Central Limit Theorem with the Exponential Distribution

library(ggplot2)

Simulation of Sample Means from the Exponential Distribution

We demonstrate the Central Limit Theorem (CLT) using the Exponential distribution.

Step 1: Define Parameters

We choose an Exponential distribution with rate parameter \(\lambda = 0.2\), which corresponds to a mean of 5.

lambda <- 0.2
mu <- 1 / lambda
sigma <- 1 / lambda
mu; sigma

## [1] 5

## [1] 5

Step 2: Generate Sample Means

We repeatedly draw samples of size \(n = 50\) from the Exponential distribution and compute their means.

set.seed(123)
n <- 50        # sample size
reps <- 5000   # number of replications

sample_means <- replicate(reps, mean(rexp(n, rate = lambda)))
length(sample_means)

## [1] 5000

Step 3: Plot Distribution of Sample Means

We compare the histogram of sample means with the Normal distribution predicted by the CLT.

df <- data.frame(sample_means = sample_means)

ggplot(df, aes(x = sample_means)) +
  geom_histogram(aes(y = ..density..), bins = 40, fill = "blue", alpha = 0.6) +
  stat_function(fun = dnorm,
                args = list(mean = mu, sd = sigma/sqrt(n)),
                color = "black", size = 1.2) +
  labs(title = "Distribution of Sample Means of Exponential(λ=0.2)",
       x = "Sample Mean",
       y = "Density") +
  theme_minimal()

Step 4: Boxplot of Sample Means

The boxplot shows the median, quartiles, and any outliers among the simulated sample means.

ggplot(df, aes(y = sample_means)) +
  geom_boxplot(fill = "orange", alpha = 0.6, outlier.color = "red") +
  labs(title = "Boxplot of Sample Means",
       y = "Sample Mean") +
  theme_minimal()

Interpretation

• The histogram of sample means shows that the distribution is close to Normal, as predicted by the CLT.
• The boxplot highlights the center (median ≈ 5), spread (interquartile range), and outliers from the sampling distribution of means.