal Distribution")
curve(dnorm(x, mean = 10, sd = 2/sqrt(30)), add = TRUE, col = "red")
hist(uniform_mean, breaks = 30, freq = FALSE, main = "Uniform Distribution")
curve(dnorm(x, mean = 10, sd = sqrt(20/30)), add = TRUE, col = "red")
hist(poisson_mean, breaks = 30, freq = FALSE, main = "Poisson Distribution")
curve(dnorm(x, mean = 5, sd = sqrt(5/30)), add = TRUE, col = "red")
# 计算均值和标准差
normal_mean_mean <- mean(normal_mean)
normal_mean_sd <- sd(normal_mean)
uniform_mean_mean <- mean(uniform_mean)
uniform_mean_sd <- sd(uniform_mean)
poisson_mean_mean <- mean(poisson_mean)
poisson_mean_sd <- sd(poisson_mean)
# 输出结果
cat("Normal Distribution:\n")
cat("Mean of sample means:", normal_mean_mean, "\n")
cat("SD of sample means:", normal_mean_sd, "\n")
cat("Theoretical mean:", 10, "\n")
cat("Theoretical SD:", 2/sqrt(30), "\n\n")
cat("Uniform Distribution:\n")
cat("Mean of sample means:", uniform_mean_mean, "\n")
cat("SD of sample means:", uniform_mean_sd, "\n")
cat("Theoretical mean:", 10, "\n")
cat("Theoretical SD:", sqrt(20/30), "\n\n")
cat("Poisson Distribution:\n")
cat("Mean of sample means:", poisson_mean_mean, "\n")
cat("SD of sample means:", poisson_mean_sd, "\n")
cat("Theoretical mean:", 5, "\n")
cat("Theoretical SD:", sqrt(5/30), "\n")
```
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