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')