Central Limit Theorem

The probability density function of the exponential distribution is given by the equation \[P(x,\lambda) = \lambda\, \exp(-\lambda\, x)\]

The exponential distributon has mean and standard deviation equal to \(1/\lambda\). Below is the plot for the probability density function the exponential distribution

We choose 50 random points from the exponential distribution and simulate nsim times to obtain sample distribution of means and variances:

means <- NULL; vars <- NULL; n <- 50
for (i in 1:nsim){
  rand <- rexp(n,lam)
  means <- c(means,mean(rand))
  vars <- c(vars,var(rand))
}

By central limit theorem, the sample means "mu" should be normally distributed.

Using the Shiny App, you can vary nsim and \(\lambda\) to see the central limit theorem in action. The app can be accessed from the Central Limit Theorem App

An example result is shown for nsim = 1000, n = 50 and \(\lambda = 0.2\). The plot below shows the distribution of the sample means (histogram) and the red curve is the standard normal variate:

\[Z=\frac{X_{\rm means} - \mu}{\sigma/\sqrt{n}}\]