I was reading Simon Singh’s The Simpsons And Their Mathematical Secrets today and he mentioned a simple method for approximating e – given a uniform RNG , e can be approximated by the average number of draws required for the sum of the draws to exceed 1. This is a neat little demonstration and easy to generate in R – taking the uniform RNG and plotting the average number of draws required to exceed a sum of one, and then replicating this using an increasing number of draws to illustrate convergence:

# Function to calculate number of draws required
function() {
  r <- runif(1,0,1)
  n <- 1
  while (r<1) {
    r <- r+runif(1,0,1)
    n <- n+1
  }
  return(n)
}
# Generate a series of draws from 2 .. 2^16 (65536)
N<-2^seq(1,16)
# Generate simulation
y <- sapply(N, function(x)mean(replicate(x,gen())))
# Plot convergence
qplot(1:16, y) + geom_line(linetype=2) + geom_hline(aes(yintercept=exp(1)),color='red')

The Simpsons and Their Mathematical Secrets by Simon Singh
My rating: 4 of 5 stars