A Demonstration of Central Limit Theorem with Exponential Distribution

Classical CLT

Let \( X_1, X_2, ... \) be iid's with mean \( \mu \) and finite variance \( \sigma^2 \). Then \( \frac{X_1 + \cdots + X_n - n\mu}{\sqrt n} \) converges in distribution to a normal \( N(0, \sigma^2) \).

Exponential Distribution

The PDF of an exponential distribution is defined by \( f(x) = \lambda e^{-\lambda x} \), where \( \lambda \) is the rate parameter, \( x>0 \). This has mean \( 1/\lambda \) and variance \( 1/\lambda^2 \).

In this shiny app, we

• Choose the number \( n \) and \( \lambda \).
• Generate \( n \) values from Exp\( (\lambda) \), using R's rexp function.
• Take the mean of \( n \) values to get \[ X = \frac{X_1+...+ X_n}{n} \]
• Make density plot with 10000 such values.
• Compare with the density plot of \( N(\mu, \frac{\sigma^2}{n}) \).

• With \( n \) increasing, the shape of the density plot resembles a bell curve which is exactly a normal distribution.

• From this demonstration, we deduce that if \( n \) increases then \[ \sqrt n \left(\left( \frac{1}{n} \sum_{i=1}^n X_i \right) - \mu \right)\xrightarrow{d} \mathcal N(0,\sigma ^2). \]

Here \( d \) means that the cumulative function of the left side converges pointwise to the CDF of \( \mathcal{N}(0,\sigma^2). \)