A Demonstration of Central Limit Theorem with Exponential Distribution

Shouman Das
U of Rochester

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 $1/\lambda^2$.

Shiny App's Description

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 $N(\mu, \frac{\sigma^2}{n})$.

Plot Results

plot of chunk unnamed-chunk-1

Conclusion

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 $\mathcal{N}(0,\sigma^2).$