The central limit theorem

An experiment

Here’s a plot of the density function for the so-called exponential distribution and an associated histogram:

par(mfrow=c(1,2))
plot(dexp, -1, 4, asp = 2)
hist(rexp(1000), asp = 1/100)

Contrast this with the normal distribution.

par(mfrow=c(1,2))
plot(dnorm, -3,3, asp = 3)
hist(rnorm(1000), asp = 1/100)

Now, suppose that we create a random variable by averaging some exponential random variables.

X <- function(x) {sum(rexp(100))/100}
dd = sapply(1:1000, X)
hist(dd)

The averaging process makes the exponential distribution look normal.